Plus One Physics Model Exam | Chapters - 1,2,3,4,5,6,7 | Xylem Plus One

Good morning, all. Are you all doing well? Are all my younger brothers and sisters ready? We are about to dive into our Physics mega marathon live. We are going to cover the first seven chapters entirely in one live session. We are going to completely set up all the concepts and all the derivations within the chapters in a single live session. If everyone is ready, as always, if you put a muscle emoji in the chatbox, we can begin our activities. Today's live session is one that can be approached very coolly. Because teaching all the chapters in one live session is difficult for you, and for us as well, we will have to finish the chapters in a limited time. It is not like that here. Here, we are only dealing with the first seven chapters in this live session. So, we can thoroughly set up all the complete detailed concepts for each chapter. Similarly, derivations, which trouble you the most, and derivations from which questions worth about 30 marks can be expected in our model exam, we can learn them neatly and accurately today. If everyone is ready, let the muscle emojis pour into the chatbox. Also, the most important thing you need to do is, you must definitely have a notebook with you. The method you need to follow for today's marathon live, and for today's and tomorrow's marathon live, is to keep a notebook for Physics in your hand. This is an undeniable fact. Then, write down only the most important keywords and key points as lecture notes. You will get all the slides, which is fine, but you should always have a lecture note that you can refer to. Therefore, while watching the live session, write down the key points and keywords that you feel are important. Also, write down the derivations in detail. Everyone must write down all the derivations, that is, all the derivations we are going to learn today. There is no compromise on this, and it will not be correct if you don't. The next thing I want to tell everyone is to share as much as possible. There is no compromise on this. Everyone should share the link of the live session with your friends as much as possible. Don't think that only I should study, and only I should get the best marks. Let everyone get good marks. So, share the live link with your friends who are thinking of taking study leave instead of going to school and finishing their studies. Regarding today's duration, you are asking. We know that on YouTube, it is possible to take a live session for 12 hours continuously. So, there will be no need to take a live session for 12 hours today because there are only seven chapters. There are no very huge chapters coming today. Fluids, etc., are coming in tomorrow's live session. Therefore, it will not take that much time. However, the live session will definitely be there until around 7 PM or 6:30 PM, because we are going to proceed with that much depth. We are going to teach you each question and each derivation with that much depth. Are you ready? Are you ready, my children? Let's start. I have shared. I am ready. No matter how late it is today, I will be with you. I will definitely write down the important keywords and derivations. All those who say they will do it, let's start by saying "one" with a fire emoji. That is, Units and Measurements, Straight Line, Plane, Laws of Motion, Work, in that order, we are going to study today. That is how we are going to approach it. If you are ready, let everyone give a fire emoji together. You must stand firm. This same energy, this same power, this same, what to say, this same enthusiasm should be from your side until the live session ends. Because it is for you that we are going to finish all the chapters in three days from now through such a study plan. Think about it. The Physics that you have studied for a year, we are going to finish in three days. It is not a matter of just finishing. We are going to teach each one of you and prepare you for the upcoming model exam. This is the most effective study plan and live class that we are providing you. Understand it properly and stay with us. Share with your friends. Let's study together and make it grand. Now, I will not waste time by talking. I am going to move on to the chapter called Units and Measurements. Do you have any idea about the weightage of this chapter? Do you have any idea about the weightage of this chapter? You can expect questions from this chapter for four to five marks in your model exam. This is one of the most important chapters that needs to be given priority, the chapter called Units and Measurements. It is also a chapter that each of you should study excellently. Let's start, children. The first thing every student studying this chapter should know is Physical Quantities. What are Physical Quantities, children? All quantities that we can measure can be called Physical Quantities. Quantities which can be measured. All the quantities we have studied in Physics so far can be placed under this title. What are they, children? Yes, we know mass, length, temperature, force, heat, work, energy, power, velocity, acceleration, distance, displacement, area, volume, everything, everything, everything can be placed under the title of Physical Quantity. Quantities which are measurable, all that we can measure are Physical Quantities. Understand this. Now, let's move on to the next thing. When we talk about measuring, if I take a measurement, or if I perform a measurement, what will be included in it? Everyone, please tell me. If I take a measurement, what will be included in it? For example, I will measure the length of this table. I will measure the length of this table. I got the length of this table as 50 centimeters. So, this is my measurement, right? What is 50, actually? Everyone, please comment. What is 50 in this measurement? Everyone, please comment. All younger brothers and sisters, comment. What is 50? Yes, you know that 50 is our numerical value. 50 is our numerical value. What is centimeter? Yes, we know that a measurement will not only have a numerical value but also a unit. So, what is centimeter? Centimeter is our unit. You all must understand this clearly. You all must understand this clearly. That is, a measurement has two main parts. One is a numerical value, and the second is a unit. The combination of these two is what is called a measurement. The combination of these two is what we call a measurement. The community, I will reset it after my part is done. Wait, pay attention here. Suppose I take another measurement. Mass. Let's assume the mass is 55 kilograms. Let's assume my mass is 55 kilograms. Exactly 55 kilograms. What is 55 here? Tell me quickly. What is 55? It is the same way. It is a measurement. It is a numerical value. Right? Yes, it is the numerical value. What is kilogram? Yes, we know it is our unit. So, a measurement has two main parts. One is a numerical value, and the other is a unit. Understand that. This is very, very, very important. Everyone must understand this. So, that is what I have given here. A measurement of a physical quantity has two parts: N and U. N means numerical value, and U means unit. Shall we look at the relationship between them? What is the relationship between this numerical value and this unit? Let's look at it. We can look at it easily. Stay with me, everyone. Stay with me. So, let's assume my measurement is five kilometers. Five is the numerical value, and kilometer is the unit. Do we measure distance only in kilometers, children? Everyone, please comment. Do we measure distance only in kilometers? Absolutely not. You know that there are other units for distance besides kilometers, right? Yes, tell me, children. We also say it in meters. How many meters is five kilometers? How many meters is five kilometers? Yes, we know that five kilometers is 5000 meters. Five kilometers is 5000 meters. Isn't this something everyone knows? Everyone knows this. You must pay attention here. The numerical value changed from five to 5000. The numerical value changed from five to 5000. That is, the numerical value increased. But the unit is one kilometer and the other is meter. So, as the size of the unit decreased, the numerical value increased. Understand that as the size of the unit decreases, the numerical value increases. Similarly, let me give another example. One kilogram. One kilogram is how many grams? One kilogram is 1000 grams. No doubt, right? One kilogram is 1000 grams. Here, the numerical value is one, and it changed to 1000. So, the numerical value increased, but the size of the unit decreased. Look, it decreased from kilogram to gram. Understand that as the size of the unit decreases, the numerical value increases. This is a very important word, a relationship. That is, N is inversely proportional to U. N is inversely proportional to U. That is, the numerical value and the unit are inversely proportional. The numerical value and the size of the unit are inversely proportional. If the numerical value increases, the size of the unit will decrease. Understand that. This is very important. Each of you must study this. You must fix this in your mind. You can expect questions from this in the exam for very few marks. Study this. Study by understanding this concept. I have given it here. I have given its exact definition here. Okay? If you are happy, comment "SET" in the chatbox if the first basic concept is set. We will move forward. We will move on to the classification of physical quantities. If you are ready, respond. Everyone, respond. Come on, guys. Give a "SET" message. I understood this excellently. You might have already studied this chapter. You might have studied this concept. However, this is the moment to brush it up. No matter how much we study, it will only be useful if we revise it. Revision is as important as studying. It is very important. Ready, set. Everyone has understood the matter clearly. We are moving on to the next thing, children. Stay with me. The next thing is that physical quantities can be classified into two types. What are they? Fundamental quantities and derived quantities. Fundamental physical quantities and derived physical quantities. This is their classification. Let's study both in depth. Fundamental physical quantities. What do we always keep as an example in our minds so that we never forget this? Yes, what is the example I always give so that you never forget fundamental physical quantities? Think about single people. Think about Sigma Male. If I am single, what does it mean? I don't need anyone else's help. I always say that, right? If I want to go to a beach, I don't call anyone. If I want to go to a movie, I go alone. If I want to go to a park, I can go alone. I don't need anyone's help. Physical quantities that have independent existence can be called Fundamental Physical Quantities. Just like single people, physical quantities that have an independent existence and do not depend on anyone are called Fundamental Physical Quantities. Understand that clearly. They are independent physical quantities. They are independent physical quantities that can exist on their own without anyone's help. They are called Fundamental Physical Quantities. Okay. Next, how many fundamental physical quantities are there? Can you comment? How many are there? To not forget the number, I always give a code. It is the name of my favorite footballer. What is that code? How many fundamental physical quantities are there? Never forget. There are seven. To not forget that, think of CR7, our Cristiano. That's it. There are seven fundamental physical quantities. Shall we see what they are? Come on, there are seven. You must study these seven. These are the seven fundamental physical quantities: Mass, Length, Time, Electric Current, Temperature, Luminous Intensity, Amount of Substance. In all my live sessions, I explain luminous intensity for those who have doubts. What is luminous intensity? It is the intensity of light. In Malayalam, it is called "Prakashatheevratha". It is the fundamental physical quantity that measures how intense the light is. Amount of substance, you would have already studied it, right? You studied concepts like mole in chemistry. That's it. So, these are the seven fundamental physical quantities. Next, the next one is fundamental units. The fundamental physical quantities you see here, the units of these fundamental physical quantities are called fundamental units. For example, what is the unit of mass? No doubt, the unit of mass is kilogram. Next, what is the unit of current? It is Ampere. What is the unit of time?

This is a second. What can we categorize these under? These are fundamental units. The units of the fundamental physical quantities we see here are what we call fundamental units. For example, if we take mass, the unit of mass is kilogram. If it's length, it's meter. If it's time, it's second. If it's electric current, it's ampere. Do you understand? The units of these physical quantities are called fundamental units. That's important. That's very, very, very important. Next are derived quantities. I always say, if you want to remember derived quantities, you just need to think about committed people. What is the specialty of committed people? They are always dependent on someone else, aren't they? They don't go anywhere on their own. If you want to go to the park, you say, "You come too, let's go together." Right? If you want to go to a movie, you go together. People who are always dependent on others, who don't have an independent existence, are generally called committed. Right? It's the same with derived quantities. They cannot exist on their own. They exist by depending on other fundamental physical quantities. Derived quantities are formed by combining more than one fundamental physical quantity. Derived quantities are formed by combining more than one fundamental physical quantity. How many derived quantities will there be? Can you tell me the number? We learned that there are seven fundamental quantities. Can you specify the number of derived quantities? Do you know? If you know, comment. All brothers and sisters, tell me. How many derived physical quantities are there? How many derived physical quantities are there? Can you tell me? I will arrange the community after my chapter is finished. I will call and arrange the community. Tell me, children, how many derived physical quantities will there be? Come on guys, everyone respond. All my brothers and sisters, tell me. We can say infinite number, because all physical quantities except those seven fundamental physical quantities can be categorized as derived quantities. Except for those seven, all other physical quantities are derived physical quantities. Understand that. What are the examples? We can simply say, look, speed, velocity, work, force, energy. If we keep saying, we'll keep saying. All of them can be categorized as derived. Understand that all of them, except those seven, are derived physical quantities. This is important. These are things you need to learn. What is the name given to the units of these physical quantities? Derived units. That is, the name given to the units of derived quantities is derived units. For example, for speed, what is it? Meter per second. What about density? Kilogram per meter cube. Next, let's write down the next one. What about area? Meter squared. What about volume? Meter cubed. These are all derived units, or rather, the units of derived physical quantities are called derived units. The name given to the units of derived physical quantities is derived units. This is very important. You all need to keep it in mind and learn it. This is very, very, very important. These are things you must learn. Are you ready, children? Did you understand clearly? What is a physical quantity? What are its classifications? Did you understand clearly with examples of each? If you understood, I say again, put a thumbs up in the chat box. Everyone put a thumbs up in the chat box. Then we can move on to the next topic. After physical quantities, what is our next important topic? System. Yes, that's it. We are going to move on to systems of units. Different units existed in different regions. So, the name we give to a set of those units is system of units. The name given to a set of units is system of units, or rather, there were different sets of units in different regions. We call them systems of units. Pay attention, children. We will examine each one in detail. So, the first one is the CGS system of units, don't you see it? The CGS system of units. It was a system of units that existed in France. So, you can understand this very simply. The most important physical quantities are length, mass, and time, as everyone knows, right? Length, mass, and time are the most important physical quantities. So, this classification is done based on the units in which these three physical quantities are expressed. It's easy. So, in CGS, length will be expressed in centimeters. C for centimeter. Mass will be expressed in grams. Time will be expressed in seconds. Centimeter, gram, second. That is CGS. Learn that code as it is. Centimeter, gram, second. That's how the CGS system works. It is a system of units that existed in France. Next, what is FPS? Tell me the same way, my dear children. F means foot. P means pound. S means second. Foot means you should get the measurement in feet. When digging a well, we talk about feet, right? Pound means people who watch wrestling will know. If you watch WWE, when each wrestler comes, their mass is expressed in pounds, right? So, pound is a unit of mass. Second, again, time is in seconds. Understand that. What about MKS? That is also a system of units that existed in France. In MKS, it's meter, kilogram, second. Meter, kilogram, second. This is very important. You need to learn it. Let me tell you how questions will come from this in the exam. The physical quantity that remains unchanged in all three systems of units is time. In all three, time is expressed in seconds. This is sometimes asked in exams. That is, in these three systems of units, the physical quantity called time is expressed in seconds. It never changes. But length changes, mass changes, but time does not change. That's important. You should know that. So, you need to identify in which units length, mass, and time are expressed in CGS, MKS, and FPS. This is sometimes asked in exams for one mark. This table should be in your mind. You all should learn this table. Okay, right. Next thing. If there are different systems of units in different regions, won't it greatly affect the mutual communication between those countries or regions? I always say, imagine I am from Britain. I go and express my mass in the unit I know. If it's Britain, it will definitely be in pounds. So, if I go to a grocery store in France and ask for one pound of sugar, will they understand? They won't understand because the unit they know is grams or kilograms. Pound is never a unit they know. So, in this way, it will greatly affect mutual communication. Therefore, we introduced a system of units that is internationally accepted, that can be conveyed and communicated anywhere in the world. Can you comment on the name of that? We introduced a system of units that is internationally accepted. Can all my brothers and sisters comment on the name they call it? What is the name they call it? Yes, great, awesome. Everyone has commented the correct answer. It is called the SI unit. What is it, children? SI unit. All those watching the live stream without liking, please definitely like the live stream. Never watch the live stream without liking. Everyone start watching by liking the live stream. What is it, children? It is the SI unit. It is called the SI system, International Unit. International System of Units is also fine, you will get marks. Okay. SI unit. It is the system of units that is accepted by everyone, internationally accepted. It has base units and supplementary units. It has base units and supplementary units. Shall we look? Yes, we are going to look at the base physical quantities and their units. The unit of length is meter. Mass is kilogram. Time is second. Temperature is Kelvin. Electric current is ampere. Luminous intensity is candela. Amount of substance is mole. Here, all my children should understand that all these units are written in lowercase letters. The names of the units are written in lowercase letters. You should never use capital letters. Many people make the mistake of putting a capital K for kilogram. Never do that. Everything should be in lowercase letters when writing the name of the units. It must be in lowercase letters. Now, coming to symbols, you need to pay attention. The majority are in lowercase letters, except for Kelvin and Ampere. Only these two gentlemen are represented in capital letters. The rest are in lowercase letters. That is, if it's meter, it's small m. If it's kilogram, it's small kg. If it's time, it's small s for denoting second. Candela is cd. Mole is mol. Why are only Kelvin and Ampere symbolized using capital letters? What is the reason for that? Those who know, please tell me. It's the name of a scientist. Kelvin and Ampere are the names of scientists. That's why we only use capital letters for the symbolization of the units that include them. We only use capital letters. Because it is the name of a scientist. Understand that. This is important. This is very important. Beyond these seven basic SI units, there are two supplementary units. Do you know what those supplementary physical quantities are? If you know, please comment. What are the two supplementary physical quantities in SI? What are they, children? If you all know, can you comment? What are those supplementary physical quantities? There is no doubt. We all know them. Plane angle and solid angle. Plane angle and solid angle. We will study both in detail. Before that, just learn their units and symbols. The unit of plane angle is radian. The unit of solid angle is steradian. Both are symbolized in lowercase letters. There is no change. You should represent both in lowercase letters. The unit of plane angle is radian. The unit of solid angle is steradian. When it comes to symbolization, pay attention, children. For plane angle, it's rad. For solid angle, it's sr. That's how they are symbolized. For plane angle, it's radian, rad. For solid angle, it's sr. This is how we symbolize them. Everyone should know. Shall we study each one in depth? What is plane angle? As we usually say in math, the angle we talk about in math, the angle denoted by theta, that angle is called plane angle. The angle included in a plane is called plane angle, or theta. Its formula is simple. You all know it, but I'll say it. The formula for plane angle theta is arc length divided by radius. Arc length divided by radius. This is the formula. Plane angle theta equals arc length divided by radius. This is our formula. What comes here, dear? Theta. What is arc length, children? Pay attention. This is called the arc. What is arc length? It's not L, dear children. Definitely. So, L divided by radius. What is radius? It's R. So, we can write theta equals L by R. What was the unit, recall and tell me? What was the unit of plane angle? Recall and tell me everyone. Plane angle means it's simply the angle formed within a plane, related to the plane. We can denote it by theta. It is equal to arc length by radius. Arc length is L. Radius is R. So, we can put L by R. Do you remember the unit? Everyone is answering excellently. Radian. Radian is the unit of plane angle. Everyone should know. Important. Right. Next, we are going to move on to solid angle. What is solid angle, children? Consider a solid sphere. Everyone, consider a solid sphere. Imagine a solid sphere like a shot put. Then, I cut off a cone-shaped portion from it. Look. I cut off a cone-shaped part from that solid sphere. When cutting off like that, won't there be an angle there? That angle is called solid angle. This is capital omega from the Greek alphabet, okay? We might have learned the symbol for ohm, ohm, the unit of resistance, right? Yes, this is capital omega from the Greek alphabet. So, that is called solid angle. If you just cut off a cone-shaped part from a solid sphere, the angle formed there is called solid angle. Learn its unit and equation. Intercepted area. What is intercepted area? Look, such a portion.

When cutting and removing, there is an area here, right? This area is what we call the intercepted area, or A. So, A divided by R squared, by the square of the radius. So, how will the equation come? Simply, it will come as A by R squared. So, the solid angle will be equal to the intercepted area divided by the radius squared, A by R squared. This is important, it can be asked for one mark or two marks. Solid angle was a question asked in a recent question paper. Comment on the unit of solid angle. Comment on the unit of solid angle. It was taught. The unit of plane angle is radian. What is the unit of solid angle, children? You all should know. The unit of solid angle is SR, or steradian. Understand that the unit of solid angle is SR, or steradian. This is very, very, very important. Either one of these can be asked, or you should have a clear understanding of the units within SI. Everyone should also study the supplementary units. These are questions that are individually asked for two marks. Everyone should understand and study. Study it in such a way that you don't even need to revise it again. Are you writing down the keywords? Are you writing down the key units? Don't just keep the book idle. Everyone is writing. I have caught everyone. After class, send me a photo of your book. Ready? I believe everyone is writing. You must write down all the key points, like the solid angle. At least write down the formula for solid angle and the unit. At least write down the most key points. Okay? You are writing, aren't you? No, no. Yes, I believe you are writing. Because you should get good marks. I have no other intention. Because you getting an A Plus now, and me getting an A Plus, you are not going to bring me a thousand rupees. You are not going to bring me any big gift packages. But after this exam, there will be a message from you. After the live session, after the physics class, after the physics exam, there will be a message from you. "Sir, we aced physics. There was nothing beyond the questions you asked in the question paper. We were able to write brilliantly. I was able to perform my best." Your comment will be there. I get nothing more than the satisfaction I get from that. That is the greatest happiness. It's not that I expect anything from you or hope for anything. "Write that formula, boy! Study the derivation, boy! Did you understand, boy?" I ask a hundred times and teach you with the sole aim that you should get good marks. Because at this moment, I will say anywhere, no one from Saianth has told us to take a marathon live of 10 or 12 hours, and no one will ever say that. We are giving this for our children. We are giving this live only with the desire that our children, our younger brothers and sisters, should get good marks. Otherwise, we could have limited the live session to just two or three hours at night. We can give you limited classes. We get our salaries on time. But you should get good marks. You are the children who trusted us. You should be able to perform well in your exam. Even at this moment, there are many children who haven't started studying. Just squeeze something out of them. That's all you need to do. That's it. So guys, let's move forward. I haven't included many questions. There are very few questions in today's live. I have included only very rare questions. I have included only those questions that are absolutely essential. Because in the live session called "Top Level Questions," all the numerical and concept-oriented questions will be taught. But derivations will not be there. Derivations will be there only today. In that live session, all the numericals and all the concept-oriented questions will be there. So, I have included only a very limited number of questions today. Understand. Find the fundamental quantity from the following. Select the fundamental quantity from the ones given below. This is a question that is indispensable. You have to study it anyway. What is the fundamental physical quantity among the following, children? Everyone should comment. Which of these is the fundamental physical quantity? You will have no doubt. Temperature is indeed the fundamental physical quantity among these. Yes, it is temperature. No doubt. It is temperature. Let's move forward. Next. One more question. Can you find the fundamental physical quantity from the following? These are all frequently asked questions. They are indispensable. That's why I have included only these two questions. What is the fundamental physical quantity among these? There is no doubt, it is mass. So, study such questions accurately. So, you should have that expectation. Understand the reality that questions oriented towards the classification of physical quantities and SI systems of units will come in my exam. So guys, we are moving forward. We are moving on to the next things. Next one is significant figures. I always say that what is frequently asked in exams from significant figures are questions of the type that are given a measurement and asked how many significant figures are there in it. You should have no doubt about that. There is no need for any kind of doubt. Such questions are consistently seen in our exams. However, my children should understand one thing. What is that? Yes, definitely. What is the definition of significant figures? You should study that. It is something you can never avoid. What are significant figures? I will explain it easily. Significant figures, when you take a measurement, it is the combination of the certain digits and the first uncertain digit within it that we call significant figures. When we take a measurement, the combination of the certain digits and the first uncertain digit within it is called significant figures. Understand it accurately. The combination of the certain digits and the first uncertain digit in a measurement is called significant figures. Understand it accurately. Now, what is this certain digit and what is this uncertain digit? I have explained it many times. I am sure many of you have heard it. However, I want to say it one more time. I want to say it one more time. Shall we stand for a moment? Let's see what is certain and what is uncertain. Pay attention, children. I will explain it easily. This is a metallic wire. Imagine this is a metallic wire. We need to measure the length of this wire. Shall we measure the length of this wire using a scale? Imagine this is our scale. I am measuring the length of this wire using this scale. Let me give the readings on the scale. Yes, I am placing it randomly. So, understand that. I am not giving precise readings or precise divisions. I am just giving it to give you an idea. Okay. Take it in that way. It's not a precise scale or readings. Come on. One, two, three, four, five, six, seven, eight, nine, ten. Right. Okay. Can you tell me the length of this wire by looking at this reading? Many people will probably give different readings. However, let me tell you. Look, some will say, "Sir, 8.1 centimeters." Some will say, "Sir, 8.2 centimeters." Isn't that right? Some will say 8.1 centimeters, and some will say centimeters. If you look closely, we can't say that, right? We can say that. Some will say 8.1, some will say 8. It's something we are not sure about, right? Definitely. So, will there ever be a change in what we call eight? It will never change. We are sure about eight. Eight will not change. Eight is eight. There is no doubt about eight. The only doubt is whether it is one or two. Isn't that right? If we go to 8.3, we can make it a bit smaller. I agree. I am putting it at two. So, let's stop here. Okay. So, some will say eight, and no one will have any doubt about eight. Eight is certain. There is no doubt that eight will not change. The only doubt will be whether it is one or two for most children. Isn't that right? So, pay attention here. Do you know what we call this eight? That is called our certain digit. Eight is the certain digit, or the digit that is sure. Eight is the certain digit. We are sure that eight will not change. What is an uncertain digit? It is a digit that we are not sure about. It can be one, it can be two. The digit about which there is doubt whether it is one or two is called the uncertain digit. The digit that is sure is called the certain digit, and the digit that is not sure is called the uncertain digit. So, the one and two here can be categorized as uncertain digits. The one and two here can be categorized as uncertain digits. The one and two here can be categorized as uncertain digits. Did everyone understand the concept accurately? Did you understand, children? So, the combination of the certain digit and the first uncertain digit in a measurement is what we call significant figures. This is called significant figures. Did all the younger brothers and sisters get that idea? If so, comment a "set" message in the chatbox. Then we can move on to the rules. Let's move on to the rules of how to find the significant figures in a measurement. Can you stand with me? Everyone, stand with me. If all the children stand with me, we can move on to the rules. Are you ready? Are you ready, children? If everyone stands with me, we can move on to the rules. Come on, quickly, stand with me. Come on, come on, come on. Yes, ready. Then, here come the rules. We are about to enter the rules. Everyone, stand with me. First rule. Pay attention. All non-zero digits are significant. Understand that all digits that are non-zero are significant. That is, my children should understand. For example, 123 centimeters. Is there zero in this? Absolutely not. Since there is no zero in this, all the digits in it are significant. So, how many significant figures are there here? How many will there be? Without any doubt, there will be three significant figures here, because one, two, and three are all non-zero. All three are significant. So, there are three significant figures. Another one, 78 meters. How many significant figures will there be? Tell me, how many significant figures are there in 78 meters? Everyone, please comment. Here too, there is no zero. Everything that is not zero is significant. So, there are two significant figures here. Next, 27 kilograms. How many significant figures are there? Six, three kilograms. There is no zero here either. Everything that is not zero is significant. So, there are four significant figures. Understand that there are four significant figures. Four significances. Did you understand the matter accurately? No doubt? That is, all non-zero digits, which are not zero, are significant. Understand it accurately. Fine. Okay, right? Did you understand it brilliantly? Now, we are moving on to the next rule. Let's move on to the next rules. Let's study in the same way. Next rule: For a number with a decimal point, let's consider a number with a decimal point. Okay. Trailing zeros are significant. That is, the zeros that come after the decimal point are what we call trailing zeros. So, all trailing zeros are.

Understand what is significant. Let me give an example. 7.0 meters. How many significant figures are there? Tell me. 7.0 meters. This zero is a zero that comes after a decimal point, right? It is a trailing zero, and it is significant. Don't doubt it. So, how many will there be? There are two significant figures here. Understand that there are two significant figures. Next example: 24.00 kilograms. How many significant figures are there? 24.00 kilograms. How many? There are four significant figures. Because the last two zeros are zeros that come after the decimal point, they are all significant. So, in the second case, there are four significant figures. Shall we look at another example? 1.00 centimeters. How many significant figures are there? All zeros after the decimal point are significant. Don't doubt it. How many are there? There are four significant figures here as well. Again, there are four significant figures. Did you understand the concept clearly? That is, understand that all zeros that come after the decimal point are significant. That is very, very, very important. Okay? Shall we move forward? Let's move to the next point. For a number without a decimal point, the trailing zeros are not significant. Consider a situation without a decimal point. When considering a situation without a decimal point, understand that trailing zeros are not significant. Example: 2000 meters. How many significant figures are there? How many significant figures are there in 2000 meters? None of these zeros are zeros that come after the decimal point. Zeros that come without a decimal point are not significant. Therefore, there is only one significant figure here. Understand that there is only one significant figure. Next, tell me: 220 kilograms. How many significant figures will there be? This zero is not a zero that comes after a decimal point. Therefore, it is not significant. Right? Therefore, it is not significant. There are only two significant figures here. Only two significant figures. Let me give another example: 3 * 10^8 meters per second. 3 * 10^8 meters per second. Is this a value everyone knows? Yes. How many significant figures are there in this? What you need to understand here is that you don't need to consider what comes in the power of ten. It is not significant. So, there is only one significant figure here. What is it? There is only one significant figure. That's it. Next, let me give another example. Now, 72 * 10^3. How many significant figures are there? 72 * 10^3. How many significant figures will there be? Without any doubt. Ah, here you don't need to worry about the power of ten. So, 72 means how many significant figures are there? There are two significant figures. Only two significant figures. Did you understand the matter clearly? Did you understand the concept? That is, trailing zeros are not significant. Understand that trailing zeros in a whole number without a decimal point are not significant. This is important. Very, very, very important. Right. Now, let's move to the last rule. Leading zeros are not significant. The zeros that come in front are what we call leading zeros. Those leading zeros are not significant. Understand that none of the zeros that come in front are significant. Leading zeros are not significant. None of the zeros that come in front are significant. Understand that none of the zeros that come in front are significant. Example: 0.024 meters. How many significant figures are there? 0.024 meters. How many significant figures are there? All these zeros are leading zeros. They are zeros that come in front. None of them are significant. Understand that none of them are significant. How many significant figures are there? Without any doubt, there are two significant figures. Next: 0.007. 0.007 meters or kilograms. Right. Kilograms. How many significant figures are there? Tell me. Yes. We know that none of the zeros that come in front are significant. There is only one significant figure here. There is only one significant figure here. Understand that. Another example: 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.00 centimeters. How many significant figures are there? The zeros that come after the decimal point are all significant. Don't doubt it. There are four significant figures here as well. Again, there are four significant figures. Did you understand the concept clearly? That is, understand that all zeros that come after the decimal point are significant. That is very, very, very important. Okay? Shall we move forward? Let's move to the next point. For a number without a decimal point, the trailing zeros are not significant. Consider a situation without a decimal point. When considering a situation without a decimal point, understand that trailing zeros are not significant. Example: 2000 meters. How many significant figures are there? How many significant figures are there in 2000 meters? None of these zeros are zeros that come after the decimal point. Zeros that come without a decimal point are not significant. Therefore, there is only one significant figure here. Understand that there is only one significant figure. Next, tell me: 220 kilograms. How many significant figures will there be? This zero is not a zero that comes after a decimal point. Therefore, it is not significant. Right? Therefore, it is not significant. There are only two significant figures here. Only two significant figures. Let me give another example: 3 * 10^8 meters per second. 3 * 10^8 meters per second. Is this a value everyone knows? Yes. How many significant figures are there in this? What you need to understand here is that you don't need to consider what comes in the power of ten. It is not significant. So, there is only one significant figure here. What is it? There is only one significant figure. That's it. Next, let me give another example. Now, 72 * 10^3. How many significant figures are there? 72 * 10^3. How many significant figures will there be? Without any doubt. Ah, here you don't need to worry about the power of ten. So, 72 means how many significant figures are there? There are two significant figures. Only two significant figures. Did you understand the matter clearly? Did you understand the concept? That is, trailing zeros are not significant. Understand that trailing zeros in a whole number without a decimal point are not significant. This is important. Very, very, very important. Right. Now, let's move to the last rule. Leading zeros are not significant. The zeros that come in front are what we call leading zeros. Those leading zeros are not significant. Understand that none of the zeros that come in front are significant. Leading zeros are not significant. None of the zeros that come in front are significant. Understand that none of the zeros that come in front are significant. Example: 0.024 meters. How many significant figures are there? 0.024 meters. How many significant figures are there? All these zeros are leading zeros. They are zeros that come in front. None of them are significant. Understand that none of them are significant. How many significant figures are there? Without any doubt, there are two significant figures. Next: 0.007. 0.007 meters or kilograms. Right. Kilograms. How many significant figures are there? Tell me. Yes. We know that none of the zeros that come in front are significant. There is only one significant figure here. There is only one significant figure here. Understand that. Another example: 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Let's square both sides, we can square both sides. So what is the square of time? T square. What is the square of two pi? Four pi square. Once the root is gone, it's just L by G. Four pi square, isn't that a constant, my children? Certainly, that is a constant. Since it is a constant, it has no dimension. So the dimension of time will come as T square itself. Equal to L divided by G. We know the dimension of G is L T raised to minus two. Look here, L here and L here, right? You should not cancel like this. You must understand, you should not strike it out on the answer sheet. Once L is gone, the remaining is T raised to minus two. When T raised to minus two goes to the top, it becomes T square. So both sides got the same dimension. The equation is dimensionally correct. Understand that the equation within it, what is the equation, is dimensionally correct.

If you are happy, comment a smile emoji. Then we can move on to our last application. Let's move on to our last application. Yes guys, we are moving forward to our last application, that is application number two. We are moving on to the final application. There are only two questions in this. It's the last application, right? The first is kinetic energy and the second, I think, centripetal force. We only need to look at these two things. I will finish both easily for you. No tension needed, we can crack it easily. Just stay with me.

Derive the expression for kinetic energy. What all does kinetic energy depend on, they are saying? Kinetic energy possibly depends on mass, speed. It depends on these two things. Look, smile, buddy! Someone said a smiley face, why aren't you all smiling? Everyone should smile and be happy. You are not right. The derivation is starting. If you can do it with me, it's very important to do it with me, okay? So, what all does kinetic energy depend on? Kinetic energy is proportional to mass, speed. Equation number one. To change the proportionality to an equals sign, what if we introduce a constant? So, kinetic energy equals K into, give powers, M power of A, V power of B. Equation number two.

The next step is to give dimensions to all of these. K is a constant, it has no dimension. Leave it, let's consider the other terms. It's kinetic energy, my children. We all know the dimension of energy, right? The dimension of energy is M L square T raised to minus two. I am putting one in the power of M for the convenience of equating. K has no dimension. Mass is M, so M to the power of A. V means speed, so L T raised to minus one, power of B. What equation number should we give? Equation number three. I am putting all the powers inside. I am putting all the powers inside, so pay attention, everyone pay attention. M power of one, L square T raised to minus two, isn't that right? M L square T raised to minus two. Here I am putting the powers inside, so M power of A, L power of B, then what do we get? T power of minus B, T power of minus B. What equation is this? This is equation number four. It is the fourth equation. Understood clearly, no doubts, right?

Now let's try to equate the powers. Look here, in the place of A, there is one. So first, I am writing A equals one. Here, in the place of B, there is two. So B equals two. We got both. Now we just need to substitute directly. That is, kinetic energy equals K into M power of one, V power of two. We can write it in the form M power of one, V power of two. Here, I will give it equation number five. You all know the value of K, right? What is the value of K? K is a constant, its value is one by two. Since we already know it, we can substitute it if we want. So, if that's the case, what will kinetic energy be? Half M V square. This is our final equation. Kinetic energy equals half M V square. This is our final equation. Clear, right?

If you have understood it confidently, only then put a "Set Anna" comment in the chat box. Everyone, if you say "I have understood this brilliantly, no doubt," then only put a "Set Anna" comment. If so, then we can move on to the last question. With this, I am going to finish this chapter. Here comes the last question from this chapter. What will that be? Yes, everyone must study it. It is the relation for our own centripetal force. It is the last one. With this, the chapter is finished. Within approximately one and a half hours, all the concepts in the chapter called "Units and Measurements" and all possible supportive questions related to it were taught, okay? There are still more questions; those slightly tougher questions can be studied in the top-level questions section. Here, the aim is to strengthen the concepts and prepare the content.

So, what all does centripetal force depend on, they are saying? Centripetal force may possibly depend upon mass, radius, velocity. It depends on these three things. Shall we start, buddy? So, force is proportional to... Let's not do it here, let's do it on a new page, isn't that more dignified? Yes. So, force is proportional to, as they say, mass, radius, velocity. Mass, radius, velocity. Let's give it as equation number one, or it was given as one. Let's bring in a constant here. The constant I am introducing here is K. K into M power of A, R power of B, V power of C. This is equation number two.

Now, shall we give dimensions to everyone? We all know the dimension of force. It is M L T raised to minus two. If there is nothing in the power, we put one. K is a constant, it has no dimension. The dimension of mass is capital M, with A in the power. The dimension of radius is L, with B in the power. The dimension of velocity is L T raised to minus one, with C in the power. This is equation number three. Now I am putting all the powers inside, so pay attention. M L T raised to minus two equals M power of A, L power of B, L power of C, T power of minus C. What equation number is this? It is four. Let's combine the powers. That is, if the base is the same, I am adding the powers, pay attention. So, M power of one, L power of one, T power of minus two equals M power of A. Sorry, guys. M power of A, L power of B plus C, T power of minus C. Understood clearly, no doubts, right? What equation number can we give this? Let's give it five.

Now, shall we equate the powers? Now let's equate the powers. Stay with me. Here, what is in the place of A is one. So I can simply write A equals one. Next, pay attention to the next one. Here, in the place of minus C, there is minus two. So we can write that, right? Minus C equals minus two. So we wrote C equals two. So we got A and we got C. Next, I am going to equate. Look, B plus C is one. B plus C is... Shall I equate it? Look, B plus C, let's equate the side now. B plus C equals... So B plus C is two, equals one. So B equals one minus two. So B equals minus one. So, isn't everything done? We got A. Look, A is one, C is two, B is minus one. Isn't everything done? Shall we replace everything, my children? Let's replace.

So, centripetal force equals K into M power of what? M power of A. What is the value of A? It is one. So, M power of one. What is next? R power of B. So, what is R power of B? The value of B is minus one. Next, V power of C. The value of C is two. This is equation number six. I am simplifying it. Can we write R power of minus one as one by R, buddy? R power of minus one equals one by R. So, if we replace it like that, F equals K into M V square by R will come. K into M V square by R. My children, tell me, what is the value of K here? Everyone, please comment. What is the value of K here? What is the value of the constant? What is the value for the constant here? What is the value of the constant here? No doubt, the value of the constant here is one. So, we can give K as one. If so, what will be the equation for centripetal force? F is equal to M V square by R will be obtained. Yes, this is our final equation. This is our final equation. Everyone must study this. Such questions are frequently asked in exams. This is very, very, very important. It is a question that must be studied. Okay, buddy? Everyone understood, right? Confident, right? Are you confident with this too? If so, shall we move on to the next?

There is one more thing to say in this chapter, that is limitations. It is something everyone should understand, that is limitations. Limitations of dimensional analysis. Let's look at what the shortcomings of dimensional analysis are, what the limitations are.

Using this method, we can never find the value of dimensionless constants. The value of dimensionless constants can never be found using this method. Also, if a physical quantity depends on more than three physical quantities, it can never be solved using this method. For example, centripetal force depends on mass, radius, and velocity. If it were to depend on a fourth parameter, it could never be solved using this method. It can only be solved using this method if it depends on a maximum of three quantities. Next, we cannot derive a formula if it contains trigonometric functions, logarithmic functions, and exponential functions. If our equation contains trigonometric functions, logarithmic functions, and exponentials, we can never do it. If sine, cos, tan, e power of x, log, etc., come, we can never derive it. So, these three are our shortcomings. These three are the limitations. Understood clearly, right? Is it set?

This chapter is over. The next chapter, Motion in a Straight Line, Jasheel Sir is waiting. Is this okay, buddy? If everyone understood, can you please respond in the chat box? If everyone understood, can you please respond in the chat box? If you put a "Set Anna" comment, I will call Jasheel Anna. Everyone, please respond. If everyone responds, we can call Jasheel Sir, and you can study the chapter called Motion in a Straight Line brilliantly. I will call Jasheel Anna. Where did Jasheel Anna go? Where is Jasheel Anna? Yes, Jasheel Anna has gone to the staff room. It's because the AC is on here. Yes, great, awesome! Everyone should study the subsequent chapters accurately, just like this. No compromises, okay? Just like this, have you all written key notes for all the remaining chapters? Those who have written short notes, please put a fire emoji. Everyone who says, "I have written down all the important keywords and derivations in this, I have written short notes," please put a fire emoji, buddy. Come on, guys, no one should give up on that. Everyone who says, "I have written down everything in this, I have written down all the important points," should also put a fire emoji. Air drop karo. Two fire emojis were put. Seems like I need to check the answers. Not yet? Need to give one more role. It's over there. I ended everything. I was taking half an hour there. Yes, yes, everyone is putting fire emojis. Not bad, right? Ah, okay. Ready, ready, is it done? Need to check if there are any answers. What were you doing all this time, man? This person was playing by writing answers. I was playing by doing answers, I even added 2025 model questions. Okay, ready. Don't we need variety? No, not anymore. Enough, teach. Okay, ready, everything is ready. Sir, send it. Send it. If you're giving, ask the children if they are well. Ask if the children are doing well, if they are awesome. What do you need? Ask for it. Send the slides, don't tell stories. What do you need, my children? Comment, comment, everyone comment. What do you need in the coming days, besides what we are giving? Do you want biryani meat? Give it. We can give biryani meat. I K C, if you want to win, what happened? It has come, it has come. The thing has arrived. Open your eyes, open your hands, there's a song like that, isn't there? There's no such song, this person is making up songs. No, no, sorry. Okay, C, let's emphasize all concepts. We just need to emphasize like this, right? Just emphasize like this. We will emphasize all concepts like this. Yes, yes, yes, yes. Okay, Anna is saying he will be back in one hour. Now, let's begin our session, my children. I believe everyone is ready. We are about to start. Yes, what is the beautiful, brilliant chapter called Motion in a Straight Line? It is chapter number two. Yes, it is chapter number two. Three used to be before, now it's two, my children. Yes, it is chapter number two. Ready, shall we start, buddy? Has everyone come after having tea? Please tell me. Has everyone come after having tea, vada, dosa, and all? Everyone, please comment. Yes, sir, we have come after eating. We are energetic. Don't ask what sir ate, sir has come after eating. Sir ate two dosas and one vada. Come on, come on, guys, shall we start? I am saying there is no time to waste, and then I am wasting time, right? Pay attention. So, we know a few things about this chapter. Still, we need to learn all the things, all the concepts brilliantly and move forward. So, if you stand firmly with me, we can discuss all the concepts and all the important questions in this chapter, my children.

Okay, ready. So, in this chapter, what we are dealing with, what we are dealing with in Plus One Physics, is mainly about Mechanics. When we say Mechanics, we can classify it into three. The study of a body at rest is called Statics. Similarly, Dynamics, as we all know, is the study related to the motion of a body. And Kinematics, in Kinematics, we do not study about the cause of motion. Dynamics is about the laws of motion, where we study about the forces that cause motion. That is what we need to know overall regarding Mechanics. Let's make the whole community set. So, what is rest? All concepts are taught from the base level. That is why I have added even such small things in this slide. Everyone must learn everything. That is, even for children who know nothing, watching this class should be worthwhile. That is our aim. The duration of the class will be until approximately seven chapters are finished. The class is likely to extend until around 8 PM or 7 PM in the evening. Please cooperate, keep watching. Look, my children, so what is rest? If the position of a body, if the position of a body (now, I am looking at this tree), if the position of a body does not change with time, then we say that body is at rest. "Position of an object does not change with time" or "with respect to time," that is what we call rest. If the position of a body does not change, then what will motion be? If the position of a body changes with time, then we say that body is in motion, right? "Position of an object" or "position of the object changes with respect to time" is what we call motion.

As we move forward, it's distance and displacement. What is distance? What is displacement? Shall we look? Now, a man is going from here to here. So, this is what we call the initial position. This is what we call the final position. Okay, buddy? This man can go like this. This man can go straight. This man can also go like this. The man can go through three paths. Ready, my children? Ready, ready? If this man goes like this, the length of the path he took, the length of the road's path, is what we call distance. So, what will displacement be? Displacement is the straight-line distance between that man's initial position and final position. That is what we call displacement. Understood, my children? So, what is distance? Simply, distance is path length. Displacement is the shortest distance between initial and final position. The straight-line distance between the initial position and the final position, what do we call that, my children? The straight-line distance is what we call displacement. Ready, my children? Did everyone understand that? Is the slide getting cut? No, the slide is not getting cut.

Okay, my children? So, learn what distance and displacement are. Distance is a scalar quantity. Displacement is a vector quantity. Distance can have positive, negative, and zero values. For displacement, sorry, I got it wrong. Distance can only have positive values. Displacement can have positive, negative, and zero values. For example, if a body started its journey from here and went this way, then we can say the displacement is positive. If it came back to the same spot, then the displacement is zero. If it went back this way, the displacement is negative. But in all these cases, the distance will be positive. Distance is path length, and length will be positive. Did everyone learn, my children? Okay, yes, yes, yes. Is it ready, my children? Did everyone understand, my children? I believe everyone understood. I believe everyone understood. Ready, ready, ready, ready, ready, ready, ready, ready, ready. Yes, yes, yes.

Let's move on to the next. Hey, my children, this is a question from the 2023 Christmas exam. Everyone must answer. "For straight-line motion, distance and displacement are equal." Is this statement true or false? Everyone say "True" and answer. If a body is in straight-line motion, are distance and displacement equal? Is this statement true or false? We know that if a body moves forward in a straight line, you will understand just by looking. A body is moving forward in a straight line. So, look, buddy, this is the initial position, this is the final position. Ready, right? Here, distance and displacement are indeed the same. This statement is true. So, if a body is in straight-line motion, what are the distance and displacement of that body, my children? They will be the same. Pay attention to another thing: it must be a straight line. What must it be? It must be in the same direction. The direction should not change. If the direction changes, the situation changes. Okay, right? So, I believe everyone understood.

So, we are moving on to speed. What is this speed? Simply put, in a unit time, what is it, my children? In a unit time, the distance covered by a body is called speed. To put it a little more simply, in one second, what is it? The distance covered by a body in one second is what we call speed, my children. Ready, my children? Did all the children understand what speed is? "The distance covered by an object in unit time." The distance covered by a body in a unit time or in one second is what we call speed. What is the equation to find speed? It will be distance divided by time. If we divide the distance covered by that body by time, we can find speed. So, since it is distance by time, speed will be a scalar quantity. Speed has no direction. The unit of speed is meter per second. The dimension of speed is L T raised to minus one. You should know all these things. The value of speed can only be positive. Distance can only have a positive value. So, speed will also only have a positive value. Ready, my children? Or we can also say it like this: the rate of change of distance is what we call speed. We can say it like that if we want.

What's next, my children? It's velocity. What is velocity? In a unit time, in a unit time, the displacement covered by a body. There, it was distance, right? If speed is the distance covered in one second, then what will be the specialty here, my children? The displacement covered by a body in a unit time is what we call velocity. So, "displacement of an object in unit time." Doesn't it have another definition? Can anyone say what that other definition is? Can anyone say what that other definition is? What would it be? There, we know that the rate of change of distance is called speed. So, what would it be here? "The rate of change of position" or "the rate of change of displacement." Both are correct, no problem. Okay, right? The rate of change of displacement or the rate of change of position is what we call velocity. Okay, buddy? In some cases, we consider position as displacement. Okay, let me leave some measure. So, the unit is meter per second. Velocity is a vector quantity. Velocity can have a positive value. If a body is going that way, it's a positive value. If it's coming back, it's a negative value. Okay. If it's a round trip, average velocity is zero. So, velocity can have a positive value, a negative value, and it can even be zero. Ready, my children? Ready? What is another equation for velocity, my children? What is another equation for velocity? Can anyone say it in terms of position? Can anyone say what another equation for velocity is, my children? Velocity is the rate of change of position, delta X by delta T. This is an equation everyone must learn. Velocity, average velocity, or simply velocity, is the rate of change of position. What is this delta X? It is change in displacement or change in position divided by change in time. Okay, did you understand this equation? Did you understand? Actually, what is change in position, my children? It is displacement. So, displacement, we can also say displacement like this: change in position. This is the initial position of the body, we called it X1. This is the final position of the body, we called it X2. So, what will displacement be? It is final position minus initial position. That is, X2 minus X1. We called that delta X, delta X. Okay, right? So, velocity is displacement by time. We can also write it like this: the rate of change of position or the rate of change of displacement. Will you forget, my children? Will you forget? No one should forget. Not even one student watching this class should forget. Velocity is the rate of change of displacement or the rate of change of position. Okay, my children? Did everyone understand what velocity is?

Now, there is a table for this, a table that everyone studies. The exam will ask, "What are any two differences between velocity and speed?" In bold letters, let's first write down the definition. Speed is the distance covered by an object in unit time, or it is the rate of change of distance. Here, what will it be? It is the rate of change of displacement. Speed is a scalar quantity. Velocity is a vector quantity. Speed can only have positive values, but velocity can have positive, negative, and zero values. Ready? Okay, right? So, we have perfectly studied speed and velocity, studied them thoroughly. If you have understood it brilliantly, let an awesome comment come in that comment box. Let an awesome comment come in that comment box. In the same way, let's move forward with all the concepts and related questions. You don't need to go anywhere else searching for model exam questions anymore. Just watch this one live session. Anna has brought questions asked in the 2025 model exam here. So, questions up to the 2025 model are set here. In the live session before the model exam, we will study all the questions in the model. We have already dealt with all the questions from the Christmas exam, not 20th Christmas, but up to the 2024 Christmas exam, during the Christmas exam. Since there was no Onam exam, that's not there. Now, when we go for the next public exam, we will deal with questions up to 2025, including the 2026 model. So, we will make all the PYQs set. Ready, my children? Yes, yes. "An object travels in the direction of..." In what direction does an object travel, my children? Does an object travel in the direction of acceleration? No. For example, if I throw a stone upwards, when I throw a stone upwards, the velocity is upwards, the body is going upwards, but the acceleration is downwards. So, the direction of a body is never the direction of acceleration. Okay, right? Then what is it? Then what is it? Velocity. Very good, very good. The direction of a body is decided by the velocity of that body. Velocity decides where the body should go. Not acceleration. For example, a car. A car, my children, pay attention. A car is a beautiful jeep. Aha, awesome jeep, police jeep. So, someone is standing here like this. Oh no! So, it will brake, right? So, the car is moving this way, right? The car applied brakes. If the car applies brakes, we all know it's a retarding force. The force will be this way. And similarly, the acceleration will be this way. So, look, velocity is this way, and the car is moving this way, but the force and acceleration are in opposite directions. The direction of acceleration is the direction of force. Okay, right? So, look here, velocity, force, and acceleration are in opposite directions if the body is in retardation. Just know that. So, remember, velocity decides the direction of a body. Next question, next question. Yes, yes. This is a question from the 2022 Christmas exam. "Give any two differences between speed and velocity." We can say many things, right? Speed is a scalar quantity, velocity is a vector quantity. Speed only has positive values, but velocity can be positive, negative, or zero. Simple, just write this much. Okay. And even if you write the definition, you will get marks. Okay, my children? Yes, yes.

So, so, come on, Mahesh Suresh. Thankappan Thankamma, everyone come, everyone come, everyone come. We need to study. We need to study things energetically and what should we do? We need to move forward, forward, forward. Ready, buddy? Ready, buddy? Okay, buddy? Yes, yes, yes. So, let's move on to the next. We are going to jump, run, run, jump to the next. All children, please pay attention. Many people are watching the live session here without liking. This is bad, okay? This is bad, okay? Press that like button. I am eagerly, I am eagerly waiting for your likes. What kind of 'girly' eagerly? It's not a strange 'girly', right? Ready, ready, ready, ready. So, next, next, what we are going to look at is instantaneous velocity. What is it, my children? Instantaneous velocity is what we are going to look at next. Yes, before that, there was uniform velocity. When I sent the slide here, I think I forgot to do something. I think I forgot to select that uniform velocity.

I think so. Let's look at uniform velocity. Shall I send that too? If anyone has any complaints, please tell me. If there are complaints about me sending it, it's okay. Shall I send that slide of uniform velocity along with it, sir? It got changed when it was sent here. It didn't get changed. He showed some reluctance to come. So, shall we just send the slide of that too, sir? Where are you, man? Where are you? Where is the uniform velocity? Where have you gone and ended up? You don't see the children waiting here. You don't see the children waiting here. Yes, non-uniform velocity. Uniform velocity. Non-uniform velocity. Yes, yes, yes. It got changed when it was unsent. It will come now, children, just a minute. So, let instantaneous be there. Let instantaneous be there. Let's look at uniform velocity. What is uniform velocity, children? A body is moving uniformly. Its velocity doesn't change. What will be the velocity, children? It will be constant. That's what we call uniform velocity. Look, when we say uniform velocity, what is uniform velocity? Look, what is uniform velocity? A body has uniform velocity. It covers displacement in equal intervals of time. Now, look at the car's motion. In the first second, 10 meters. In the second second, 10 meters. In the third second, 10 meters. In the fourth second. That is, it's moving 10 meters every second. So, comment on the car's velocity, everyone, comment. What will be the velocity of this car? What is the velocity of the car at each and every point? Take any point. Take any point. Here, you found the car's velocity as displacement by time, 10. Displacement by time, 10. Displacement by time, 10. Displacement by time, 10. Displacement by time, 10. Displacement by time, 10. No matter which point you take, the car's velocity doesn't change. The car's velocity is 10 meters per second. It's a constant. Then we say it's uniform velocity. Simple. What is uniform velocity? It should be equal displacement in a given time interval. Equal displacement in equal intervals of time. In the first second, 10 meters. In the second second, 10 meters. In the third second, 10 meters. In the fourth, 10. In the fifth, 10. In the sixth, 10. Every time, at each and every point, there is no change in the body's velocity. The body's velocity will be constant. Then we can call it uniform velocity. A body has uniform velocity. It covers. Is it to be kept inside the cover? No. It covers. The displacement it covers is equal displacement in equal intervals of time. You should know it. It's something you need to know. Don't we say "Thambi"? Yes. Ready. Ready. Then, what will be the question from 2020 sales? There was a question from here, children. They will directly ask us questions from here. They will directly ask us what is uniform motion or uniform velocity, children. Learn one point. Learn one point. Uniform velocity. What is uniform velocity? Uniform speed in a particular direction. What is said? Uniform velocity is uniform speed in a particular direction. It's going in a particular direction. The direction doesn't change. For a body moving in a particular direction, its uniform speed and uniform velocity will be the same. Okay, man? Uniform velocity is also called uniform motion. We also call uniform velocity by another name. That is uniform motion. So, if asked in the exam, when is uniform speed and uniform velocity the same? We can say that if a body is moving in a particular direction, then uniform speed and uniform velocity are the same. Uniform velocity is uniform speed in a particular direction. It's going in a particular direction. Okay. Now, if we take circular motion, it's wrong. Circular motion. If a body is in circular motion, it has uniform. It will be uniform speed. Not uniform velocity. Because why? Why? The direction of velocity changes. If the direction of velocity doesn't change, then uniform speed and uniform velocity are the same. Ready, right? Remember that. Remember that. There will be confusion in circular motion. In circular motion, speed is constant, velocity is not constant. Okay, children? Ready? Does that point click? Did that point get through? Questions can be asked like this from here. Questions can be asked like this. So, listen carefully. Now, what will be non-uniform velocity? We know. Look. In the first second, 10 meters. So, in the first second, the body's velocity is 10 meters per second. In the second second, he traveled 20 meters. So, in the second second, the velocity is 20 meters per second. In the third second, he traveled 40 meters. He is in accelerated motion. The velocity is changing. So, look, children. In the third second, his velocity is 40 meters per second. Then we say that here the velocity is not constant. So, what will be non-uniform velocity? Simple. What we learned there is equal displacement in equal intervals of time. So, what will be here? You can say it yourself. You can say it yourself. What will be non-uniform velocity? Just add 'un' to equal and it becomes unequal. Unequal displacement in equal intervals of time. If you add 'un' to that definition, that definition is complete. That's how you should learn definitions. There's no point in memorizing. Ready, children? Okay, man? Did everyone understand this? Define uniform motion. It's a question already discussed in 2020. Leave it. Okay. Then, this is very, very important. I see a possibility of it being asked in the exam. It's important. It's important. Everyone should learn it. What is instantaneous velocity? Simple. Simply, instantaneous velocity is the velocity at any instant of time. Velocity at any instant of time. If a body is moving like this, and I can find the body's velocity even at the moment it's passing by, that is instantaneous velocity. Velocity at any instant of time. If I can calculate the body's velocity at any time it's moving, that velocity is called instantaneous. Instant. Don't we say "instant coffee"? So, the time interval will be very small. If you can find the velocity of a body in a very small time interval, a tiny time interval, that velocity is called instantaneous velocity. Children, learn it. Velocity at any instant of time. Velocity at any. Just learn it like this. If instantaneous velocity is asked in the exam hall, just learn it like this. Velocity at any instant of time. Ready, children? Did everyone understand? There is also an equation to learn here. So, what is instantaneous velocity? Velocity of a body at any instant of time. Okay. Yes, yes, yes. Now, pay attention. There is also an equation to learn here. That equation is very simple. You just need to write this definition. You have to learn the equation here. We know what the equation for velocity is. We know the equation for velocity is delta x by delta t. Okay? You know this. This is called average velocity. Average velocity. So, when does average velocity become instantaneous velocity? The time interval should be very small. When the time interval becomes very small, average velocity becomes instantaneous velocity. So, here we take a concept. Our time interval should be very small. Our time interval should be very, very, very, very, very small. It should approach zero. It shouldn't become zero. If it becomes zero, we can't find it. Our time interval should be very small. So, our time interval should be close to zero. The time interval delta t tends to zero. Delta t tends to zero. Delta t should approach zero. The time interval should be very small. That's what it means. So, what is the name of the person used in mathematics for this? Comment. We call it by a name in mathematics. What is it called? Afanas Anna teaches this. Limit. If something approaches something else, very close, very close, very close. That concept is called limit. That concept is called limit. So, what we use in mathematics is limit. LIM. Just remember it like lime soda. Limit delta t tends to zero. So, what should we do to make average velocity instantaneous velocity, children? Just add the limit. Limit delta t tends to zero. What does this mean? The time interval in this equation is very small. That's what it means. Learn it. Everyone learn the equation. Okay. Ready, right? Ready, right? I will tell you another equation. I will tell you another equation. Velocity is equal to dx by dt. dx by dt. You can write this too. You don't have to write both. dx by dt means nothing. If the time is very small, dt is a small change. Delta t is a change. Delta t is a change. Dt is a small change. Oh, my hand slipped. Delta t is a change. Dt is a very, very, very, very, very, very, very small change. Infinitely small change. Okay? Delta t is change. Dt is a small change. Just remember this much. So, it's a small time interval. There's a small displacement. That's dx by dt. So, you can write any one of these equations. There's no compulsion to write both. It hasn't been asked like that in the exam so far. If asked in the exam to write the equation in terms of limit, then you have to write this. It hasn't been asked like that so far. And it might be asked. So, all children should learn this equation. What is instantaneous velocity? Velocity at any instant of time. Equation? Let's finish the work and go. Okay. Ready, children? Ready, children? Everyone understood, I believe. Then let's move to the next. Come on, come on, come on. Is everyone okay? Ready? Shall we move to the next? Yes. Question from 2020 model. The speedometer of a car shows 70 kilometers per hour. This represents. The speedometer of a car shows 70 kilometers per hour. What is this, children? Everyone say. Everyone say. Is this average velocity? No. Instantaneous speed. Don't say instantaneous velocity. Don't say it. Because it's the speedometer. It shows speed in its name. Not direction. So, what will it be? This is instantaneous speed. Everyone is a good child. You are all giving the answer excellently. So, we have locked velocity. We are now going to acceleration. May velocity be excellent for all children. If you understood it excellently, give a comment saying "Understood, sir." Then we are going to the next. That is acceleration. Acceleration. Okay. Okay. You asked for my name. My name is J M K Mash. Isn't it, man? You meant the teacher's name. J M K is known. Then come on, come on. We are going to learn acceleration beautifully. There, we learned that velocity is the rate of change of position, or the rate of change of displacement. That's what we called velocity. So, what is acceleration? An apple or a mango is falling down. Its velocity is changing. There is a special characteristic to the change in velocity. Every second, its velocity increases by 9.8. So, what will be the acceleration of that body? I will say the acceleration of that body is 9.8 meters per second squared. That is, the change in velocity of a body in one second, or in a unit of time, is called acceleration. So, how can we define it? Acceleration is the rate of change of velocity. Say it once, children. What is acceleration? Acceleration is the rate of change of velocity. Right? The rate of change of velocity. So, tell me its equation. Wherever you see "rate of change," just divide by time. Wherever you see "rate of change," the equation will have time. It's the change in velocity. So, acceleration is the rate. Rate comes when you divide by time. It's the rate of change in velocity. Rate of change of velocity is what we call acceleration. Acceleration can be defined as the rate of change of velocity. Okay, children? Isn't it simple? Ready, ready, ready. Yes. Okay. Note. Note. Retardation. Many children have a misconception about retardation being negative acceleration. Many children have a misconception that if the sign of acceleration is negative, it's retardation. This is a definition that shatters that. If asked in the exam, what is retardation? You just need to write this much. Retardation is when the speed of a body decreases. Listen, I'll say it again. The speed of a body decreases. Speed of the body decreases. It is said to be in retarded motion or decelerated motion. When the speed of a body decreases, we call it retardation or decelerated motion. Okay, right? Ready, right? Just remember this point. If the speed of a body decreases. Everyone will understand. If speed decreases, retardation. If speed increases, acceleration. Simple. Now, a car. Take a car. This car. The car saw a child. Applied brakes. We know the car's velocity is v. After applying brakes, the car will stop. Okay? So, the velocity is in this direction, right? Then, where will the acceleration be? Everyone say. Everyone say. Where will the acceleration be? If the acceleration is in this direction, it will hit and pass this child. Won't it? So, this body needs to be stopped. The velocity of this body needs to be reduced. The speed of this body needs to be reduced. To reduce the speed, we know the force must be in this direction. So, the acceleration will also be in this direction. Got it, everyone? Understood, everyone? If a body is in retarded motion, acceleration and velocity will be in opposite directions. Okay, right? Now, if you throw a stone upwards. If you throw a stone upwards, everyone knows it's retardation. So, the velocity is upwards. So, it's certain that the acceleration due to gravity must be downwards. That's why we say acceleration due to gravity is always acting downwards. Whether it's a stone falling down or a stone thrown upwards, acceleration due to gravity is always downwards. Did everyone understand, man? Did you understand, man? I believe you understood. Then, Christmas question. Define acceleration. Unit and dimension. What is acceleration? Children, say. What is acceleration? We know acceleration is the rate of change of velocity. The unit is meters per second squared. Unit. The unit is meters per second squared. The dimension of acceleration is M T to the power of minus two. Ready, right? 2023 question. Okay. Man, I'm going. My mind is not ready. I'm getting hurt. Don't cry like this, man. Don't cry like this. I'm here. We are all here. Why are you crying like this? It's nothing. Make your mind set and sit here. If you want, jump once. Listen to Maria's comment. Maria, what happened? Maria is sad. What happened? See, my mind is not ready. I'm going to get hurt. Hari, who hurt you? Was it me? Are you the teacher? Go and hit him. No. I said, go and run a round or two. Run around the house. Run a round or two. Make your mind set. Yes. Ready. Ready. So, children, shall we move to the next? Yes. Let's move to the next. No time to waste. Moving to the next. Define acceleration. We will define it. We all know it. Question from 2027. What is retardation? Improvement and 2023 model. Retardation is when the speed of the body decreases. The body is in retarded motion. Velocity and acceleration are in opposite directions. Ready, children? Everyone okay? Man, I told you, if your mind is not ready, get out and run. Run a couple of rounds again, and it will be fine. Ready, ready. Come on, come on. Uniform acceleration. What is uniform acceleration? Children, so, what is uniform acceleration? Everyone pay attention. Everyone pay attention. If a body has uniform acceleration, we know what uniform velocity is. Equal change in displacement, or equal displacement in equal intervals of time. I studied at Punnapra Engineering College. Okay, ready, ready, ready, ready, ready. Okay, right? So, what is uniform acceleration? We know acceleration is the change in velocity by time. Change in velocity by time. So, can we say what uniform acceleration is? Equal change in velocity in equal intervals of time. Say it once, children. Say it once, children. What is uniform acceleration? Equal change in velocity. The change in velocity should be equal. The time should also be equal. Then the acceleration will be the same. Yes. Uniform acceleration. An example is the acceleration of a freely falling body. It's uniform acceleration. Equal change in velocity. The velocity is increasing by 9.8 every second. Equal change in velocity. Equal intervals of time. Sir, I need a song. I will sing a song. If not today, then tomorrow. If not possible, then the day after tomorrow. A great song. So, everyone understood uniform acceleration, I believe. We are moving to the next. What will be? Define uniform acceleration. Question asked in March 2018. Non-uniform acceleration. What is non-uniform acceleration, children? If it's equal there, add 'un' here. Unequal. Yes, yes. Oh, Maria, it's nothing. Maria, people will say things like that. Let it go. Study well, Maria. I'm with you. Don't you have my assurance, Maria? What else do you need? Look, look. Non-uniform velocity. Non-uniform velocity. Non-uniform velocity. Sorry, non-uniform acceleration. What is non-uniform acceleration? Unequal change in velocity. Unequal change in velocity in equal intervals of time. What will be the change in velocity there? It's unequal. If the change in velocity is equal, it's uniform. If it's unequal, it's non-uniform. Did everyone learn? Did everyone learn? Did everyone learn? Unequal change in velocity in equal intervals of time. Yes. I am from Alappuzha. Ready, children? Ready, children? Everyone understood, I believe. Non-uniform acceleration. Like we learned instantaneous velocity, if a question like this comes, it's over. So, will it be confusing in the exam? Then the teacher will come here. Why did you ask? Our teacher told us that if it's instantaneous, you can just finish it like this. That's all. So, what is instantaneous acceleration, children? Acceleration at any instant of time. What is it? Acceleration at any instant of time. Excellent. So, let's get the equation. Let's get the equation. We all know the equation for acceleration. It's change in velocity by time. If the time interval is small, we can use lime soda to show it. Limit delta t tends to zero. Delta t is approaching zero. It won't become zero. It's approaching zero. Delta t tends to zero. Okay. There's plenty of time. Tonight, there's time. Man, everyone, we need to consider it. We must have learned all this. That's why we are going like this. Okay. Yes. So, acceleration. Another equation for acceleration. Pick it up. Instantaneous acceleration. Another equation is dv by dt. So, if it's dx by dt, then dv by dt. These are excellent equations that everyone should know and learn. They are not special equations found only in the syllabus. They are found everywhere. Yes. We are moving to the next topic. Equations of motion. Okay. What we have learned so far, from the sections asked so far, they might ask a two-mark question. But from here onwards, we can expect up to a three-mark question. The weightage of this chapter is from a minimum of six marks to eight marks. Questions from this chapter will be asked for at least six marks. Okay. Yes. So, let's look at the equations of motion. What are the equations of motion? Say, children. What are the equations of motion? They will never ask "What are the equations of motion?" in the exam. Equations of motion simply mean representing the motion of a body mathematically using velocity, position, time, and acceleration. That's what we call equations of motion. There are three types of equations of motion. There is a position-time relation. There is a velocity-position relation. And there is a velocity-time relation. There are three types of relations. The derivation is actually asked from the graphical method. The analytical method has not been seen to be asked. Simply, if asked to write the velocity-time relation, you can find it from the graph. Okay, right? So, the graphical method is important. So, we should know each equation. Problems will be asked in the exam based on this. We can expect up to three marks from here. We are working out the problems, don't worry. Look, children. The first equation of motion is the relation between velocity and time. Velocity-time relation. It's a simple relation. That's all, children. The final velocity is initial velocity plus at. We can learn its derivation from the graphical method. Okay, right? We derive it in three ways. There is the analytical method, the graphical method, and the calculus method. Calculus method means using integration, etc. That will not be asked. Okay. Lamborghini or something. Ginny or whatever it is, what does it matter to us? It's a car. That's all. So, everyone pay attention. Ready. Then, the next relation is velocity and position. That relation is a simple relation. Everyone learn it. Yes. Is equal to. Oh, what happened to me? Is equal to ut plus ut plus half at squared. This is position-time. Oh, sir, I don't see the position here. We define this as displacement. We know displacement is x2 minus x1, change in position. Some people use v0 for initial velocity. So, v0 will come. Plus half at squared. Some schools teach it like this. Some schools teach it like this. You can learn either. You can learn whichever you like. Whichever you like. Did they teach you this in school? This one or that one? Which one? Comment, children. Which equation did they teach you in school? The first equation or the second equation? Everyone comment. Which equation? S is equal to ut plus half t squared, or x equals minus x, that is displacement. Yes, yes, yes. The second equation was taught in school. In some places, it's the first equation. That's why I've given both. Okay, right? Ready, ready, ready. Come on. Next, velocity-position relation. Oh, it got changed. Oh, it got changed. Oh, it got changed. This should have come here. It got changed. It got changed. Man, it got changed. Man, it got changed. The velocity-position relation. We know v squared is equal to u squared plus 2as. This is the relation between velocity and position. In school, they sometimes teach it like this: v squared equals u squared. Instead of u, v0. U0, not u0. v0 squared plus 2a into x2 minus x1. Position. Now, if the initial position is zero, we can call this x. We can call this 2ax. If the initial position is zero, we can call this x. You can call this x. Everyone learn these three equations. If everyone understood, shall we do one or two or three questions based on this? Is everyone ready? Is everyone prepared? Is everyone prepared, children? Everyone? Okay, right? So, what is the relation between velocity and position? We know v squared is equal to u squared plus 2as. This is the relation between velocity and position. In school, they sometimes teach it like this: v squared equals u squared. Instead of u, v0. U0, not u0. v0 squared plus 2a into x2 minus x1. Position. Now, if the initial position is zero, we can call this x. We can call this 2ax. If the initial position is zero, we can call this x. You can call this x. Everyone learn these three equations. If everyone understood, shall we do one or two or three questions based on this? Is everyone ready? Is everyone prepared? Is everyone prepared, children? Everyone?

Are you ready? From here, they will ask in the exam. Stopping distance. Yes, yes, yes. From here, they will ask in the exam. What will they ask us? They will ask for stopping distance. So, don't we have to write it? Don't we have to write it? Don't we have to write it? Yes, we have to write it, we have to write it. Guys, in the 2025 model, they asked us, "What is the relation between stopping distance and the initial velocity of the vehicle?" The equation for stopping distance has been asked. So, we will draw a picture like this. There is no compulsion to draw a picture. I am drawing this picture only because I am a bit of an artist. There is no compulsion for you to draw. A beautiful car. We won't let go of our jeep. The easiest car to draw is a jeep. Our jeep is a police jeep. Brother, like this. Yes. So, the initial velocity of this jeep is U. The initial velocity of the jeep is U. Just like before, here in the front, a child is standing. Oh dear, poor child. What will Uncle Police do? What will Uncle Police do? Uncle Police will apply the brakes, right? Uncle Police applied the brakes. The child was not hit, but was stopped by applying brakes. Oh my, look. This distance is called stopping distance. Okay, right? So, what will be the final velocity here, children? Tell me, what will be the final velocity of this body? You can tell yourselves. The final velocity of that body is zero. It's zero. It's zero. Okay, right? Then let's take the equation of motion. Here we know the initial velocity, we know the final velocity, and we need to find the displacement. The retardation of this body is A, or simply the acceleration is A. It will be in the opposite direction. Then, then, everyone tell me, what is the equation? We know that V squared equals U squared plus. Do we know time? We don't know time. So, there is only one equation without time. V squared equals U squared plus 2AS. There is no time in this. All other equations have time. V equals UT plus half AT squared, and V equals U plus AT. Both have time. There is only one equation without time. What is it? V squared equals U squared plus 2AS. From the equation of motion, we know that V squared equals U squared plus 2AS. Here, the initial velocity is U squared. The final velocity is zero. So, we can put zero for the final velocity. Final velocity is zero. Zero squared equals U squared plus 2AS. So, 2AS equals minus U squared. Then S equals. Then S equals minus U squared divided by 2A. Are you ready, children? This is the equation for stopping distance. If we want, we can say one more thing. We know that since the acceleration and velocity are opposite, since the acceleration and velocity are opposite, if we take the acceleration as negative, and if we take the displacement as positive, we can also write the equation like this. Learn that too. Or, there will be a minus in front of U squared divided by 2A. So, minus and minus will cancel out, and it will become 2A. Writing it in both ways is the correct answer. When doing a problem here, you should pay attention to one thing. If you are using this equation, you must put a negative sign for acceleration in the problem. If you are using this equation, we already got this when we put the negative sign, right? If you are using this equation, you should not put a negative sign for acceleration. We are doing the problem. This is our problem. Turn it off. I have turned it off. If our throat goes away tomorrow or the day after, guys, nothing will happen. Guys, it's okay if it gets a little warmer. I will put it on slowly. So, look, 2023 model. 2023 model. If that equation was asked in the 2025 model, and if the 2023 model asked us a problem from there, it's a simple problem. Don't be scared by looking at it. Everything is given. You just need to put it in. Okay, children. A car is moving straight. A car is moving straight on the highway. The car's velocity is 126 kilometers per hour. So, the velocity of the car is 126. 126 kilometers per hour. To convert kilometers per hour to meters per second, what should we do? Multiply by 5 by 18. If you have a calculator, take it. Divide 126 by 18. What do you get? I think it's six or something. Six. The answer is 35. The answer for everything is final. Yes, 35. I know it will be online. 35 meters per second. Yes. The initial velocity is given. The stopping distance of that car, oh boy, the stopping distance of that car is given. The stopping distance of the car is 200 meters. 200 meters. What is the retardation of the car? Right? We need to find the retardation of that car. And how much time did it take for the car to stop? We need to find two things. What are the things we need to find, children? We need to find the retardation of that car. And how much time will it take for it to stop? We need to find these two. What will we do? If you know this equation, you can do it. You can do it very simply with this. Otherwise, we can find it using the equation of motion. You can do whatever you like. You can memorize the equation if you want. I will never encourage that. Let's do it with the equation of motion. We know the initial velocity of the car. We also know that the final velocity of the car is zero. We know the displacement. We know the initial velocity. We know the final velocity. We know the displacement. We don't know the time. Which equation can we take to find the acceleration? There is only one equation without time. Just like before, we can take it. V squared is equal to U squared plus, what is it, children? 2AS. Isn't everything given? The final velocity is zero. The initial velocity is 35. It will be 35 squared. Plus 2 into. We need to find the acceleration. We know that the displacement is 200 meters. If you have a calculator, tell me the answer quickly. We can say. What can we say? Yes, 2A into 200. A into 200. Minus 35 squared. Are you ready? Then acceleration is minus 35 squared divided by 2 into 200, which is 400. Comment the answer for this, children. What will be the answer? Yes, it's correct. 6.125. 6.125 meters per second squared is the right answer. It's correct. So, it came negative. Acceleration. Acceleration came negative. Okay, right? Is it 3? Is it 3? Isn't it 6? Isn't it 6 point? Everyone comment. Everyone comment. Quickly, quickly, comment. Quickly, quickly, quickly. Yes, acceleration is found. Now, to find the time, what should we do? To find the time, we know that V equals U plus AT. V equals U plus AT. The final velocity is zero. The initial velocity is 35. The acceleration has just been found. What is it? Minus 6.125. Minus 6.125. 3.06. It's 3.06, everyone is saying. So, it must be that. It must be that. It's 3.06, everyone is saying. So, it must be that. 3.6. I am not doing it. I am trusting you. 3.06 meters per second squared is ready, ready, ready. So, here we have 3.06. You have to do this too. You have to do this too. Minus 3.06. Time, let's call it T. So, look, if we bring this brother here, won't it become plus? So, 3.06 equals T into T. T equals 35 divided by 3.06. Children, what is the answer? What is the time? Everyone tell me. Everyone tell me. What is the time? It seems to be 11.2. Three. Approximately 11 point two. 11. 11.2 seconds. How is this question? If it's a question from a good family, comment "good family". Comment "good family" if it's a question from a good family, children. Isn't it a great question, children? 11.4? Is it 11.43? Yes, yes. It's not 11.2, it's 11.43. 43. You are having trouble with the calculator, aren't you? Are you doing this with a calculator in your hand? Yes, 11 point. If I divide all this, I will have to sit here. Yes, ready, ready. Come on, come on. Comment, comment, comment. It's a beautiful question. Okay, yes. Next question. Next question. Next question. We will do the next question after the next topic. It will be better. So, let's move on to the next topic. Are you ready? Is everything okay so far? So, if you are asked a problem using the equation of motion, simply write the equation of motion, then put in the given data, or write down the given data first. Then you will understand which equation of motion to use. Isn't it a good family question? Then let's go. We still have problems with the equation of motion. Let's look at acceleration due to gravity. After that, we can do problems from here. Ready? There are only two more topics. Mainly acceleration due to gravity and then graphs. We need to derive the equation from the graph. You said it's simple, but it's the most important topic. It's something that requires a lot of time. Ready? Everyone look, children. We can wrap up this chapter today in about three-quarters of an hour or half an hour. Sir, is there rest? We are not taking rest. So, look, let's see what acceleration due to gravity is. We know that the Earth pulls all objects towards its center with a force. That force is called the gravitational force of attraction. The acceleration that a body gets due to that force is called acceleration due to gravity. Okay, right? Ready, right? Yes. So, remember its value is 9.8. It does not depend on the mass of the body. It does not depend on the mass of the body. Its direction will always be downwards. The direction is always directed towards the center of the Earth. The direction will always be downwards. Directed towards the center of the Earth. Remember all this. Okay. So, if the direction of going downwards is negative, and if we take it as negative, then the acceleration due to gravity will be minus 9.8 meters per second squared. Ready, right? Or, the acceleration we need to take is minus 9.8 meters per second squared. Whether the body is going downwards or upwards, it doesn't matter. If the body is going upwards now, the acceleration will be downwards. It will be 9.8. It's minus 9.8. In schools, they might have taught you to take positive upwards and negative downwards. It's not like that. No matter where the body goes, whether it's going upwards or downwards, the acceleration due to gravity is always acting downwards. If you take the direction of going downwards as negative, the value of g will always be negative. Okay. Sir, do we need to memorize all the equations? The equations we need to memorize are these three equations. And the equation for instantaneous velocity and instantaneous acceleration. Later, when we go to graphs, the slope is velocity. The area of the graph is displacement. What is the slope of the velocity-time graph? Sorry, the slope of the position-time graph is acceleration. The area of the velocity-time graph is displacement. These are the things we need to learn as equations in this chapter. The rest can be derived. You should not get confused by memorizing all the equations. You only need the main three. Only the three equations are enough. We can derive everything from those three equations. Ready? So, shall we do some problems? Let's set it up. Yes, shall we do some problems? Look, children, a question from the 2025 model. A question from the 2025 model. So, we have given some importance to the model question. A player throws a stone upwards. The speed is given as 29.4 meters per second. What is the direction of acceleration during the upward motion of the ball? Everyone tell me, children. Isn't it a good question, children? Isn't it a question from a good family, children? A stone is thrown upwards. Tell me the direction of acceleration, children. The direction of acceleration. It has been taught in schools. What is it? When going upwards, the acceleration is positive, and when coming downwards, it's not like that. Acceleration due to gravity is always acting downwards. Yes, very good, very good. So, let's write the answer, children. In what word? Downwards, right? In the downward direction. Okay, right, children? Okay, right, children? Okay, right, children? Ready. Question B. What are the velocity and acceleration of the ball at the highest point of its motion? At the very top, what is its acceleration? What is its velocity? We know that if a stone is thrown upwards, it's very important. If a stone is thrown upwards, it goes up, up, up, up, up, up, up, up, up. At the very top, its velocity becomes zero. But it will have acceleration. The derivation is almost over. The derivation is almost over. So, at the topmost point, we can say its velocity is zero, and it will have acceleration due to gravity. We know that is minus 9.8 meters per second squared. It's okay to write it as 9.8. It's also okay to write it as g. They are simply asking for the magnitude. Ready, right? I hope everyone understood. If you understood it properly, let's move on to the 2022 model question. Next, the 2022 model question. An object released from rest is accelerated downwards under the influence of gravity. In this case, it's good. It's a fantastic question. A ball is falling downwards. Yes, it's falling downwards. An object is falling downwards. What is it saying? It's freely falling. So, the initial velocity is zero. It's falling downwards. So, when it falls downwards, I assume. I assume. I took this displacement as H. Or, it's okay to take displacement as S. I took it as H. I took the downward direction as negative. Ready, right? We can only do this if we know all this. We can only do this if we know all this. The final velocity is V. The final velocity is V. So, shall we write the equation of motion? Let's take the first equation of motion. V equals. V equals. Tell me, children. We can take acceleration as acceleration due to gravity. Let's take it as minus G. Since it's downwards, we took it as minus. Since it's downwards, we took it as minus. So, what will happen to the first equation of motion? V equals U plus AT. What will happen? Everyone must tell me. Everyone must tell me. We are taking the downward direction as negative. We are taking it as negative. So, the velocity will be negative. The velocity will be negative. So, what will we get here, children? Everyone tell me. Everyone tell me. Yes, minus V. Initial velocity is zero. Initial velocity is zero. Don't look at anything. Zero. Instead of A, put G. Put T. There is no positive or negative for time. So, what do we get? Minus V equals GT. Ah, cancel it out. Cancel it out. Cancel it out. We got the first equation, didn't we? Didn't we get the first equation perfectly? We don't need to memorize these equations. You just need to apply them based on the conditions. V equals GT. Great, great. Then the next one. The next equation. Tell me the next equation. V squared equals U squared plus 2AS. Isn't this the next equation? This is the next equation. V squared. We know V squared. It's negative. It will be minus V squared. Since the initial velocity is zero, we can put zero there. Plus 2 into. Children, put it in. Into acceleration due to gravity, we know it's minus G. It's minus G. Similarly, displacement, we know it's minus H. It's downwards. So, displacement is also negative. We can put it in. Quickly, quickly, put it in. So, what do we get? V squared equals. These minus signs and these minus signs will cancel out, won't they? So, we get 2GH. Great. Wow, how beautiful it looks. How beautiful it looks. Wow, it's amazing. You don't need to memorize these derivations. You just need to understand and do them. Understand and do them. Yes, okay, okay. Now, if there is any negative sign, if you take the downward direction as positive, then if you put positive everywhere, you will get the same answer. You will get the same answer. Don't worry. Next. What else, children? Yes, S equals. Last and final. S equals UT plus half AT squared. Isn't it? Displacement, what can we put, children? We can put minus H for displacement. Minus H equals. Initial velocity is zero. So, it will be zero. Plus half into. We can put minus G. We can put T squared. Isn't it great? So, what is the answer? What is the answer? Everyone tell me. Minus H equals minus half GT squared. Cancel, cancel, kill it. So, what do we get, children? H equals half GT squared. Did everyone understand this? Did everyone understand this? Write these three equations. Don't go memorizing these. Just understand. If the initial velocity is zero and it's falling freely, and the downward direction is negative, then we took the displacement as negative and the acceleration as negative. We took the velocity as negative. We did nothing else. We just put the equation in. I hope everyone understood. Then, the 2020 model question. The 2020 model question. A body with zero velocity and still accelerated. A body has zero velocity, but it has acceleration. We can say. What is it? A body projected upwards. Tell me, children. A body projected upwards. At the topmost point. At the topmost point. The velocity becomes zero. Gravity. Gravity is an example. A body projected upwards. A body projected upwards. To the topmost point. Write "to the topmost point". Or, "top point of the body". Body projected upwards. You take rest, then keep the phone aside. Look, March 2017. Free fall is an ____ accelerated motion. Children, everyone answer. Free fall, we all know it's uniform. It's uniform. Learn to understand and love physics. If you love physics, physics will love you back. You don't expect to get love back for everyone you give love to, but in the case of physics, I assure you. If you sincerely love physics, physics will give you back what you need. When will you understand? You will understand when the exam results come. Physics did not betray us. Physics was with us. So, sincerely love physics for these two years. Physics will give you what you need. Physics will give you what you want. A good job, a good car, a good house, everything will be given to you. Ready, children? Yes, let's move on. Okay. From this topic, I think it's the last question. It's a Christmas question. Write the expression for the velocity with which it reaches the ground. The velocity when a body reaches the ground is asked. We have already done it. I am not wasting time doing it. Here it is. V squared equals 2GH. V squared equals 2GH. Let me copy this. Haha. How beautiful. How comfortable. How beautiful. How comfortable. The same thing. So, I will copy this. I will copy this. Oh, it's cut. It's cut. I can copy it. The same thing as before. I am not wasting time. I will paste it the same way. We need to find the velocity. We need to find the velocity. So, what will the velocity be, children? V equals root 2GH. See? We have learned everything important that is likely to be asked. Now, let's move on to graphs. We have half an hour left. We can set up the graphs in half an hour. Okay, children? Okay. Yes, yes, yes. Okay. Sir, it's not good to bite. It will be a waste of three years. No one asked you to bite. If you bite, it won't be three years, but it will be the same for your whole life. So, don't try to bite anything. Write what you know. Write what you know. Even if you fail, it's okay. Just write what you know. Write the exam with dignity. That's all. Just write what you know. Okay? Graphical representation. Look, graphical representation. Look. We are studying three graphs. Position-time graph, velocity-time graph, acceleration-time graph. Among them, the acceleration-time graph is not very important. There is only one graph and one equation. In this, we are studying a few, three or four graphs. In this, we are studying three graphs. In this, there is only one graph and one equation. In this, there are two equations. In this, there is one equation. Shall we look? Shall we look? Come with me. Okay, ready, ready. Won't V become zero? Look, write the expression for the velocity with which it reaches the ground. It's falling downwards, boy. It's falling downwards. So, how? How will the velocity become zero? It will never become zero. We are talking about the point before it touches. I know it becomes zero when it touches. I know that very well. The energy will be wasted there. We are taking the point just before touching. When a coconut falls from a palm tree, doesn't it have a lot of velocity? It's not falling at rest. After falling, it will be at rest. I know that too. We are talking about the point just before falling. You need to learn the steps of the Carnot engine. Your sir will explain it to you in detail. The Carnot engine. The derivation for the efficiency of the Carnot engine has not been asked so far. Only the equation and problems based on the equation have been asked. Okay. Children, let's study each body. Let's study each body. Let's study each thing. We are going to study and move forward. Yes, yes, yes. Yes, yes, yes. Yes, yes, yes. Okay, okay. It's better to change displacement to position. That's why I am changing displacement and putting position there. If you write displacement, it's not wrong. It's not wrong. But for better understanding, it's better to put position instead of displacement. Okay, right? For a little better understanding, we are putting position here. Okay, right? Ready, ready, ready, ready, ready, ready. Yes, now let's look at each graph. Children, the first graph is when the body is at rest. No matter which point we look at, no matter which point we look at, the displacement of the body, or the position of the body, there is no change. So, we will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. It's very, very important. If a body is at rest, what will be the shape of the position-time graph? Everyone comment. Comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the shape of the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position is still five. No matter what time we take, there is no change in its position. We will say the body is at rest. So, what will be the graph we get, children? Everyone comment. What is the shape of this graph? Everyone comment. Yes, we get a straight line. We get a straight line. What will be the specialty of the straight line? Yes, parallel to the x-axis. Ready, right? We get a straight line parallel to the x-axis. Ready, right, children? Just writing "straight line" is not enough. You have to write "parallel to". You have to write "parallel to the x-axis" or "time axis" to get marks. Now, the next one. The body is in uniform motion. Or, in uniform velocity. We know that uniform motion is equal displacement, or equal change in position, in equal intervals of time. In the first second, the body's position is zero. Sorry, at zero seconds, the body's position is zero. In the first second, the body's position is five. In the second second, it's also five. In the third second, it's five. In the fourth second, it's five. That is, the body's position is not changing. We know that if a body is at rest, the position does not change with respect to time. There is no change in position. No matter what time we take, the position of this body is five meters. So, in the first second, its position is five. In the second second, its position is still five. In the second second, its position

Okay, here is the translation of the provided Malayalam text into English, following all the specified rules.

Draw, this is, this is in the first second, this is in the second second, this is in the third second, this is in the fourth second. Can everyone try to tell the shape of this graph? Can anyone try to tell the shape of this graph? What is the shape of this graph? If the shape of this graph is a little clearer, it should be drawn like this too. It should be drawn like this too. It should be drawn like this too. What is the shape of this graph? What is it, children? Yes, how do we study graphs? We should study graphs like this. Just study as I say. Nothing else. You just need to study graphs in this way. What graph is this? What shape is this, children? If the body is in uniform accelerated motion, we get a parabola. What parabola? Tell me, children, everyone tell me, parabola. Yes, that's correct, everyone, parabola. Parabola, parabola, parabola. Okay, children? Okay, dear? Ready? We get a parabola. Positive parabola. Positive parabola. Ready, ready, ready. We can draw our graph like this. Yes, we can draw it like this. So, if you place a pot like this, it is accelerated motion. Now, if it is retarded motion in the same way, what if it is retarded motion? If it is retarded motion, the graph will come like this. Look like this. The same graph will come like this. It will come like this. If you draw the total, it will look like this. It will come like this. This is retardation. This is the graph of retardation. This is also a parabola. This is also a parabola. If you turn the pot upside down, the NCERT textbook has this figure. So, it is difficult to understand, that's why sir drew it like this. Okay, even if used like this, it is retardation. If you turn the pot upside down, or if you place the pot on top, it is retardation. What is its specialty, look at it. In the first second, it traveled this much distance. In the second second, the distance decreased. In the third second, it traveled only a little more distance. In the fourth second, it traveled very little distance. In each second, its displacement is decreasing. Because it is in retarded motion. I believe everyone understood. Uniform retardation. If it is uniform acceleration, let's see. Position diagram. Slope. The slope of the position-time graph. Look, children. If anyone asks you the slope of any graph, learn it. If anyone asks you the slope of any graph, learn it. The change in the one on the Y-axis divided by the change in the one on the Y-axis. Who is on the Y-axis? Their change divided by the change in the one on the X-axis. This equation, that is, y2 - y1 by x2 - x1. We know the slope equation is y2 - y1 by x2 - x1. That is, the change in y divided by the change in x. Or, to put it more simply, divide what is on y by what is on x. Who is on y? Displacement. Who is on x? Time. So, displacement by time is velocity. So, will you remember, children? Slope equals. You just write this down. Slope equals delta y. Here, who is delta y? Position. So, delta x. And who is on x? Time is on x. So, delta t. Delta y by delta t, we know, is velocity. So, what did we get? Slope equals. Slope equals velocity of the body. Let me tell you one more thing, children. If this angle is theta, and if we take tan theta, and if we take tan theta, what will we get, children? Only velocity. If you take tan theta, look at this triangle. I will color this triangle. Look at this triangle. Consider this triangle. Look at this triangle. If you take tan theta, opposite. Opposite is delta x. Adjacent is delta t. Yes, we got only velocity. So, if you take tan theta, you get velocity. If you take the slope, you get velocity. Actually, slope is tan theta, which is learned in mathematics. The slope of a graph is tan theta. Ready? Then, let's go to the next graph. Let's go to the next question. Let's go to the model question. Let's go to the model question. Everyone pay attention. We are doing the model question. Everyone pay attention. Let the answer also come. What is it? Velocity. We will draw the graph with this. We will draw it beautifully. We will draw the graph. Look, draw the x-axis and y-axis. Here, we know it is position. Here it is our time. Look, one, two, three. So, time is one, two, three. One, two, three. Now, now, next position? Zero, two, four, six. Two, four, six. So, children, look here. What do we get here? Two. What do we get here? Two. What do we get here? Ready? Did everyone understand? Did everyone understand? The graph looks like this. A straight line. Now, what do we need to find? The slope, right? The slope is velocity. So, we find the slope. To find the slope, we need two points. I am taking one point as this. Here, zero, zero. I am taking the next point as this. The point there is three, six. So, find the slope, children. The slope is y2 - I am writing the equation. Y2 - y1 by x2 - x1. Y2, 6 - zero divided by 3 - zero. Six divided by three. What answer will we get? Two meters per second is the velocity. Is it okay? Did everyone understand? Did everyone understand? Did everyone get 2 meters per second, children? Yes, very good, very good. Everyone got 2 meters per second. Then, it is a 2022 model question. Pay attention. Draw the position-time graph of a body having zero acceleration. What is it, children? Zero acceleration? If acceleration is zero, there is no change in velocity. When is acceleration zero? Acceleration is zero when the change in velocity is zero. That means velocity is constant. Isn't it uniform motion? Saying a body has no acceleration means it is in uniform motion. If it is uniform motion, don't look at anything else. Get a straight line inclined with the time axis. Right? Don't look at anything, children. The answer is found. This is position. This is time. This is the graph. Yes, okay. Okay, come quickly. Carbon, you come quickly after buying carbon. You have five minutes. I will give you five minutes. Go quickly, slowly. Okay? I saw. I saw carbon. Go quickly and come. It is a 2019 model question, children. Pay attention. The position-time graph of a particle. The mass is given as four kilograms. What is the force acting on the particle from zero to four seconds? From zero to four seconds. What is the force? Oh, dear children. This is not a position-time graph. If it were a position-time graph, it would be a straight line inclined with the time axis. Remember, there is uniform motion. Velocity is constant. Here, the velocity is zero. In both these cases, the body has no acceleration. Here, it is in uniform motion. It is in uniform motion. If it is in uniform motion, we are sure there is no acceleration. Right? Right? Right? Right? Right? Yes. So, we can say the force on this body will be what? You tell me, children. Because velocity is constant, velocity is constant. We can say velocity is constant. If velocity is constant, we can say acceleration is zero. Therefore, the force is also zero. Ready? Did everyone understand, children? Did everyone understand? Did everyone understand? Yes, yes, yes. Let's go to the next one. Yes. It is a 2019 model question. Draw the motion of a body. Motion of a body. Along the positive axis. Positive acceleration. Nothing. Positive graph. Yes, don't be scared by what is said. This is our x-axis. The x-axis is drawn. Here, what has come? What is it? Positive axis. Actually, it is x, I think it changed when I copied. Look, it is the positive x-axis. With positive acceleration. Don't look at the money. This is the graph. We did everything, didn't we? See, if you learn the basic graphs, you can do any question. We are going to the next one. Velocity-time graph. We are also studying velocity-time graphs. If the body is at rest, children, pay attention. If the body is at rest, we don't need to draw the graph. Why? Because if the body is at rest, the velocity is zero. Oh my. What will the graph be? The graph will be the x-axis. Is it finished? Now? What else? Uniform motion. What is uniform motion? Velocity is constant. Uniform motion means velocity is constant. So, in the velocity-time graph, velocity is constant. So, I am taking this. Let's take velocity as two. Ten meters per second. In the first second, in the second second, in the third, fourth, fifth. Everywhere, velocity is constant. What will we get, children? Tell me the answer. We get a straight line. Yes. A straight line. What will it be? Parallel to. Parallel to what axis? The x-axis or the time axis. Ready? Ready? Yes. Next. Next is uniform acceleration. Uniform acceleration. Ready, ready. How will we get the uniform acceleration graph, children? Everyone tell me. Yes. Uniform axis. Equal change in velocity. Equal interval of time. So, we can say, in the first second, velocity is five. In the second second, velocity is ten. In the third second, velocity is fifteen. Draw the graph yourself. In the first second, five. In two, ten. In three, fifteen. Again, get a straight line. So, how will the graph be, children? What will we get? Get a straight line inclined. With what axis? What axis will it be? The x-axis or the time axis. Ready. You need to know all these things. Uniform acceleration. Nothing else. Nothing else. Now, the slope of this graph and its area. Then a couple of problems. Then we can move to the final section. Ready? Set? Slope of the velocity-time graph. Slope of the velocity-time graph. I already gave you the idea. If anyone asks for the slope of any graph, do delta y by delta x. Children, look. Shall we find the slope? What is on y? Velocity is on y. So, change in velocity by time on x. Change in time. Change in velocity by change in time. We all know it is acceleration. So, what did we get? We got acceleration. Now, as I said before, theta. We know tan theta is the slope. So, if we take tan theta, what will we get, children? Acceleration. This is an equation everyone must learn. The slope of the velocity-time graph is acceleration. Now, we need to learn one more thing in this. What is it? Area under the velocity-time graph. We also need to learn the area of the velocity-time graph. We all know what area is. Area. I will give you a technique for finding the area. Multiply what is on y by what is on x. Y into delta x. Or simply, x. Multiply y by x. Multiply y by x. Okay? Y into x. Or delta x. So, look. Area. Who is on y? What is the area of this graph? The length of this graph, the breadth. Length is velocity. Breadth is time. So, v into t. What did we get? v into t. We all know what velocity into time is. That is displacement. It is displacement. So, what did we get? The area is velocity into time, which we know is displacement. It is displacement. Everyone learn it. Everyone learn it. The area of the velocity-time graph, children, is displacement. It is displacement. Ready? Ready? Yes. Now, let's work out some model questions. Let's work out some model questions. Are you ready? Let's work out some small, small, small questions. Pay attention, everyone. It is a 2023 model. The area under the velocity-time graph gives. We all know it is displacement. It is displacement. Next is a 2021 model question. This graph is given. We are asked to find the displacement from five to fifteen seconds. Find the displacement of the body in the time interval 5 to 15 seconds. So, look. If we take 5 to 15 seconds, what does this shape look like, children? If it is 5 to 15 seconds, it is a rectangle. What is the length of this rectangle? What is the length of this rectangle? It is 20. Right? Right? What is the breadth of this rectangle? What is the breadth of this rectangle? This is the breadth. From five to fifteen is ten. Ready? So, we know area is displacement. Area is displacement. Just take the length into breadth. So, displacement equals 20 into 10. What will we get, children? 200 meters. Isn't it area, sir? Shouldn't we put square meters? Look at the units taken here. The unit for length is meters per second. The unit for breadth is seconds. So, if we multiply these, what will we get? We will get meters. We will get meters. Yes, I have come. Correct, correct. He came after buying carbon candy. Went to the shop. We don't have any candy, dear. Look at the 2020 model question. 2020 model question. The speed-time graph is given. Calculate the distance traveled by the particle at t = 0. Okay. Calculate the distance. We need to find the distance. We need to find the distance. From t = 0 to 10 seconds. Nothing. We just need to find the area. We just need to find the area. We just need to find the area. Area equals. What is this? What is it, children? This is a triangle. Triangle. Triangle. Triangle. Triangle. What is the area of a triangle? Half into base into height. So, half into, children. What is the base, dear? The base is 10. Isn't this the base? Base is 10. Look at the graph. What is the height, children? What is the height, children? The height is 12. 12. So, 10 into 12. What answer will we get, children? 60. 60 meters. Excellent, excellent. We got 60 meters, children. The answer. Now, the last question. Then we can move to our problem. We can move to our derivation. Draw the velocity-time graph and speed-time graph of a body thrown vertically. Throwing a stone upwards. Velocity-time graph and speed-time graph. This is a graph everyone should learn. It is an important graph. It is very likely to be asked. First, we throw a body upwards. It falls down. We need to draw the velocity-time graph for this. First, let's draw the velocity-time graph. So, we know that for a velocity-time graph, we draw two axes beautifully and draw a cross. Then, on one side of the cross, we need to put velocity. On one side, we can put time. Now, pay attention. I am throwing a body upwards. The question says it is thrown vertically into the air. If thrown upwards, think about it. It is thrown upwards. Initially, it has a velocity. So, initially, the body has a velocity. Let's call this initial velocity 'v'. As it goes up, children, what happens? As it goes up, the velocity decreases. So, as time increases, velocity decreases and decreases. Look, as time increases, velocity decreases and decreases. Right? So, at this time, this body has reached the top. So, at this time, let's call this time t1. At this time, the body has reached the top. Velocity decreased and decreased and became zero. Now it is falling down. We know that when velocity goes up, velocity is positive. The velocity of a body going up is positive. That's why it is drawn on the positive axis. Now, what is special when the body falls down? When it falls down, the velocity increases. The velocity increases. Don't draw it like this. The velocity increases in the negative direction. In which direction will it increase, children? In the negative direction, the velocity increases. When it reaches time t2, it has reached the bottom. At time zero, it started. At time t1, it reached the top. At time t2, it reached the bottom. Simple graph. Just draw a line. If you draw a line, the velocity-time graph is obtained. Why does it go up and come down? So, remember this. If only going upwards, draw up to here. Now, if only coming down is asked in the exam, draw up to here. Draw a line. If it is free fall, draw the axis here. This is free fall. This is free fall. This is upward motion. That's all. Now, let's also look at speed-time. Let's also look at speed-time. Let's look at speed-time. Like before, we can draw speed here. Here it is speed, not velocity. It is speed. Here is our time. What is the specialty, children? Like before, the speed is high. Initially, we have speed. Speed. Speed decreases and decreases and decreases. Speed becomes zero. This is our previous t1. Now, it has reached the top. Speed is zero. Now, when it falls down, the speed increases. The speed increases. Speed has no negative or positive. Speed is always positive. Speed is always positive. Speed increased. This graph is easy to learn because we can understand it very simply. Look, speed decreased and decreased to zero. Then speed increased and increased. Yes. We can call this time t. This is the graph. I believe all children understood. This is an important graph. Everyone must learn this graph. Everyone learn it. Now, the graph of uniform accelerated motion. Acceleration-time graph. Nothing. It will be a straight line. It is the acceleration diagram. If a body has uniform acceleration, the acceleration-time graph will be a straight line. Now, let's also look at the area of this graph. Let's also look at the area of this graph. I will give you a technique to find the area. The technique to find the area is to multiply what is on y by what is on x. Who is on y? Acceleration. Who is on x? Time. What is acceleration into time? It is not velocity. It is change in velocity. So, the area under the acceleration-time graph. What will we get? Change in velocity. We know that acceleration is change in velocity by time. So, acceleration into time means change in velocity. Ready? Ready? Because time is zero, delta t was written as zero. Yes. Now, let's move to the last topic of this chapter. Let's move to the last topic. Let me tell you one more thing. It is a 2020 model question. Acceleration-time graph of an object released from the surface of the earth. Just like in 2022, a question was asked in reverse from what is always asked. What is it? We need to draw the acceleration-time graph of an object released from the surface of the earth. So, here is acceleration. Here is time. We know the body is falling down. Acceleration will be negative. Don't look at anything else. It is uniform accelerated motion. Negative acceleration. You can write 9.8 here if you want. 9.8. Okay? Simple, isn't it? Simple straight line. When I drew it, it doesn't look like a straight line. It is a straight line. This is the acceleration-time graph of a freely falling body or a body projected upwards. Now, we come to our last topic. Sir, give me five minutes. Give me five minutes. Give me five minutes. Let me do this. It's simple, children. Nothing. Just two or three steps. It's very simple, but important. You will definitely get a question from here in the exam for three marks. The question will be asked from the last two of these three graphs. Ready, children? A body has initial velocity. Okay. So, in our velocity diagram, we don't usually draw the velocity diagram from here. Usually, the velocity diagram is drawn from here. Because the body has initial velocity, the graph should be drawn from a little higher. The body has initial velocity. After time t, after time t, the body's velocity became v. Okay. So, because the body is in accelerated motion, we get a straight line. Usually, we draw it like this. We draw it like this. Because the body has initial velocity, the graph should be drawn from a little higher. Okay. I am taking two points from this graph, children. Pay attention. I am taking two points. This point is zero, zero. This point is t, v. Remember your home TV. If you put TV on top. Is the velocity-time graph? To find the velocity-time graph, just take the slope. If we take the slope of this graph, we know the slope equation is y2 - y1 by x2 - x1. Now, the slope, we know, is the slope of the velocity graph. It is already known to us. It is acceleration. This is y2. This is y1. So, v - zero divided by t - zero. What did we get? Oh my, oh my, oh my. Not v - zero. I am who? You are who? Won't you say that? Here it is zero, zero. Not zero. It is u. So, here, v - u. It will come. Forgive me. So, what did we get? Multiply t this way. Then what will we get? at = v - u. Or v = u + at. Did all children understand, children? Did all children understand? Isn't it a simple derivation, dear? Isn't it a simple derivation, dear? Shall we move to the next one? Position-time relation. Position-time relation. Position-time relation. Pay attention, children. Same graph. Same graph. But a small difference. I am taking the area here. I have split it into two. A rectangular portion and a triangular portion. Now, pay attention. The breadth of this rectangle. Sorry, breadth. Or the base of our triangle. Let's call it t. Okay, dear? Now, the length of our rectangle. Let's call this length. Let's call the length of the rectangle u. Isn't it u? Yes. Now, look at this triangle. What is the height of this triangle? What is the height of this triangle? We know that from here to here is v. From here to here is u. So, what will this be, children? v - u, right? So, the height of the triangle is v - u. Now, let's do it. So, there are two triangles, right? Two areas, right? The area of the rectangle. This. This is the area of the triangle. So, the total area of the graph. The total area of the graph is the area of the rectangle plus the area of the triangle. Area means displacement. We all know. The area of the first region is length into breadth. Length into breadth. Let's call it length into breadth. The second is a triangle. The area of a triangle is half into base into height. So, displacement. Displacement equals. Instead of length, let's put u. Instead of breadth, let's put t. Half into, children. What is the base of this triangle? Isn't it t itself? So, put t. For h, we can put v - u. Now, we know v - u. What is v - u? Isn't it at? We did it just before, dear. What is v - u? It is at. Remember 'at'. Put this in place of this. Our answer is ready. Our answer is. What did we get? s = ut + half at squared. Write it beautifully with a double line and a cross. Okay. Last derivation. Last and final derivation. Our chapter is about to end, children. Everyone pay attention. S is the velocity-time graph. I am taking the area here. Pay attention. I am taking this area. What is this, children? This is a trapezium. What is a trapezium? It is the name given to a quadrilateral with two parallel sides. It is called a trapezium or lambakam. It must have two parallel sides. Two parallel sides. Let's call the first parallel side 'a'. It is v. Let's call the second parallel side 'b'. It is u. So, a and b are two parallel sides. The distance between these two parallel sides is called the height of the trapezium. So, we can call the height of the trapezium t. Because this distance is t. Here, we are finding the area of the trapezium. We all know the area of a trapezium is half into. If you don't know, learn it. a + b into h. The area of a trapezium is half into the sum of the two parallel sides into the height. Or the distance between the two parallel sides. Put this equation and get the answer. Put this equation and get the answer. We know area means displacement. Half into. Half into. What is 'a', children? It is v. What is 'b'? It is u. What is the height? It is t. Now, we need to find t from the equation of motion. What is it? Isn't it at? So, we can simply say. v - u by a. If v = u + at, then t = v - u by a. Right? Put this here. The answer is ready. The answer is. We get 2as = v + u into v - u. We know a + b into a - b is a squared minus b squared. So, v squared minus u squared. This 2a, multiply it here. So, in the next step, we can write 2as = v + u into v - u. The answer is ready. a + b into a - b is a squared minus b squared. So, v squared minus u squared. In some schools, they will use v naught instead of u. So, try to write it like that, or the teacher will not give marks. It is not a model exam, it is the final exam. Did you do anything? Are you ready, children? Did everyone understand? This is our chapter. It is finished. Look at the repeated questions from 2018, 2019, 2020, 2021, 2022, 2023 models. Wow. It was asked in 2021, 2022, 2023, 2024. It can be asked in 2025, 2026 too. The same question. Exactly the same question. No change. No change. What was done just now. What was done just now. This is the derivation of the equations of motion that we studied together. It is the most important derivation. Yes. This ut + half at squared was also asked. It was asked in the 2024 model. It was asked in the 2024 model. There were two questions in the 2024 model. I think it was asked for four marks. Or five marks. Two derivations were asked. This is a section where you can guarantee at least three marks. It will definitely be asked. You have understood it. You must definitely study and go. So, these are the things we need to deal with today. There are many more questions, but we need to do them. Topper.

We all have to deal with the top-level question. The toppers will be coming in the topper series. Are you ready? Not even until evening, but a 2022 model. After getting that done too, I'll leave, or else I'll feel sad. Velocity, oh dear, what is this? Using the graph, derive an equation for displacement in terms of v. No, that's not it, it's the displacement equation, sorry. This is a question we dealt with in 2020. S = ut + 1/2 at^2, right? Isn't that it? Using the graph, derive an equation for displacement in terms of initial velocity. It's indeed S = ut + 1/2 at^2. So, that's all our work. We're done. Next, we have motion in a plane. Slide. It's in the slide. Nothing needs to be done, just wipe it. What's this, it's all smeared. Let's wipe it with some paper. Let it be fresh, my children. Go eat something heartily and energetically. Don't leave the phone open there and go, but okay? You eat food, and by then, I'll finish my two chapters. Okay. What time will it be over? When I come, will it be over? There's no time until 5:00 PM. Brother, we can take it lavishly. Children, are you well? Are you ready? Shall we move on to motion in a plane? Please get me a glass of hot water and go. Certainly, children. It's motion in a plane. I'll drink it slowly after it cools down. Children, we are about to enter the chapter called motion in a plane. You are energetic. We have finished two chapters. The first two chapters were finished brilliantly. We finished teaching everything within those two chapters in three hours. Now, motion in a plane. The next two chapters also have significant weightage. Chapters with a weightage of eight or nine marks, very important chapters, are what we are going to study and crack. Are you ready? Yes, I believe everyone is ready. So, without delaying, shall we move forward? Let's move forward. Before that, I want to show you something. I don't know if you can see it. I just went home and came back from a visit. I went to see someone and came back. I just checked Instagram. On Instagram, a child has meticulously written the lecture notes for the chapter called Units and Measurements. I told you to write lecture notes during our live classes. The child's name seems to be Theresa. I don't know if you can see it, but she has meticulously written the lecture notes. She has sent me the photos. You all should write lecture notes like this. Write the derivations. Otherwise, it's no use saying I watched many live classes and tried to understand many things. It's only useful if you write parallelly. I believe you all will do it. If you have written the previous chapter meticulously with derivations, everyone in the chat box, do a fire emoji. Then we can move on to the important things in motion in a plane. Are you ready? Sir, don't we need to eat? You can eat. You can eat in parallel, or even if you take a 10-minute break to eat, there's no problem. You can just watch it at a fast pace during that time. Okay. Have you eaten? It's not time yet, is it? It's only 12 o'clock. We have time. We can eat slowly. Okay. Isn't it fine to eat at one or one-thirty? I'm starting to drink hot water intermittently. I'm starting to understand that my voice is gradually going away. It's not usually like this. The sound has started to change slightly. If I don't drink hot water, I'll be in trouble. Are you ready? So guys, we are moving forward. We are entering the chapter called motion in a plane. Stay with me, everyone. Stand by my side. Just like how brilliantly you studied the last two chapters, let's study this chapter too. It's a chapter with many derivations. I need everyone's support, guys. First, there's an introduction. In the introduction, I will teach you what motion in a plane is. In the previous chapter, we dealt with motion in a straight line. Motion in a straight line means one-dimensional motion. We studied motion along one axis, right? There, we only need one coordinate to indicate the position of an object. Let's take it as x or just x. But when it comes to motion in a plane, we need two coordinates to indicate the position of an object. We need x and we need y. If it's three-dimensional motion, we need x, y, and z. Did you understand? Did you understand clearly? Shall I give you an example to get more clarity? Imagine I tied a thread like this. And I placed an ant on that thread. What type of motion is the ant's motion? The ant can either move forward along the thread or come backward. That's all the ant can do. That is, the ant can only move along one axis. What type of motion is that, children? That is one-dimensional motion, motion in a straight line, which we studied in the previous chapter. Now, let's come to this chapter. Take the same ant and place it on a piece of paper, on a piece of cardboard. Where can the ant move? It can move forward, backward, to both sides. Right? That is motion in a plane. Here, we have two axes, x and y. What if we take a butterfly instead of an ant? A butterfly can move forward and backward, up and down, and sideways. What is that? We can consider that as three-dimensional motion. Did you understand the concept clearly? Are there any doubts? What we are going to study here is motion in a plane. We are going to deal with it in this chapter. I believe you are ready. Here, we are studying two-dimensional motion. The first topic we are going to study and discuss is scalar and vector quantities. First, scalar quantities. Physical quantities can be broadly classified into two main types: scalar quantities and vector quantities. What is a scalar quantity? To put it very simply, if a physical quantity has only magnitude, then we can call it a scalar. It only has a measure, only magnitude. If it doesn't have direction, then the name we call it is scalar quantity. Shall I give an example? Time. If you ask what time it is, what will you say? It's exactly 12:13. We only state the measure, right? We don't say 12:13 towards east, or towards west, or towards south. We don't add direction. Right? Right? Yes. Now, if we go to a grocery store, we say, "Brother, I need one kilogram of sugar." Right? We only state the mass. We only say one kilogram of sugar. Do we add direction with it? One kilogram of sugar towards the south. He will make me take it towards the south. Right? No. We never add direction with mass. Isn't that common knowledge? So, physical quantities that have only magnitude are called scalars. Examples are mass, time, speed. What are these, children? Recognize that these are all scalar quantities. Next are vector quantities. What do they have? Vector quantities have both magnitude and direction. They need one more thing. What is it? They must obey vector laws. They must follow vector laws. Which ones are they, children? Can those who know comment? Which are the vector laws? Can someone comment? We are studying them in this chapter itself. Which ones are they, children? Yes, the triangle law of vector addition and the parallelogram law of vector addition. Don't you all remember? So, when a physical quantity also obeys those vector laws, it becomes a vector quantity. So, it must satisfy three things: it must have magnitude, it must have direction, and it must obey vector laws. Examples are displacement, velocity, acceleration, force, momentum. What are these? Understand that these are all vector quantities. Is it crystal clear? Do you have clarity on what is scalar and what is vector? If so, put a heart emoji in the chat box. We will work out questions related to this. An important question is waiting for you. It's a classification question. If you are ready, please respond. Then I'll drink some water. There is only one important question. That question is: Classify the given physical quantities as scalars and vectors. Generally, questions from this are likely to come in the exam. The question is to classify the given physical quantities into scalar quantities and vector quantities. Can we do it easily? It's something we can do very simply, very easily, right, children? Tell me. Without any doubt. Momentum is a vector quantity. Work is a scalar quantity. Force is a vector. Acceleration is a vector. What quantity is power? Power is scalar. Charge is scalar. Weight? Weight is actually the gravitational force that the Earth exerts on us, isn't it? Force. Weight is actually a force. Therefore, weight is a vector. What is energy? What is energy? Energy is scalar. Did you understand the concept clearly? Is it great? Are you happy? Okay. You should all learn to classify meticulously like this. Such classifications are asked in exams, you know. Whatever physical quantity is given, you must clearly understand whether it is scalar or vector. Fine. Shall we move forward? Next is the representation of a vector. When we say vector, I always tell you, who is the cartoon character we should keep in mind? Who is the cartoon character that should come to our mind when we hear vector? Comment everyone. Tell me. Which cartoon character should we bring to our mind when we hear vector? Without any doubt. When we say vector, think of our Luttappi. Don't you remember Luttappi's spear? Yes. You all must remember the spear, right? Yes. Just think of that spear. Imagine that spear is our vector. Doesn't this spear have a length? What is the name we call that length? Quickly comment. This spear is the vector. What does its length represent? Quickly tell me. This spear is called the vector. What does its length represent? Without any doubt, its length represents magnitude. What? Magnitude. It represents the measure, the magnitude. What does its arrowhead represent? Yes, you must understand that too. Its arrowhead represents direction. Is it clear? The two main properties of a vector are magnitude and direction. We can represent a vector by drawing an arrow. So, the length of the arrow or the spear represents magnitude. Its arrowhead represents direction. Understand that. That's important. Very important. Now, shall we move on to the types of vectors? The first one was asked in the March 2021 exam. What are equal vectors? What are equal vectors? The two main properties of vectors are magnitude and direction. So, if both are equal, then the name we call such vectors is equal vectors. If the magnitude and direction of two vectors are equal, we can call them equal vectors. In the figure, can you see vectors A and B? Are both of them in the same direction? Look. Both are in the same direction. Look at the length of both. Both have the same length too. Right? Therefore, we can say they are equal vectors. They are equal vectors. Understand that. Clear? Next. Negative vectors. What is a negative vector? It will have the same magnitude, but the direction will be opposite. If two vectors have the same magnitude but opposite directions, we can call them negative vectors. What can we call them? They are called negative vectors. Clear? Did you understand clearly? The name we call them is negative vectors. If this is called A, the name of the vector with the same magnitude in the opposite direction is called vector -A. Because it is in the opposite direction, it is called a negative vector. Next, null vector. It's a very important question. You must also study its properties. What is a null vector? We study null sets in math. A null set is empty. Null means empty, meaning nothing. We can take it like that. So, a null vector is also like that. It's a vector with nothing. Its magnitude is zero. It has no magnitude at all. Magnitude is zero. It has no specific direction. So, it has no magnitude, and it has no specific direction. Such vectors are called null vectors. A vector with zero magnitude and no specific direction is a null vector. What are its examples? The displacement of a stationary object. Does a stationary object have displacement? No. The displacement vector of a stationary object is a null vector. The velocity vector of a stationary object? That is also a null vector. If an object is stationary, it has no velocity. So, the velocity vector is an example of a null vector. Understand that. Let me ask you another question. The acceleration of a stationary object. If I give it as the third point, the third example, is the acceleration of a stationary object correct or incorrect? Will you agree with me or not? Can you comment yes or no? I am giving acceleration of a stationary object as the third example. Will you agree with me or not? Will you agree with me? We can never do that, right? Because why? Now, I threw this object upwards. At this peak point, it is at rest. It is not moving. Right? After going up, it will stop here at rest. But even at this point where it is at rest, it has acceleration. What is the name of its acceleration? It is called acceleration due to gravity, right? Even at this point, the object has acceleration. Understand that it has acceleration. So, we cannot put that point here. Study the two examples. Next, its properties. Very important. Before stating the properties, brothers and sisters who are watching without liking the live, please definitely like the live. Your love is what you should show on that like button. Everyone, please like it. Okay. Now, if someone comes after some time, remind them to like. Someone comment to remind everyone to like. Ready. So, let's move on to the properties. First property: If a null vector is added to any vector, the vector remains unchanged. That is, you said you know. Let me see. Vector A + Null Vector. What is the answer? Quickly tell me. Vector A + Null Vector. What will be my answer? Comment everyone. What will be obtained? If you add a null vector to any vector, you get that vector itself. That is, it is equal to vector A. Another example: Vector B + Null Vector. What is that? It is vector B. Understand that. If you add a null vector to any vector, you get that specific vector as the answer. Okay, right? Next property. Multiplication of a vector by a scalar. Pay attention. Multiplication of a vector by any scalar quantity. This is important. That is, let lambda be a scalar quantity. I am multiplying it with a null vector. What answer will we get if we multiply a scalar by a null vector? The answer will be a null vector. If you multiply a scalar by a null vector, the answer you get will be a null vector. Understand that. Next, the dot product of any vector with a null vector will be zero. This is very important. If we take the dot product, the answer is never a vector. The answer is a scalar. Therefore, the answer will be zero. Understand that. Similarly, Vector B dot Null Vector. What will that be? That will also be zero. Vector B dot Null Vector. That will also be zero. So, if you take the dot product, the answer you get is zero. Understand that it is zero. Next, the cross product of any vector with a null vector. It will be a null vector. That is, Vector A cross Null Vector. What will it be? If you take the cross product, the answer is a vector. So, you get a null vector. So, Vector B cross Null Vector. What is that? That is equal to a null vector. Did you understand clearly? Did you understand brilliantly? So, are all the properties clear? If a vector has zero magnitude and no direction, what can we call it? We can call it a null vector. I have explained all its properties clearly. If it's okay, if it's clear, put a thumbs up in the chat box. Everyone, put a thumbs up. Everyone, put a thumbs up, children. Everyone, all children, put a thumbs up. Then we can move on to the next topic. This is finished. We are going to move on to the next topic. Next is the topic of orthogonal unit vectors. What is an orthogonal unit vector? Actually, there was a thing called unit vector here. When I imported it, something went wrong. Let me import the unit vector. Sorry, my mistake. When I sent the file, the unit vectors are not visible. Let me fix it. Just a minute. Because it is very important. Without it, it won't be right. So, I'll fix the unit vector thing. Just a minute. Just one minute is enough. Time is up. I got the thing. Yes, that's the unit vector. The next important topic is the unit vector. Actually, before studying the unit vector, I want to teach you something else. Shall we move on to that? That is, how to find the magnitude of a vector. Shall we see? So, I am writing vector A. Vector A = 3i cap + 3j cap + 2k cap. Can you comment what its magnitude is? If you can, can you comment? What will be its magnitude? If you can, please comment. What will it be? Without any doubt. It's easy. Look. Put a root. Inside the root, square the coefficients of i, j, and k. Just square the coefficients of i, j, and k. So, you get 2^2 + 3^2 + 4^2. That is equal to root of 4 + 9 + 16. So, what is 4 + 9 + 16? It's 13 + 16, which is 29. So, root of 29. Right? Shall I give another example? Similarly, everyone pay attention. Another vector. Let's say vector B is now vector B = 2i cap + 4j cap + 3k cap. Can you find its magnitude? Everyone, try this. Everyone should know this. What is the magnitude of this given vector? Everyone, try to see. It's easy, children. First, put a root. Inside the root, square the coefficients of i, j, and k. So, 2^2 + 4^2 + 3^2. What is the answer, children? Root of 4 + 16 + 9. What is the answer? 16 + 9 is 25. 25 + 4 is 29. You get root of 29. Didn't you get it? Did everyone understand? This is how you find the magnitude of a vector. This is how you find the magnitude of a vector. Okay, right? Clear? Did you understand clearly? Now, what is a unit vector? If the magnitude of a vector is one, then we can call such vectors unit vectors. If the magnitude of a vector is found to be one, what can we call such vectors? Unit vectors. I'll give an example. Now, pay attention to vector A. Vector A = (1/√3)i cap + (1/√3)j cap + (1/√3)k cap. Can you find the magnitude of this vector? Everyone, try this. Try this. If you just watch, it won't happen. Write it down in your book. Try writing it. What will be the magnitude of this vector? Let's find the magnitude of vector A. Root of (1/√3)^2 + (1/√3)^2 + (1/√3)^2. What is the answer you get? Root of 1/3 + 1/3 + 1/3. That is equal to root of 3/3. The answer will be equal to one. The answer is one. This is an example of a unit vector. It is an example of a unit vector. Understand that this given thing is an example of a unit vector. This is very, very, very important. It is asked in exams. You need to study it. This is an example of a unit vector. Another thing. If a vector is given, sometimes the question of how to find its unit vector is asked. Let me tell you that too. Look. There's a small equation for that. So, the unit vector of A is called A cap. There's an easy way to find it. Look. A cap = Vector A divided by Magnitude of Vector A. This is how you find the unit vector of A. Another example. If you want to find the unit vector of B, it's simple. Vector B divided by Magnitude of Vector B. That's the process. Whatever vector's unit vector you want to find, just divide that vector by its magnitude. That's all. Isn't it easy? Did you understand it confidently? So, let me do an example in the same pattern. Okay. We have a vector. We have its magnitude. So, it's easy to find its unit vector, isn't it? What is the equation? A cap = Vector A divided by Magnitude of Vector A. So, how do we write it, children? Everyone pay attention. Vector A is 2i + j + 2k. So, 2i cap + 3j cap + 2k cap divided by the magnitude. The magnitude is root 17. So, we can write root 17. Or, if you want, we can split it and write it. That is, 2/√17 i cap + 3/√17 j cap + 2/√17 k cap. Did you understand clearly? Isn't it easy? This is how we find the unit vector. This is how we find the unit vector. Okay, right? Clear? Did you understand clearly? Now, can you find the unit vector of vector B? Everyone, try to find the unit vector of this vector B. Everyone, try to see if you can do it. Try writing it down. You have a book, right? Try writing it. We have vector B, and we have its magnitude. So, it's easy to find the unit vector, isn't it? What is the equation? B cap = Vector B divided by Magnitude of Vector B. I am writing it. Comment whether you got it. Because you might not be able to comment the answer. So, see if you got the answer I wrote. Vector B is 2i + 5j + 3k. So, 2i cap + 5j cap + 3k cap divided by the magnitude. The magnitude is root 29. So, vector B, or the unit vector of B, is equal to (2/√29)i cap + (4/√29)j cap + (3/√29)k cap. Everyone who got this answer, put a "set" message in the chat box. If you say, "I got this answer," put a "set" message in the chat box. All brothers and sisters, come on guys, respond. If you got this answer, comment "set" in the chat box. Did everyone get it? Great. I believe everyone got it. Such questions are likely to be asked in exams. You need to study them. It's very important. Got it, sir. Okay, my dear. Ready. Everyone must have got it. Guys, we are moving forward. Next, we are going to enter orthogonal unit vectors. What are orthogonal unit vectors? What is an orthogonal unit vector? Actually, i cap, j cap, and k cap are our orthogonal unit vectors. Since they are unit vectors, what will be the magnitude of i cap, j cap, and k cap? Comment. What will be their magnitude? They are unit vectors, right? Without any doubt, the magnitude will be one. The magnitude of these will be one. Understand that. Because they are unit vectors, the magnitude will be one. Next thing: What is their purpose? Unit vectors used to indicate direction are called orthogonal unit vectors. Unit vectors used to indicate direction are called orthogonal unit vectors. i cap is used to indicate the x-direction. j cap is used to indicate the y-direction. k cap is used to indicate the z-direction. The unit vectors used for this are called orthogonal unit vectors. Did you understand clearly? Did everyone understand clearly? Is it set? Are you happy? I believe you are happy. Right. Okay. Moving forward. Next, let's move on to the addition and subtraction of vectors. How can we add and subtract vectors? Let's move on to that. Addition of vectors. Now, everyone, you tell me the answer. We have two vectors. This is one vector. This is another vector. Both these vectors are in the same direction. Here, we can...

Let's call this vector A. Let's call this vector B. These two vectors are in the same direction. Pay attention to the magnitude I am writing. Let's assume the magnitude of this vector is five units. Five units. Let's assume the magnitude of this vector is six units. Then, when these two are added, in which direction will the resultant vector be? Everyone comment. In which direction will the resultant vector be when these two are added? Everyone comment. In which direction will the resultant vector's direction be? There is no doubt. Since these two vectors are in the same direction, the resultant vector will also be in the same direction. There is no argument. What will be the magnitude of that resultant vector? Subtract the magnitude of the smaller one from the magnitude of the larger one. Because both are in the same direction, add the magnitudes of both. Don't subtract. I have a tendency to say that, but that's not it. Add both. Add the big and the small. Add the magnitudes of the two vectors. Because why? Both are in the same direction. If both are in the same direction, add the two magnitudes. Five A plus R, what is that? Isn't it 11? So, what is the answer? It will be 11 units. What is it? 11 units. Okay, right? Understood clearly? If both are in the same direction, the resultant will also be in the same direction. The magnitude will be equal to the sum of the magnitudes of the two. Fine. Next example, okay? When two vectors act in opposite directions, what if two vectors are in opposite directions? So, let's assume this is one vector. Let's assume this is another vector. How do these two act? Both are in opposite directions. This is vector A. This is vector B. Let's assume these two are in opposite directions. Let's take its magnitude as six units. Let's take its magnitude as two units. Then, what is the magnitude of our resultant vector? What will be its direction? Can you tell me? Yes. Let's assume this is our resultant vector. In which direction will its direction be? First, tell me that. In which direction will its direction be? Which is the larger vector among these? Vector A is the larger vector. The resultant will be in the direction of that larger vector. In the direction of the larger vector among these, who will be there? The resultant vector will be there. So, in which direction will the resultant vector's direction be? It will be this way. No doubt. It will be in the direction of the larger vector. Now, what will be the magnitude of the resultant vector? Subtract the magnitude of the smaller one from the magnitude of the larger one. If both are in opposite directions, subtract the magnitude of the smaller one from the magnitude of the larger one. So, six minus two, what is that? Isn't it four? So, what is the answer? It will be four units. The answer will be four units. Understood clearly, right? No doubt, right? Everyone understood, right? Happy, right? Confident, right? If all the younger brothers and sisters are confident, we can move on to the next one. The next one is the subtraction of vectors. We cannot subtract vectors directly. Instead, we do negative addition. I will explain. That is, if we have vector A minus vector B, this operation is not possible. We cannot subtract vectors directly. This is not possible. We use vector A plus minus vector B. This is what we do. That is, instead of directly subtracting two vectors, we do vector A plus minus vector B. We can do vector A plus minus vector B. Is it clear? Understood clearly, right? No doubt, right? No one has any doubt, right? Happy, right? Everyone happy, right? So, if that's the case, shall we try to work this out? Let's try to work this out. This is our vector A, let's assume. This is our vector B, let's assume. Okay. This is vector A and vector B. No doubt, right? Everyone look. I called this vector A. This is our vector B. Its magnitude is suppose two units. Its magnitude is suppose five units. We cannot do vector A minus vector B. Instead, we do vector A plus minus vector B. We do vector A plus minus vector B. So, how do we do it? So, there is no change in vector A. I copied vector A as it is and pasted it here. There is no change in vector A. Plus, children, what is special about minus vector B? You all know vector B. See, what is minus vector B? What change needs to be made to this vector B? What change needs to be made to this vector B to get minus vector B? Can you comment on that? Everyone comment. Minus vector B means what change needs to be made to this vector B? Yes. The magnitude doesn't need to be changed, but the direction will need to be changed, right? Absolutely, we need to change the direction. Put the direction in the opposite. There is no other difference. Put the direction in the opposite. The magnitude doesn't need to be changed. This is what is called minus vector B. The magnitude is still five units. When adding these, you all know the characteristics of the result, right? Haven't we already learned that? Yes. Let's assume this is our resultant vector. In which direction will the result be? In which direction will the result be? No doubt. The direction of the resultant will be in the direction of the larger vector among these. The larger vector is vector B, right? In that direction, our resultant will be. So, in which direction will the resultant vector's direction be? It will be this way. No doubt. It will be in the direction of the larger vector. Now, what will be the magnitude of the resultant vector? Subtract the magnitude of the smaller one from the magnitude of the larger one. If both are in opposite directions, subtract the magnitude of the smaller one from the magnitude of the larger one. So, five minus two, what is the answer? Isn't the answer three? So, the answer will be three units. The answer will be three units. Understood crystal clear, right? Not a bit of doubt, all younger brothers and sisters understood, right? The answer will be three units. Happy. Okay. Shall we move forward? Yes, guys, we are moving forward. This is addition and subtraction. The next thing is, suppose our vectors are inclined at a particular angle, what do we do? If they are tilted vectors, how do we add them? We have learned the method of adding upright vectors. Now, if our vectors are tilted vectors, shall we see how we add them? There are two methods for that. The first is the triangle law of vector addition, and the second is the parallelogram law of vector addition. We will use these two methods. First, the triangle law of vector addition, and second, the parallelogram law of vector addition. What is the triangle law? What is the parallelogram law? Shall we see? Come on, children, pay attention. First, the triangle law of vector addition. This is a question asked in the 2023 Christmas exam. I will explain it easily. You just need to stay with me. So, this is a vector. This is a vector. Pay attention. This is a vector. Look at the next vector. This is another vector. This is another vector. Okay, right, fine. The name I give to this first vector, pay attention. This is our vector A. Look at the name I give to the second vector. Vector B. What is special about these two vectors? These two vectors are in the same order. What does this same order mean? Isn't it one after another? It comes after vector B. It comes one after another. So, you can say that they are in the same order. Can we consider these two vectors as two adjacent sides of a triangle? Right? We can consider them as two adjacent sides of a triangle. If so, what will be the third side of that triangle? Yes, you must understand. The third side of that triangle will be what we call our resultant. There is another special characteristic. How should that third side be? It should be in the opposite order. Yes, it should be in the opposite order. That third side should be in the opposite order. If A and B are in this direction, the resultant is in the opposite direction. So, let me say it one more time. According to the triangle law, if two vectors are considered as two adjacent sides of a triangle, then the sum of those two vectors, or the resultant vector, will be the third side of that triangle. It will be the third side. There is another special characteristic. The order of the resultant will be in the opposite order to the order of A and B. Understand clearly. If two vectors are considered as two adjacent sides of a triangle, then the resultant will be the third side of that triangle. That resultant will also be in the opposite order. Understood clearly, right? If this is okay, put a fire emoji in the chat box. So, we can simply write vector R equals vector A plus vector B. If it's okay, everyone comment with a fire emoji. So, this is its theory. There is nothing beyond what I have said. Let's move on to the next one. Let's move on to the parallelogram law of addition. If you are ready, respond. Yes, Clingen. From the idea of Carbon Carbon, a message comes from Clingen. Everyone, a fire emoji, then we can move on to the next one. Parallelogram law of addition. It's easy, guys. The topics from now on are very simple. You can learn them easily. So, let's move on to the parallelogram law of addition. Come on, children. Everyone stay together. Come on. What does the parallelogram law say? There is no change, just like this. No change. Two vectors. Here, I took two vectors. This is the first vector. This is the next vector. Here too, I am considering two vectors. This is our second vector. Okay. This is our second vector. You can see it, right? No problem, right? No. Can you see these two vectors beautifully? Yes. Shall we give them names? I am going to give them names. This is called our vector A. This is called our vector B. What is special about these two vectors? These two vectors are vectors originating from a common origin. Look, aren't these two vectors originating from a common origin? Absolutely, both are vectors originating from a common origin. Right. Okay. Can we consider these two vectors as two adjacent sides of a parallelogram? Right? We can consider them as two adjacent sides of a parallelogram. Then, I am going to complete that parallelogram. Stay with me. I am going to complete the parallelogram. I am completing that parallelogram. After completing the parallelogram, what will I get? Yes, we can definitely draw a resultant. After completing the parallelogram, we can draw a resultant. I will indicate this precisely with a dotted line, or you will get confused. So, I am indicating the completion of that parallelogram using a dotted line. Okay, right, fine. After completing the parallelogram, I am drawing a resultant. What does that resultant indicate, children? Quickly tell me. Yes, the diagonal. The diagonal of the parallelogram indicates the resultant. That is, the diagonal we draw for that parallelogram indicates our resultant vector. Let me say it one more time. If two vectors are considered as two adjacent sides of a parallelogram, then the diagonal we draw for it after completing that parallelogram, that diagonal will be our resultant. Here, R is our resultant. Understand clearly. If vectors A and B are considered as two adjacent sides of a parallelogram, then the diagonal we draw for it after completing that parallelogram, this is the diagonal, right? Absolutely, this is the diagonal. Understand that the diagonal indicates our resultant vector. So, vector R equals vector A plus vector B. If this is confident, comment with a "Set Annayi" message in the chat box. I am okay with this. I have no doubt. I am very much confident about this. Everyone comment with a "Set" message. All younger brothers and sisters, comment. Then we can move on to the next topic. That is, resolution of vectors into rectangular components. That is, how do we resolve a vector? That's what we are going to see. How can we resolve a vector? You should answer before me, or even before me. You should try. Right. Here, I am considering an XY plane. This is an XY plane. Pay attention. Let's consider this as an XY plane. Y X. Is my X a bit crooked in this XY plane? Did X get crooked? No, it's not a big deal. It's okay. In the XY plane, I am taking a vector. This is my vector. Let's call this vector A. This is my vector A. This is my vector A, let's assume. This is vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my vector A, let's assume. This is my vector A. Let's call this vector A. This is my

The direction of the resultant means we have to find alpha. Understand that what we need to find here is alpha. Fine, okay. So, for that, we are going to consider a triangle. Yes guys, we are going to consider this triangle. We are considering this triangle. This is the triangle we are considering. Ready? So, what are we going to take here? We are going to take tan alpha. Tan alpha, children, tan theta means opposite side by adjacent side. Here, tan alpha means what is the opposite side? It is SA. What is the adjacent side? It is OA. So, tan alpha is equal to, pay attention, instead of SA by OA, we can write OA as OP + PA. So, if that is the case, tan alpha is equal to SA by P, can we give A instead of P? We can give A + PA. Tan alpha is SA by A + PA. We are going to replace everything. Instead of SA, we found B sin theta. We found SA as B sin theta. We are going to substitute PA as B cos theta. Pay attention. We are going to substitute B sin theta instead of SA. So, tan alpha is equal to B sin theta divided by A + B cos theta. If that is the case, alpha, or the angle, will be equal to tan inverse of B sin theta divided by A + B cos theta. Did you understand it clearly? We can give this equation number as one, or two, or three. This is equation number four. This is equation number five. Okay. I am confident about this. If you say you have no doubt, can you comment a heart emoji in the chat box? If you say I understood this, I have not even a tiny bit of doubt, I am very much confident, can you put a heart emoji in the chat box? Then we can move on to the next. Are you ready? If you say I understood this clearly, I don't have even a tiny bit of doubt, put a heart emoji, and we will move on to the next topic. The most crucial topic in this chapter is projectile motion. We are going to move on to that. Projectile motion, we are going to learn its derivations. Time of flight, maximum height, horizontal range, equation of projectile, we are going to learn all of them. We will deal with all the derivations in detail. Are you ready? So, what is projectile motion? Simply put, if I throw an object with an initial velocity, under the influence of Earth's gravitational field, under the influence of gravitational force, if I project an object with an initial velocity, it can be called projectile motion. Let me say it one more time. I have a stone in my hand. Throwing this stone is a projectile motion because if we project an object with an initial velocity under the influence of Earth's gravitational force, its motion is called projectile motion. Throwing a stone, throwing a ball, firing a bullet, throwing a javelin, these are all examples of projectile motion. Understand that these are all examples of projectile motion. The shape of the projectile's path is a parabola. It is very important. The path of the projectile is parabolic. The path is also called trajectory. Apart from the word path, the word trajectory is also used. Okay, so understand what a projectile is. Understand that the shape of its path is a parabola. Here, I have represented a projectile motion. Pay attention. I have represented a projectile motion, like throwing a stone. Fine. The initial velocity with which the object is projected is U. The angle of projection is theta. The angle at which we throw it, the angle at which it is projected, that angle is called the angle of projection. It is theta. Fine. So, let's break U into two components. Didn't we learn to resolve vectors? The horizontal component is called U cos theta. What we call UX is the same as U cos theta. Didn't we get confused? UX is the same as U cos theta. What is the vertical component? It is U sin theta. The horizontal component is U cos theta, and the vertical component is U sin theta. These two have a big role. U cos theta is what makes the stone move forward. U sin theta is what makes the stone move upwards. Is it clear? Because this is how the stone moves, in a parabolic path. So, understand that the horizontal motion is caused by U cos theta, and the vertical motion is caused by U sin theta. Clear? Another thing. This U cos theta, U cos theta will always be the same. Throughout this motion, U cos theta will be the same. But how is U sin theta? U sin theta varies. It is a varying parameter. U sin theta varies. It changes. It keeps changing. Throughout the motion, it will not be the same. It will be a varying parameter. That is, let's take the peak point. Here, the value of U sin theta will be zero. Do you understand? As U sin theta goes up, it decreases and decreases, and at the peak point, U sin theta will be zero. Understand that. So, there are two components of velocity. The horizontal component of velocity is U cos theta. It is what drives this projectile forward. What takes it upwards? It is our vertical component of velocity, or U. Sorry, what drives it forward is U cos theta, or the horizontal component of velocity. The vertical component of U sin theta is what drives the projectile upwards. Understand that. Fine. Next, U cos theta is the same. U sin theta keeps changing. Now let's move on to other parameters. The first parameter is R, or horizontal range. Suppose I threw a stone from here. The horizontal displacement between the place I threw it from and the place the stone fell is called the horizontal range. Suppose I threw a stone and it fell in the next field. The distance between the place it fell and the place I threw it from is called the horizontal range. Next parameter. I threw a stone from here, and it reached the ground. The time taken for that stone to reach the ground. The time taken for the stone to fall to the ground is called the time of flight. Capital T. Horizontal range means horizontal displacement. Time of flight means the time taken for the stone to fall to the ground is called the time of flight. Next one, maximum height, or H. The maximum height reached by the stone is called the maximum height. The maximum height reached by the stone is what we call maximum height, or H. Did you understand the matter clearly? Not even a tiny bit of doubt, understood thoroughly, right? Understand and remember these parameters. Okay? So, the angle of projection is theta. The initial velocity is U. There are two components: U cos theta, the horizontal component, and U sin theta, the vertical component. U cos theta will always be the same. U sin theta will change. At the top, the value of U sin theta will be zero. Horizontal range means horizontal displacement. The time taken for the stone to fall to the ground is called time of flight. The maximum height is called maximum height. Let me mention one more thing. The acceleration of this stone will be the acceleration due to gravity. It acts downwards. Understand this point too. Fine. Okay. In all these years, what has been asked from this chapter are the derivations. We are going to learn those derivations. In all these years, the derivations asked from this chapter are what we are going to learn, understand, and become confident about. Are you ready? Are you ready guys? If you are ready, come on. The first derivation is time of flight. What is time of flight? Let's see. Suppose I threw a stone. Suppose it took a time T to reach the top. How much time will it take to reach the ground from the top? Suppose I threw a stone upwards. Suppose it took five seconds to reach the top. Then how much time will it take to reach the ground from the top? Tell me quickly. Suppose I threw a stone, and it took five seconds to reach the top. Then how much time will it take to reach the ground from the top? Without any doubt, it will take the same T seconds. It will take the same five seconds. Time of ascent and time of descent will be equivalent. Understand that time of ascent and time of descent will be equivalent. So, what will be the total time? That is equal to T + T, which is equal to 2T. So, time of flight is equivalent to 2T. It will be equivalent to 2T. Understand that. Fine. Okay. We are considering the first half of the motion here to find T. Let's assume we are considering this first half of the motion. Okay. The kinematic equation that will help us is V = U + AT. Can we give this as equation number one? Let's write its vertical component. So, Vy = Uy + AyT. Equation number two. Children, what is the vertical component of velocity? The vertical component of velocity is U sin theta. What is the final value of U sin theta? In this peak point, we are considering only the first half. What is U sin theta at this peak point? What is U sin theta at the top? U sin theta is zero. So, what will be the final velocity of U sin theta? It will be zero. Initial velocity is U sin theta itself. Let's give a whole square. What is the acceleration? We were told that acceleration acts downwards. So, pay attention. It acts downwards. So, let's give it as -g. So, what do we get? U sin theta - gT. Equation number three. I am taking this -gT to the left side. Then what do we get? gT is equal to what can we write? U sin theta. Equation number four. Can we find T from this? We got gT. Is it not very easy to find T from it? It is very simple. So, T is equal to what can we write? U sin theta. U sin theta divided by g. It will be U sin theta by g. Did you understand it clearly? Is there any doubt? Let's give this as equation number five. We know that time of flight is equivalent to 2T. So, time of flight is equal to what will come? 2 U sin theta divided by g. Equation number six. It took seven steps to reach the final answer. Time of flight is 2 U sin theta by g. If you are confident about this, can you put a fire emoji in the chat box? I understood this perfectly, I have not even a tiny bit of doubt, I am very much confident. Can you put a fire emoji in the chat box? Everyone put a fire emoji. Great. Someone is writing the derivation perfectly. I can't read the name. The derivation is written correctly. Okay. U sin theta by g. Okay. Energetic, right? If you are confident, put a fire emoji. Then we can move on to the next derivation. Horizontal range. We learned time of flight. Shall we move on to horizontal range next? Let's move on to horizontal range. We are throwing the same stone. We are throwing the same stone. The horizontal displacement traveled by that stone is what we call horizontal range. The horizontal displacement traveled by the stone is what we call horizontal range. Here, what we need to find is R. Okay? This is not a displacement. We all know that displacement is velocity into time. What is the velocity that makes this stone move forward? What is the velocity that causes horizontal displacement? Isn't it U cos theta? We said that what drives it forward is U cos theta. So, displacement is velocity into time. Here, horizontal displacement is horizontal range. Velocity is U cos theta. Time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the time of flight. Let's substitute directly. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight. Let's substitute directly. Children, let's substitute. So, R is equal to U cos theta into time of flight T. Let's give this as equation number one. So, displacement is equal to velocity into time. Here, the horizontal displacement is the horizontal range. The velocity is U cos theta. The time is the total time, time of flight

Understand the matter. If we say that this uniform circular motion is an example, then the motion of satellites, etc., can be considered by us as uniform circular motion. Now, we are going to study the parameters related to this. The first one is angular velocity. What is angular velocity? What is ordinary velocity? Velocity means displacement by time. Here, what is angular velocity? Angular displacement by time. The displacement here is angular displacement by time. What will be the unit? The unit will be radians per second. The unit is radians per second. Learn this. Next is time period. What is time period? The time required to complete one revolution. That is, the time required for an object to complete one revolution along a circular path is called the time period. It is the time taken by an object to complete one revolution. Its formula is 2π/ω. What will be its unit? The unit will be seconds. Also learn frequency. What is frequency? Frequency is 1/time period. The unit is hertz. That is, frequency is the number of revolutions per second. The number of revolutions per second is called frequency. The time taken to complete one revolution is called time period. The time taken to complete one revolution is called time period, and the number of revolutions completed in one second along a circular path is called frequency. Understand that the unit of time period is seconds and the unit of frequency is hertz. Fine. Next derivation. There are only two derivations left. The next derivation is the relation between linear velocity and angular velocity. We are going to derive the relation between linear velocity and angular velocity next. The relation between linear velocity and angular velocity. Here, you can see an object exhibiting circular motion, right? Yes. So, let's consider two points, P and Q. The angular displacement is Δθ. The radius of the circular path is capital R. Here, we can consider the velocities as v and v' if needed. Okay, right? Everyone knows this. Pay attention, children. Angular displacement Δθ = What can we write? Angle = Units and measurements, didn't I teach you? The equation, the equation of motion, you must learn it. You must definitely learn it. I will teach you top-level questions related to it. You must learn the equation of motion anyway. How can we write angular displacement Δθ? We learned in units and measurements that plane angle = arc length / radius. Here, the arc length is PQ, and the radius is capital R. So, we can write Δθ = PQ / R, because arc length / radius. Here, the arc length is PQ, and the radius is capital R. What is the formula for angular velocity ω? It is Δθ / Δt, right? Is Δθ clearly understood? There are no doubts, right? No, I believe so. Here, R is constant. We are taking it out. So, pay attention. ω = 1/R * (PQ / Δt). This is equation number four. This is equation number four. 1/R * (PQ / Δt). Equation number four. Now, everyone pay attention. Assume the time period is very small. Δt → 0. Assume the time is very small. Assume Δt is very small. If Δt is small, can we cover such a large distance? We definitely cannot cover it. Only a very small distance, right? We can only cover a very small distance. Definitely, only a very small distance can be covered. Look, this much distance might be covered. So, what can we call this distance? Yes, this distance can be taken as dx. We can call this distance dx or Δx. Since it is a very small distance, I called it dx. That is, PQ is approximately equal to dx. Or, we can replace Δt with dt. Understand clearly. So, I am going to write this. If the time is very small, how will the equation change? ω = 1/R * (dx / dt). So, ω = 1/R * v. This is the final equation. Equation number five. Equation number six. This is equation number seven. This is our final equation. Did you understand it confidently? If everyone understood, please comment "Set" in the chat box. If everyone understood this and is confident, please comment "Set Anna" in the chat box. If all the younger brothers and sisters comment "Set Anna," we can move forward. We will move forward. Next, we are going to study centripetal acceleration. What is centripetal acceleration? Let's move on to that. Before that, there is only one question. Only one question. Which is the correct statement among the following? In uniform circular motion, velocity and acceleration are radial. Incorrect. Both are not radial. Velocity and acceleration are tangential. Both are not tangential. Velocity is tangential and acceleration is radial. That is correct. The fourth one is also incorrect. Option three is the right answer. Option C is the right answer. Shall I move to the next one? Next one is centripetal acceleration. If an object exhibits uniform circular motion, the acceleration it experiences is called centripetal acceleration. If an object exhibits uniform circular motion, the acceleration it possesses is called centripetal acceleration. I will teach you by drawing a diagram. No problem. I haven't brought a diagram to draw, but I have a desire to teach you by drawing. That's why. So, stay with me. So, an object is exhibiting circular motion. I will mark two points on its path. Two points on the circular path. Let's name them P and Q. Let's name them P and Q. Ready? Next, what will be the velocity vectors at these points? We definitely know they are tangential. So, this is the first velocity vector. This is the second velocity vector. All these velocity vectors will be tangential to the circular path. No doubt, right? It's something taught, right? Definitely taught. Shall we name them? This velocity's name is v. This velocity's name is v'. Okay, right? Also, the angular displacement can be given as Δθ. Let's consider the center as O. Linear displacement also exists besides angular displacement. What can we give as linear displacement? Linear displacement can be treated as Δx. This is Δx. So, the linear displacement is Δx. Is everything okay? Also, the radius of the circular path is capital R. Capital R is the radius of the circular path. You all must learn to draw this diagram with all these things. This is important. We can only draw like this in the exam. You all must know how to draw a diagram like this. Next thing, these velocity components, right? I am going to draw another diagram by taking only these. Stay with me. I will draw with the same color. So, I drew the velocity vector v here. Next, I drew the velocity vector v' here. Can you see both velocity vectors? I have marked both precisely. I have drawn both precisely. I drew v and v'. Fine. Okay, right? Yes, I believe so. This is the velocity vector v. And this one is the velocity vector v'. Fine. This is v and this one is v'. Okay, ready. If so, according to the triangle law, what will be the third side here? Can you tell me quickly? Everyone try to tell me. What will be the third side? No doubt, it will be Δv. The third side will be Δv. So, you might have big doubts in your minds, like how it became Δv. It's easy, I will tell you. What is change in velocity? Change in velocity is final velocity minus initial velocity. If so, if so, tell me. Final velocity = Change in velocity + Initial velocity, can we write it? Yes. Final velocity = Change in velocity + Initial velocity. We can write it. That's the same thing. According to the triangle law, look. This is our initial velocity. This is the change in velocity. If we add these two, the resultant is v'. According to the triangle law, two vectors are considered as two adjacent sides of a triangle. Considered in the same order. Then their resultant is the third vector. v' is the resultant in the opposite order. Understood? Fix this logic in your mind. Fine. Next, what will be this angle? This will also be Δθ. Shall I tell you how? Look. v' is tangential to this radius vector. Clear? v' is tangential to the radius vector we draw. v is tangential to the radius vector we draw. If the angle between these two radius vectors is Δθ, then the angle between these two tangents will also be Δθ. That's why the angle is Δθ. Understand this. Now, shall we give it a name? I am going to give it a name. Pay attention. I am entering the naming ceremony. What color is this point? Yes, I gave this G. I gave this H. I gave this I. Okay, named. GHI. Right. Now, I am going to straighten this triangle and draw it. I am going to straighten it properly. It will look like this. Because these two are similar triangles, so straightening it won't cause any problem. I am straightening it. Look. This is G. This is H. This is I. I straightened it. Now, these two are similar triangles. These two are similar triangles. If they are similar triangles, then their sides will be in the same ratio. Let's look at the sides. I took the side called HI. Corresponding to it, what is the side in this triangle? It is PQ. HI or QP. We can say HI is similar to QP. HI / QP = Next, pay attention. Next, pay attention. The side similar to IG is O. The side similar to GI is OP. So, GI / OP. This is taken as equation number one. Children, what is HI? Pay attention quickly. What is HI? It is Δv. So, Δv / QP. What is QP? QP means Δx. Δx = What is GI? It is v, the velocity. What is OP? It is equivalent to R. So, v / R. v' / R. This is taken as equation number two. Let me take this Δx to the right. Let me take Δx to the right. Pay attention. So, Δv = Δx * (v / R). This is equation number three. Understand clearly. It is very important. Let me divide both sides by Δt. I am going to divide both sides by Δt. So, Δv / Δt = (Δx / Δt) * (v / R). This is equation number four. Children, change in velocity by time is acceleration. Isn't the acceleration here called centripetal acceleration? Centripetal acceleration. Displacement by time means velocity. So, v * (v / R). Understand clearly. I am writing the replacements on the side. Δv / Δt means centripetal acceleration, or ac. Δx / Δt means velocity, v. I have replaced both. If so, what will be the final answer for centripetal acceleration? We will get v² / R. Centripetal acceleration = v² / R. Everyone must learn this. It is a very important equation. This is one of the final equations. Equation number five. Equation number six. Now, I am going to replace v. What is v? It is Rω. I am going to replace it. So, centripetal acceleration = (Rω)² / R. We will get = R²ω² / R. Equation number seven. Let me cancel the common terms. R and the square cancel. We got another equation for centripetal acceleration. Rω². You must also learn this, children. You must also learn this. This is also a very, very, very important equation. Equation number eight. This is equation number eight. Are you confident? If you are confident and everyone understood this, please comment a heart emoji in the chat box. With this, our centripetal acceleration is finished. Now, there is only one numerical problem left to study. The last numerical problem. I cannot avoid teaching this. That's why I am teaching it. Although the remaining questions are moved to the questions section, this question must be taught. So, I am going to move on to that. Are you ready? Shall I call Jaseel Anna? Jaseel Anna will come, I believe. Where is Jaseel Anna? Jaseel Anna might be roaming around somewhere nearby. If Jaseel Anna is called, he will come running. Come running, Jaseel Anna. Come running, Kannan. Come running, Kannan. Like being called. So, children, next is the insect's question. Pay attention. An insect is trapped in a circular trap. The radius of the trap is 12 cm. In the last seven revolutions, it completes in 100 seconds. What we are asked to find is the linear speed and centripetal acceleration. It's easy to do. There is an insect, a bug. It is trapped in a circular trap. It is moving in circles, that is, exhibiting uniform circular motion. The radius of the trap is 12 cm. It completes seven revolutions in 100 seconds. So, shall we find each one? It's simple. First, I will write the given data. Radius = 12 cm. When converted to meters, it is 12 * 10⁻² meters. 12 * 10⁻² meters. Ready? Okay. Now, let's find the time period. If it completes seven revolutions in 100 seconds, what is the time period? It is 100 / 7. That is equal to 14.28 seconds. I have already written it down. That is, the time taken for one revolution. If seven revolutions take 100 seconds, the time taken for one revolution is 14.28. Yes. We can find the angular velocity here. ω = 2π / T. Formula. 2π is 3.14. T is 14.28. What will be the answer? It will be 0.44. 0.44 radians per second. Now, can we find the linear speed? v = Rω. For R, we put 12 * 10⁻² and for ω, we put 0.44. Substituting, children, what will be the answer? It will be 5.28 * 10⁻² meters per second. We got the linear speed. 5.28 * 10⁻² meters per second. We can also find the centripetal acceleration in the same way. That's it. With that, centripetal acceleration is done. It is Rω². R is 12 * 10⁻². ω is 0.44. Put a square on it. The answer will be 2.32 * 10⁻² meters per second squared. This is the centripetal acceleration. You must also understand its direction. Centripetal acceleration always acts in the radially inward direction. It acts in the radially inward direction. With this, our chapter is finished. What is the next chapter? Work, Energy, and Power. Jaseel is waiting here. I will go and have some light food and come back. Have some light food and come back. Jaseel will definitely take about an hour and a half class, I believe. I will stop when you come. I will jump and come. Anyway, Motion in a Plane is finished. I will put a tick mark for Motion in a Plane in our community. Then everyone should put fire emojis. Because others who are watching the live stream should also know. Yes, everyone should put fire emojis. We will move to the next chapter grandly. It's our fourth chapter today, right? Yes, it's our fourth chapter today. We are going to move on to Work, Energy, and Power. After that, I will have food and come back, and we can study Laws of Motion. We can study banking of roads. Are you going to study collision now? Collision will be awesome. One-dimensional, two-dimensional. Work energy. Variable force. Oh my god. Work energy theorem. Variable force. He said he will teach it. If he teaches that, then it's fire, children. I will go and have food and come back. I believe you all have had food. Those who haven't, go and have food for five minutes and come back. Come back. Okay. It will be awesome. Come on, come on, come on, come on, come on, guys. Come on, guys. Let's start. We are not wasting any time. Yes, yes, yes, yes. Children, it's a new chapter. Work, Energy, and Power. So, shouldn't we welcome it like this? Everyone put fire emojis and set the chat box on fire. It's a new chapter. All children, set the chat box on fire. Set it on fire quickly. We are going to study. Okay, I believe you are ready. Yes. Without wasting any time, I will silence my phone. It will be a big disturbance to the class. Ready. Shall we start? Yes, yes, yes. Come on. So, we are going to study Work, Energy, and Power. We are going to start. Everyone pay attention. What are we discussing first? Work. See, I am moving Anna. I am dragging Anna. Then Anna has displacement. When we apply force, if there is displacement in a body, we say that we have done work. So, when we apply force to a body and the body is displaced, we say that we have done work. Right? So, we need to define this work, right? This is a horizontal plane. You can see a body placed here. Yes. Let the mass of the body we placed be m. I am applying force like this. Yes, the force I am applying is F. When I apply force, the body's displacement is in this direction. Yes, the body's displacement is in this direction. So, I am splitting this force into two components. If the angle it makes with the displacement is θ, then I am splitting the force into two components. We know what components we get. Yes, one component will be here. We can call this F cos θ. Ready, children? One component will be upwards. What can we call this component? Children, we can call it F sin θ. Now, I will ask you one thing. Is F cos θ or F sin θ responsible for the body's displacement? Everyone comment in the chat box. Is it F cos θ or F sin θ that is responsible for this body's displacement? Everyone comment in the chat box. Who will it be? Definitely, it is F cos θ that displaces this body. So, F cos θ is what creates work there. So, can we say this work like this? Work = Product of force in the direction of displacement. F cos θ means what? Force in the direction of displacement. That is, F cos θ * displacement. We can write this as F S cos θ. F S cos θ. This is the equation for work. So, what is the equation for work, children? What is the equation for work? Force * displacement * cos θ. How can we define work? Work is the product of what? Force in the direction of displacement and displacement. Okay, right? What we learned before is that work is force * displacement. Now, there is a small change. Force in the direction of displacement and displacement. If you multiply the force in the direction of displacement by the displacement, you get work. Has the equation become clear to everyone, children? Has this equation become clear to everyone? Yes. Our work is a scalar quantity. Force is a vector quantity. Displacement is a vector quantity. But our work is a scalar quantity. When we multiply two vectors, if we get a scalar quantity, what is that vector multiplication called? Anna taught us vector addition in the last chapter. Right? Taught us how to add vectors. Right? In the next two chapters, that is, in Work, Energy, and Power, and in System of Particles and Rotational Motion, what are we going to study? We are going to study vector multiplication. If multiplying two vectors gives another vector, it is a cross product or vector product. If multiplying two vectors gives a scalar quantity, work is a scalar quantity. Because I applied force and the body was displaced in this direction. In this direction. So, work does not have a specific direction. So, work is a scalar quantity. So, if multiplying two vectors gives a scalar quantity, what is that vector multiplication called? It is called dot product or scalar product. So, in terms of dot product or scalar product, how can we write it? Work is the dot product of force and displacement. F · S. What is it, children? F · S. That is, F S cos θ. I can write F S cos θ as F · S. That's all. Nothing else. This thing can also be written like this. So, what is F · S, children? F · S means F * S * cos θ. I can write F S cos θ as F · S. That is, when I multiply two vectors, if I get a scalar quantity, that vector multiplication is called dot product. We don't have dot product, we have dot product. We have given the dot. Or we can say, what can we say? Scalar product. Ready, children? Ready, children? Next, what is vector product? We will study it there. Where will we study it? In System of Particles. I believe everyone understood. Ready, believe you are ready. If the equation for work is set, all children, send a "Set" message. Work is understood, sir. What is the unit of work? Children, everyone tell me. What is the unit of work? We all know the unit of work is Joule. Joule. Or kilogram, kilogram meter squared per what, children? Second squared, right? What is the unit of force? Newton meter. We can say anything. We usually give Joule. It is a scalar quantity. Work is a scalar quantity. Let's look at the dimensions of work. Children, look at the dimensions of work. We know that the dimension of work is ML²T⁻². Yes, kilogram meter squared per second squared. Ready. Okay. Know these things about work. Now, there is a question related to this. You are looking at the 2022 model question. As I said, we are dealing with all the model questions asked up to 2022. We have already dealt with all the model questions for Motion in a Straight Line. This time, we are going to deal with all the questions for Work, Energy, and Power up to 2025. I believe everyone is ready. Okay, right? Okay, right? Ready, right? Yes. Look at the next question. What is meant by work done by a force? What is work done? Simple. What is work done? Work done is the product of what? Force in the direction of displacement and displacement. If we apply force to a body, and if there is displacement in that body, we say that work has been done. Work is the product of force in the direction of displacement and displacement. If you write the equation for work, the entire task is done. You will get full marks for the model. At least write the equation. Ready, children? Yes. If so, let's move on to the next one. It is the 2022 model question. Yes. Force and displacement are given. Find the work. Okay, this is scalar multiplication. How to do scalar multiplication, sir will teach you. Then we will proceed. It is a dot product. We have two vectors. Vector A. We represent a vector using i-cap, and also j-cap, and also k-cap. i, j, k are unit vectors. i, j, and k are unit vectors. x, y, and z will be any number, one, two, three, any number you like. Any arbitrary number. i, j, k are unit vectors. So, we represent a vector in this way. Sometimes x1 might be zero. Sometimes y might be zero. Sometimes x and y might be zero. It depends. So, if we write it in a general form, a vector has an i-component, a j-component, and a k-component. That is, the vector is the direction. i represents the x-direction.

J represents the Y direction, and K represents the Z direction. Similarly, another vector is B. Another vector I have is B. I represent B like this: A_i_cap + B_j_cap + C_k_cap. There are two vectors, vector A and vector B. The angle between vector A and vector B is theta. If so, we all know what the dot product of A dot B is. We know A dot B is equal to AB cos theta. There is another equation for A dot B. Learn that one too. Another equation for A dot B is simply, if you look, multiply the components of I, multiply the components of J, multiply the components of K, and add them all up. Then what will you get, children? What will you get? The component of I is A_x. Everyone should learn the equation A_x * B_x. You don't need to memorize it. I will solve the problem for you, then you will understand. A_y * B_y + A_z * B_z. Remember all of this. This is the dot product. So, if you are asked to find the dot product of two vectors, you just need to multiply the components of I, multiply the components of J, multiply the components of K, and add them all up. Add them all together. Are you ready? Are you ready? Then let's do this question. We are given the force. I will write it here. The force vector is given as 3i_cap + 4j_cap - 5k_cap. Similarly, the displacement vector is given. The displacement vector is 5i_cap + 4j_cap + 3k_cap. Are you ready, children? We need to find the work. Work is equal to F dot D. What is the technique to find the dot product? Multiply the i's together. Similarly, multiply the j's together. Similarly, multiply the k's together. Let's multiply them. We need to find the angle between the force and displacement. To find the angle, we first need to find the dot product. We are finding that dot product. Look, children, first let's find the dot product. After that, we will find the angle. So, find the dot product. 3 multiplied by 5 plus 4 multiplied by 4 plus minus 5 multiplied by 3. So, what will you get, children? 15 plus 16 minus 15. Can anyone tell me the answer? What is the answer? 16 Joules. 16 Joules. Okay. But the question did not ask us to find the work done. Usually, the question asks to find the work done. Here, it did not ask to find the work done. Then what is it asking to find? What is it asking to find? It is asking to find the angle. So, how do we find the angle from this equation? How do we find the angle? Here is our equation. To find the angle from this equation, what do we need to do? We know that A dot B is equal to AB cos theta. From here, what will cos theta be, children? cos theta is equal to A dot B divided by AB, right, children? What will theta be, children? Theta is equal to cos inverse of A dot B divided by AB, right, children? This is the formula to find theta. Okay. So, naturally, we have already found A dot B. Now, what do we need? We need AB. What is A and what is B? I will tell you. A represents the magnitude of vector A. I simply call A the magnitude of vector A. How do you find the magnitude of vector A? Take the square of this, take the square of this, take the square of this, and add them all up. Then what will we get? This is actually something that comes in motion in a plane. A_x squared + A_y squared + A_z squared. Everyone remember this. This is the formula to find the magnitude of a vector if a vector is given. Square the components of i, j, and k of that vector, add them up, and take the square root. You need to know this. Are you ready? Now, let's come to our question. We already need to find the angle. So, we know the formula to find the angle. First, let's find the magnitude of F and D. Let's find the magnitude of F. Shall we find it? Let's find it. Root of. Similarly, find the magnitude of F, children. Root of. What is 3 squared plus 4 squared plus minus 5 squared? Take it. This is a tricky question. I should have taught them how to find the angle. I will teach them. Yes. So, look. 3, 4, and 5. So, look. 3 squared. 3 squared plus 4 squared plus 4 squared plus minus 5 squared. Are you ready, children? So, what will you get? Root of 9 plus 16 plus 25. So, what will you get? Root 50. Wow, that's good. We got root 50. Now, similarly, we can find the magnitude of displacement. Magnitude of displacement is root of. Root of. What will you get? 5, 4, and 3. By God, it's the same. So, 5 squared plus 5 squared plus 4 squared plus 3 squared. What will you get, children? What is 5 squared? 25. What is 4 squared? 16. What is 3 squared? 9. Here too, we get root. So, we have found the magnitude of displacement and the magnitude. What did we get? Root 50. Now, what is the formula to find the angle, children? We know that the angle is cos inverse of A dot B by AB. Here, it's not A dot B, but F dot D by FD. The job is done. If you put this in, you will get the answer. I hope everyone got the answer. If you got it, everyone say you got it. F dot D is what? FD is already found to be 16. FD is 16. Divided by root 50 into root 50. So, what will you get? We got the answer. Cos inverse of 16 divided by 50. Actually, this is a top-level question. It came by mistake. I thought it was a formula to find work done. Then I realized it was asking for the angle. It's okay. Should I write that? No, that's enough. Cos inverse. There is no other work. So, look at what I taught for that. Look at what I taught. But this is a good question. So, we don't teach these top-level questions. Okay, right? Are you ready? Now, types of work. Only types of work. Don't worry. We spent so much time on just one question. Now, we will quickly move on to types of work. We know the equation for work is Force into Displacement into cos theta. We can classify work into three types: positive work, negative work, and zero work. Simply put, positive work is when I lift this gas cylinder upwards. When I lift the gas cylinder upwards, in which direction am I applying force? Upwards. I am applying force upwards. The gas cylinder is moving upwards. So, if the force and displacement are in the same direction, we say that the work done by us is positive. What about the angle? The angle between the force and displacement should be between 0 and 90 degrees. If it is 0, it's okay. If it is 90, it's not okay. If it is 90, the work will be zero. Okay, right? So, positive work, simply put, is when the displacement occurs in the same direction as the force applied. If it is not so, we can say that the angle will be between 0 and 90 degrees. If it is 0, it's okay because we all know the graph of cosine. If anyone doesn't know, just look at it. This is the graph of cosine. This is our cosine graph. Look, children. What angle is this? This is 0. This is 90. This is 180. If you look, you will understand. Between 0 and 90, the value of cosine is positive. Between 90 and 180, the value of cosine is negative. If cos 90 occurs, the value of cosine will be zero. So, we are actually classifying work based on this. Okay, right? So, positive work. Force and displacement are in the same direction. I am pulling a body, or I am pushing a body. Sorry, I am pushing a body, or I am pulling a body. In these conditions, the work will be positive. Are you ready, children? Is it okay? Pushing and pulling are all positive work. Or, a body is falling downwards. What is the work done by gravity? Doesn't gravity pull it? So, what is the work done by gravity? It is positive work. Now, next, negative work. Look, I am pulling this person. I am pulling this person. This person is still young. The playfulness hasn't gone away. I didn't see the smile. I saw the mischievous smile. Look here, this person. I am pulling this person. So, the displacement of this person is in this direction. They are smiling. How nice. The displacement, children, is in this direction. Now, below this person, there is a force. What is that force? Isn't it friction? Isn't it the frictional force? We know that frictional force opposes motion. It opposes relative motion. So, below this person, there is a frictional force. In which direction will that frictional force be? It will be in this direction. The frictional force is in this direction. Okay, children? So, the work done by the frictional force here is negative work. The displacement is in this direction. The frictional force is in this direction. So, if the force and displacement are in opposite directions, it is negative work. If they are in the same direction, it is positive. Are you ready, children? Did everyone understand? Did everyone understand? So, when is work negative? When our force and displacement are in opposite directions. To put it more poetically, the angle between us should be between 0 and 180 degrees. If it is 180, it's okay. If it is 90, it's not okay. So, work done by friction, work done by viscosity, all of these will be, children? What will these be, children? Negative work. Negative thinkers do negative work. Now, next is zero work. Zero work. I am pushing this wall. If I push it today and tomorrow, will the wall move? No. So, what will be the work done by me? It will be zero. So, if there is no displacement of the body, then the work done will be zero. Or else, a person is carrying a basket of fish. A person is carrying a basket of fish. Okay, children? The fish inside is an oily fish. It is an oily fish. So, in which direction is this person moving? Okay? So, the displacement of this person is in this direction. The person is lifting the basket of fish upwards. So, in which direction is the person applying force? Upwards. The person is applying force upwards with this hand. So, the force applied by the person is upwards. The basket of fish is moving in this direction. The angle is 90 degrees. The work done is zero. Okay, children? So, in this case, the work done will be 90. Now, the work done by gravity is also zero. The gravitational force is in this direction, and the displacement is in this direction. Here, the work done by gravity is also zero. And the work done by us is also, children? Zero. So, if someone is carrying a basket on their head, do they need to be paid? If you say no, you will get hit, okay? They are doing work against friction, etc. So, they need to be paid, right? It's not good. Okay, are you ready, children? So, I hope everyone understood that. Or else, you can say. Sir, I will pay you for lifting the basket onto your head. We won't pay for walking like this. Because, you see, Mr. Rasheed taught us. If you walk, you are not doing any work. So, we won't pay. So, we won't pay half the money. If you say that, you will get hit on the head, okay? Don't do that either. Yes, did everyone understand? So, work is zero either when the angle is 90 degrees or when there is no displacement. From here, I can guarantee you that there will definitely be a question on types of work in the exam. So, let's do some questions. It would be better to do all types of questions. Let's do all model questions excellently. Pay attention. 2024 model question. Check whether the work done is positive, negative. First, tell me, what is the work done by gravity on a body falling downwards? We know the gravitational force is downwards. The displacement of the body is downwards. If the force and displacement are in the same direction, we say the work done is positive. Did you get it? Did you get it? Now, the second one. Work done by friction. We know that in the case of friction, if the displacement is in this direction, the frictional force will be in which direction? In the opposite direction. So, work done by friction is negative. Look at the 2022 model question. Write down any one condition in which the work done is zero. Tell me any one condition. Tell me any one condition for work done to be zero. Children, what is the condition? Either the displacement should be zero. Either the displacement should be zero, or the angle should be 90 degrees. To give an example, what would it be? Push or pull a wall. You push or pull a wall. Here, the work done will be zero. Or, work done by a load-carrying man. What is the work done by a man carrying a load? It will be zero. Now, the second one. Write any one example of negative work. What is an example of negative work? Or, what is the work done by friction? Children, what is the work done by friction? The work done by friction is negative work. Are you ready, children? Didn't we do the model questions excellently? Now, the 2021 model question. Tell me the answer quickly. Work done by friction. We know it's negative. Next, work done by centripetal force. Didn't Mr. Ann teach you centripetal force? In the previous chapter, we all know centripetal force is towards the center. Towards the center of the circle is the centripetal force. It is in this direction. Now, didn't Mr. Ann show you the direction of velocity? He drew the velocity tangent. I still remember. In the derivation of centripetal acceleration, Mr. Ann showed something like this. It's tangent. The direction of velocity. We know that the direction of velocity and the direction of displacement are the same. In the first class, I taught you about motion in a straight line. What determines the direction of motion of a body is the velocity. So, the direction of velocity determines the direction of displacement. Tell me, tell me, children. What will be the work done by centripetal force? The angle is 90 degrees. Work done by centripetal force is zero. Did everyone get it? Zero. Then, the next one. Work done by gravitational force on a freely falling body. We already said that. For a freely falling body, it is positive. If the body is going upwards, the work done is negative. Pay attention. Work done by horizontal rod. Work done is zero. We have already discussed all of this. Next, next, 2019 model question. Look, children. Work done by gravity. Water is being brought up from a well. What will be the work done? Tell me, children. Tell me, children. Yes, we are lifting a body upwards. We are applying force in this direction. The displacement of the body is also in this direction. We are doing work against gravity. So, the work done by us is positive. Pay attention. When we lift an object from bottom to top, the work done by us is positive. The work done by the gravitational force is negative. This is something to note. Now, question B. Work done by applied force and frictional force on a body moving on a rough horizontal plane with uniform velocity. Both are asked, right? Applied force, and also gravity. So, the work done by us is positive. The work done by gravity is negative. Children, next, work done by applied force and frictional force on a body moving on a rough horizontal plane. A body is moving on a horizontal plane. This is the applied force. The body is moving in the direction of the applied force. So, the displacement and the applied force are in the same direction. But friction? We know that the frictional force is opposite to it. So, tell me, children. What will be the work done by us? The work done by the applied force will be positive. The work done by friction will be negative. Excellently, all cases. Now, there is no other case. From these questions, the concept of positive, negative, and zero work is understood. There is a possibility of asking a question worth two marks from this section. It was not asked in 2025. So, perhaps in this 2025 model, we can definitely expect it. Sir, sir, I can't see. I am here. Maria, Maria, Maria. Are you happy, Maria? Maria, Maria. Are you happy, Maria? Maria, Maria. Are you happy? Maria, Maria. Are you happy? Yes, yes, yes. Children, energy. Energy. Energy. Come on, children. What is energy? Energy is the ability to do work. Is it the ability to do work? Yes, sir. Minds are thinking. What is being hunted? Isn't that something else? Alien? What are they saying? Oh God. Look here, children. Are you happy? Maria, Maria, are you happy? Maria, Maria, are you happy? Yes, yes, yes. Children, energy. Energy. Energy. So, you understood what energy is. There are actually two types of mechanical energy. What are they? We know potential energy and kinetic energy. We are going to study kinetic energy. We can study potential energy later. So, what is kinetic energy? Kinetic energy is the energy that a body possesses due to its motion. The kinetic energy of a body due to the state of motion. That is what is called kinetic energy, children. There is an expression for this kinetic energy. Everyone should learn it. Look here, children. I have a beautiful body in my hand. I apply force to this body. The body is initially at rest. The initial velocity is zero. When I applied force to the body, the body moved. It displaced from here to here. The displacement of the body is S. The final velocity of the body is V. Then, can we find the work done there? We all know that work is equal to force into displacement. The angle is 0 degrees. Are the force and displacement in the same direction? Are the force and displacement in the same direction? So, theta is 0 degrees. Okay, we wrote it. Now, we are going to change the force. Force is equal to mass into acceleration. We can write into displacement. Okay, children? Now, everyone pay attention. We are going to change the displacement. From the equations of motion, Mr. Ann taught you that V squared is equal to U squared plus 2AS, right, children? From here, can we find the displacement? The final velocity is zero, and the initial velocity is zero. So, V squared is equal to 2AS. From there, displacement is equal to V squared divided by 2A. Okay? If we put this in, we get the answer. Then, work is equal to M into A into V squared by 2A. We can cancel A and A. So, work is equal to MV squared by 2. Then, what will this work become, children? Because we applied work, it moved. So, the work we apply will become, children? The work we apply will become, children? Kinetic energy. So, everyone learned it. What is the equation for kinetic energy? Half MV squared. This is the same derivation that we are going to do in the work-energy theorem. This same derivation will be done there. There, there is initial velocity. That's all. Okay, yes. I am also fine. I am also happy. Are you ready, children? I hope everyone understood. The derivation of kinetic energy was asked in the 2021 model. Children, they asked to derive kinetic energy in some year. I don't remember if it was in the Christmas exam. It was asked in some exam. I don't know exactly. I forgot. If I find it, I will tell you. 2021 model question. The magnitude of kinetic energy is K1. If the velocity is doubled, what happens to the kinetic energy? We know that kinetic energy is half MV squared. Half MV squared. So, what happened? If we call the current kinetic energy final, the current kinetic energy K dash, what will we get, children? Half into mass remains unchanged. Velocity is doubled. So, we can write 2V. 2V whole squared. So, kinetic energy dash is equal to half into M into 4V squared, right, children? Half MV squared is kinetic energy. So, it is four times the kinetic energy, isn't it? So, we can say that kinetic energy dash is equal to four times the initial kinetic energy. So, what will be the answer? What will be the answer? Answer is 4K. Answer is 4K. 2021 model. A simple question. A question that can be done simply and excellently. Next, we will have more application-type questions. From this section, everyone should learn. It is an important section. Relation between kinetic energy and momentum. We are going to study the relation between kinetic energy and momentum. I hope all the children are okay. I hope all the children are ready. Okay? Okay? Deen sir, how many minutes do I have? Should I finish? No, I was asked to proceed according to that. It's okay, right? Then, then, relation between kinetic energy and momentum. Children, it is important. Application-type questions are always asked from here. I will tell you the technique. We know the equation for kinetic energy. Kinetic energy K or E is half MV squared, right? Yes. We know that. To bring momentum here, we are going to do some small manipulations here. We multiply mass from the top and bottom. Just for fun, for curiosity, we multiply mass from the top and bottom. So, we can write kinetic energy is equal to half into M into M squared. Here, V squared will come. Below, M will come. This M and M will become M squared. Now, mass into velocity is momentum. So, M squared V squared will become momentum squared. So, I will write this. Kinetic energy is equal to half into M into V whole squared. Can't we write it? Can't we write it? M squared V squared can be written as MV whole squared. So, the answer is kinetic energy is equal to, everyone learn the equation. Kinetic energy is equal to half into P squared by M. Or, P squared by 2M. Everyone should learn this important equation. Okay, right? Shall I tell you a condition? Sir, shall I tell you a small condition? Sir, I will tell you a small condition. If mass is constant, or if momentum is constant, then we can say that kinetic energy is inversely proportional to mass. So, we have two bodies in our hand. The momentum of these two bodies is constant. The momentum of the first body and the momentum of the second body are equal. Okay, right? Then, we will be asked, who has more kinetic energy? Then, we should remember this equation. Kinetic energy is inversely proportional to mass. If the momentum is constant, and the momentum of two bodies is equal, then kinetic energy is inversely proportional to mass. If the mass decreases, the kinetic energy increases. Now, there is another relation. From this equation, we are going to find momentum. Find momentum from this equation. What will we get from here? Here, we multiply 2M to this side. So, we can write P squared is equal to, children, what will you get? P squared is equal to, children, what will you get? P squared is equal to 2M into kinetic energy. Can we find P from there? P is equal to root of 2M into kinetic energy. Keep this in your pocket, it will be needed. We are keeping this in our pocket. Here too, a question is asked. The kinetic energy of two bodies is the same. That is, if the kinetic energy of two bodies is constant, can we say that momentum is proportional to, children? Momentum is directly proportional to root M. Very, very important. So, if kinetic energy is constant, if mass increases, momentum increases. Now, if momentum is constant, if mass increases, kinetic energy decreases. It is inversely proportional and directly proportional. Everyone should learn this equation. Important. You will see questions from here. Do you see? See? 2022 March. Light body. A body. Equal momentum. Kinetic energy. A light body and a heavy body. Their momentum is equal. Then, who has more kinetic energy? Tell me, children. There is a light body. Not a light boy, a light body. There is a light body and a heavy body. Children, who has more mass? The heavy body has more mass. The question says momentum is constant. Momentum is constant. Tell me, children. What is the equation we know when momentum is constant? When momentum is constant, kinetic energy is proportional to root M. Isn't it the other way around? Isn't it the other way around? Kinetic energy is inversely proportional to mass. Isn't it? We wrote it just before. Look, kinetic energy is inversely proportional to mass. When momentum is constant. Or, you can find it by doing the equation. Kinetic energy is proportional to M. We wrote it before. Kinetic energy is inversely proportional to mass. Or, P squared is equal to 2M into kinetic energy. So, P squared is proportional to M into kinetic energy. If P is constant, then M into kinetic energy is constant. So, kinetic energy is inversely proportional to M. Are you ready? Are you ready? Then tell me, children. Tell me, children. Who has less mass? That person has more kinetic energy. So, we can say that the kinetic energy of the light body is greater than the kinetic energy of the heavy body. Okay, children? Who has more kinetic energy? The heavy body has more kinetic energy. Sorry, the light body has more kinetic energy. Are you ready? It is inversely proportional to mass. Mass.

The body with less mass has more kinetic energy. Did everyone get it, children? Yes, yes. Light body, ready, ready, ready, ready. Then, let's move on to the model question. Model question from 2018. Heavy body and light body, same question, right? No, it's a case of momentum. See, momentum is the opposite. Same question, same question. Here, kinetic energy was asked. Here, momentum was asked. Heavy body and light body, same kinetic energy. Here, kinetic energy is the same. Kinetic energy is the same. So, if kinetic energy is constant, who has more momentum? We know, children, momentum is directly proportional to the square root of mass, right? So, the mass of the light body is less than the mass of the heavy body. Therefore, whoever has more mass will have more momentum. So, can we say that the momentum of the heavy body is greater than the momentum of the light body? This is the reason. This is enough to write. This is enough to write. So, who has more momentum? Heavy body. The heavy body has more momentum. Are you ready, children? Did everyone understand? You should do such questions excellently. I believe you will do it. I believe it is okay. Yes. Then, let's move on to the next one. Next question. Two bodies of mass m1 and m2 have the same linear momentum. What is the ratio of their kinetic energy? Linear momentum is the same. The ratio of kinetic energy is asked. Okay, right? That is, if momentum is constant, if momentum is constant, we know that kinetic energy is inversely proportional to mass. Is there any doubt for anyone? If the momentum of two bodies is constant, we can say that the linear momentum is the same. Linear momentum is the same. So, kinetic energy is inversely proportional to mass. So, the kinetic energy of the first body is inversely proportional to the mass of the first body. The kinetic energy of the second body is inversely proportional to the mass of the second body, right? Divide both. Divide both, children. Divide both. Then what do we get? K1 by K2 is equal to 1 by m1 divided by 1 by m2. Is there any doubt? No. There are two fractions. Divide them. Multiply by the reciprocal. So, we got the answer. K1 by K2 is what, children? m2 by m1. I hope you understood this excellently. If it is okay, everyone say "Okay, sir." If you understood this question, say "Understood, sir," all children. Ready, ready, ready. Yes. Then, let's move on to the next one. Work-energy theorem. This is a three-mark question. This is a question that is likely to be asked from this chapter for three marks. The weightage of this chapter is six marks. The weightage of this chapter is six marks. Ready. So, for three marks, here is the question. They will ask you to state the work-energy theorem. They will ask for the derivation and proof. So, what is the work-energy theorem? It's nothing, children. A body is moving like this. A car is moving like this. So, this car, as we all know, has kinetic energy. What if I give a push to the back of this car? I give a push to the back of the moving car. Did the car's speed increase? No. Let's take a bicycle. A person is pedaling a bicycle. Then I go to the back of the bicycle and give a push. I give a force. I give work. Won't the kinetic energy increase? Its speed will increase. Kinetic energy will increase. Simply, this is work-energy. The work we do is converted into energy. Work done is equal to change in kinetic energy. So, initially, the bicycle is being pedaled. Initially, the kinetic energy of the bicycle is 200 joules. When I did work, it became 400 joules or 500 joules. So, the bicycle that was moving at 200 joules is now moving at 500 joules. Where did that 300 joules come from? I gave it. So, 500 minus 200, the change in kinetic energy will be the work done. So, learn the statement. What does the work-energy theorem say? Work done is equal to change in kinetic energy. Or, the change in kinetic energy of a body is equal to the work done. Either way is correct. Okay. Eat food and come quickly. Let's see. Everyone pay attention. Let's find the work done. Here, our body, our beautiful body. Earlier, when we derived work-energy, the initial velocity of both bodies was zero. But here, there is an initial velocity. The body has an initial velocity. Okay, right, children? So, we apply a force to a body. The body's velocity changes. It travels a distance x. The displacement is s. It travels a distance s. Its velocity changes from u to v. Okay. Here, we are finding the work done. Work done is equal to, we know, here also force times displacement. Force and displacement are in the same direction. Ready, right? So, children, tell me. The equation for force, we know, is m times a. So, m * a * s. Now, look, children. Look, children. Now we are going to change the displacement here. From the equation of motion, we know that v squared is equal to u squared plus what, children? 2as. So, can we call v squared minus u squared as 2as? Here, we need to find s. s is equal to, children, divide 2a to this side. So, s is equal to v squared minus u squared divided by 2a. Put this in equation number 2. Put equation number 1 with equation number 2. We get the answer. So, what did we get? Work done is equal to m * a * (v squared minus u squared) divided by 2a. We can cancel a and a. So, what do we get? Can we write it like this? dw is equal to m * (v squared minus u squared) divided by 2. Is this 2 for both of them? So, can we write it like this? Work done is equal to 1/2. We can make it 1/2. We can make 1/2 mv squared. 1/2 mv squared minus 1/2 mu squared. Children, what is this? Final kinetic energy. What is this? Initial kinetic energy. So, what did we get? Work done is equal to, we can write, final kinetic energy minus what, children? Initial kinetic energy. That is, change in kinetic energy. So, work done is equal to, we can write, change in kinetic energy. This is the work-energy theorem. Did everyone learn it? Did everyone learn it? Very, very important. Very, very important. Work-energy theorem. Work-energy theorem is very, very important. Okay, right? Ready? Should we teach the work-energy theorem for variable force at a top level? We will teach it at a top level. For now, we are not teaching it today because we need to finish. We need to learn something there, right? Okay. Ready, ready, ready, ready. So, only that much. Sir will teach at a top level. 2022 model question asked. State and prove work-energy theorem. State and prove. What is the statement? Work done is equal to change in energy. How many marks was the derivation asked for? Three marks. Oh my god. In the March 2025 exam, it was a three-mark question. State and prove work-energy theorem. It was also asked in the 2023 model. State the work-energy theorem for a constant force. Only the statement is needed. Derivation is not needed. So, if it is just the statement, work done is equal to change in energy. Okay, right? No, no. This is based on marks. If this is a two-mark or three-mark question, it is definitely good to write the derivation to get good marks. Okay. 2022 improvement, 2020 model, etc., are repeated questions. It is a question that has been repeated many times. So, let's move on to the March 2023 exam. Let's come to the March 2023 exam. There is a car and a lorry. Car and lorry have equal kinetic energy. Which one has greater momentum? We know that if kinetic energy is constant, momentum is the square root of mass. We learned this just before. So, between the car and the lorry, is the mass of the car or the mass of the lorry greater? We know that the mass of the lorry is greater and the mass of the car is less. So, who will have more momentum? Which one? Which one will have greater momentum? So, we can say that the lorry has more momentum, and the car has less momentum. Ready, right, children? Next, its part B question. State and prove work theorem. March 2021. It is something that is important to study. The work-energy theorem, children. Are you ready? Are you ready? Then, look at the next question. Look at the question from 2017. Look at the question from 2017. It can be called a question from a very good family. What is question A? Energy by a body by motion is what? What is the answer to the question? Everyone in the comment box. Energy obtained due to the motion of a body. Okay, let it be. Okay. So, look. So, what is said? So, what is said? Energy by a body. Energy by a body due to motion. Tell me. Energy due to motion is what? Kinetic energy. Very good. Okay, right? Look at question B. Look at question B. Look at question B. The mass of a body is 5 kg. Initially, it is at rest. We apply a force of 20 Newtons. To a body of mass 5 kg, we apply a force of 20 Newtons. What is the kinetic energy acquired by the body at the end of 10 seconds? Is this a difficult question? After 10 seconds, what will be its kinetic energy? Oh, oh. Then let's do it. We will write down everything. We will write down everything. Who is playing with ghost photos there? Let him be removed. Let him be thrown away. I don't like you. I am going to remove you. For now, I am timing you out. You come back after 24 hours. Bye, bye, bye, bye, bye, bye. Children, pay attention. Everyone pay attention. Everyone pay attention. Let's write down the given values. Mass of the body. What is the mass of the body, children? 5 kg. Mass of the body. Ready, right? The force we apply to the body is 20 Newtons. The force we apply to the body is 20 Newtons. The initial velocity of the body is zero. The question doesn't give it. We know that the body of mass is initially at rest. Initially at rest. That is, initial velocity is zero. Time is given as 10 seconds. Time is given as 10 seconds. What needs to be found, children? What needs to be found is the kinetic energy after 10 seconds. How to find it? Kinetic energy is 1/2 mv squared. But here, there is no velocity. We will find it. We will find the velocity. This question is a confluence of three chapters. This one question. Confluence of three chapters. First, laws of motion. Using laws of motion, we need to find acceleration. Using that acceleration, we need to find velocity. What is that? What is that? What is that? Using the equation of motion from our chapter on straight-line motion, we will find the velocity. When we put that velocity into 1/2 mv squared, we will get kinetic energy. What is that? Power of work. So, from the fourth chapter, laws of motion, we can find acceleration. Acceleration is obtained by dividing force by mass. Force is given in the question as 20. Mass is given in the question as 5. So, we got acceleration as 4 meters per second squared. Excellent stuff. This is the fourth chapter. Fourth chapter. Fourth chapter. Acceleration is found. Initial velocity is known. Time is known. Acceleration is known. To find the final velocity, from the equation of motion. Equation of motion. I will write it here. According to the equation of motion, we all know that v is equal to u plus at. Okay, right? We need to find the final velocity. Initial velocity is zero. Acceleration is 4. Time is 10. Let's write the answer. 40 meters per second. Did you get it, children? 40 meters per second. The velocity is 40 meters per second. Now, let's find the kinetic energy. Oh my heart. Kinetic energy, as we all know, is 1/2 mv squared. 1/2 mv squared. So, 1/2 times mass is 5. Velocity is 40 squared. 40 times 40. We canceled it out. We got 20. We got 20. 4000 joules. If everyone got the answer, say "Got it." "Got it, uncle." Tell me. The answer is 4000 joules. It became 100. How did it become 100? I won't play this game. Isn't it 4000 joules? Isn't it 4000 joules? Isn't it 4000 joules? Oh, why are you children behaving like this? Isn't it a simple question? Yes. Using laws of motion, we found acceleration. Isn't that the second chapter? Using the second chapter, we found what? We found velocity. Similarly, using the fifth chapter, we found kinetic energy. It's a mixed question. It's a question where all chapters are mixed. Now, the next question. Work done by the net force is equal to 1/2 the net force. Is it true or false? Options. Children, is it true or false? Is it true or false? Everyone tell me, children. Everyone tell me. The answer is correct. It is correct. Didn't we write that just before? Work done is equal to change in kinetic energy. So, the next thing we are going to study is potential energy. Potential energy. There are two types. Actually, what is potential energy? It is the energy a body gets due to its position or strain. That is what we call potential energy. So, what is potential energy? It is the energy a body gets due to its position or strain. For example, this glass is sitting here. If I let it go, it will definitely fall down. Right? If I let this glass go from here, what will happen? It will fall down. Isn't it certain? So, this body now has what energy? Potential energy. This is the energy it got due to its position. Okay, right? Similarly, do you see that figure? The arrow is drawn like this. Yes, yes, yes. It is stretched like this. Right? It is stretched like this. This arrow has what? Energy. It is the energy it gets due to strain. Now, I strained this bottle. I changed the shape of the bottle. So, this bottle has energy. That's why when I release my hand, the bottle returns to its original shape. That is the potential energy due to strain. If you stretch a spring, the potential energy due to strain. The potential energy due to strain, which is created when its shape changes, is called potential energy due to strain. Okay, right, children? Potential energy can be classified into two. A body can get potential energy due to its position. Similarly, a body can get potential energy due to strain. So, what is potential energy? It is the energy created due to the position of a body or due to strain. That is called potential energy. Are you ready? We are doing both. We are doing gravitational potential energy. Similarly, children, we are also finding the potential energy of a spring. Let's see. First, gravitational potential energy. A body of mass m. A body of mass m. I am taking this body upwards. I am taking this body upwards. The force I am applying is F. The force I am applying is F. Okay, right? I apply a force F and lift this body and place it here. Okay, right, children? Okay, right? How much height is this? This displacement is called h. Okay. I am going to find the work done here. What will be my work done? We know that work done is equal to force times displacement times cos theta. Cos theta. I applied force upwards. The body moved upwards. Displacement. Sorry, the angle is zero degrees. Force and displacement are in the same direction. Angle is zero degrees. Now, what will be the force? We know that to lift this body upwards, the force I apply must be equal to what? We know there is gravitational force downwards. The force I apply must be equal to the gravitational force. If the gravitational force is equal, then only I can lift the body. If I want to lift a body, I must apply at least a force equal to the gravitational force to be able to lift the body. Okay, right? So, I say that the force I apply is mg. And how much distance did I lift it upwards? I lifted it to a height h. The angle is zero degrees. So, cos zero is 1. So, we can say cos zero is 1. So, we can write that the work done by us is mgh. This work is converted into potential energy. This work we do is what becomes potential energy. Potential energy is mgh. Did everyone understand this? If it is okay, just learn the equation. You don't need to learn the derivation. Potential energy of a body is mgh. The potential energy of a body is mgh. mgh. Actually, this is the change in potential energy. I am not going into that for now. Because when it is on the surface of the Earth, it has potential energy. It has gravitational potential energy. When it is brought upwards, that energy increases. Actually, h is the change in energy. I am not going into that. Simply, mgh is potential energy. Ready, ready, ready, ready. For the purpose, let's consider it as zero. Okay. Are you ready, children? Everyone learn mgh. The potential energy of a body at height h is mgh. mgh. mgh. We can move on to the next one. Yes, let's move on to the next one. Everyone pay attention. What is the next thing? It's a question. Question from the 2023 model. Write down the type of energy present in each of the following. Tell me, children. What is the energy in flowing water? Everyone tell me. Everyone tell me. I will cry. I will cry here. I will cry. You know the problem is very serious. So, everyone answer. What is the energy in flowing water? Flowing water. What is the energy in flowing water, children? Kinetic energy. Very good. Okay, right? What is the energy in the spring of a clock? Potential energy. Very good. What is in a rolling body? What is in a rolling body? Kinetic energy. A hammer is raised and kept. A hammer is lifted and kept. What energy is it? Potential energy. Is it set? Everyone is answering. They are sharp. They are good children. May God bless all of you with good marks. Children, let's move on to the next one. This is a question that is specifically seen in this model. These conservative and non-conservative forces and related energy. So, pay attention. What is a conservative force? Just note down what I am saying well. Conservative force is a force that does not depend on the path, but only on the initial and final positions. Such forces are called conservative forces. For example, if I bring this mobile phone from here to here, or if I take it straight from here, or if I move it around and around and bring it here, the potential energy from here to here will be the same as it will be if I bring it like this. If I bring it like this, the energy will not change. No matter how I bring it, the change in energy will always be constant. It depends only on the initial position and the final position. That is called a conservative force. So, simply put, a conservative force does not depend on the path. Example: Gravitational force. Does it depend on the path it comes from? No. Okay, right? Similarly, electrostatic force. You will learn it next year. Electrostatic force. Force between charges. Electrostatic force is also a conservative force. It is a force that does not depend on the path. It does not depend on the path. Okay, right, children? So, the work done in a round trip is zero. That is, if I bring this body from here to here. Then the work done by gravity will be zero. What will be the work done by gravity? In a round trip, if I move it around and around and bring it back to the initial position and the final position. If the initial position and the final position are the same, then we know that the displacement is zero. Therefore, the work done will be zero. So, the work done will be zero. So, children, pay attention. A conservative force is a force that does not depend on the path. It only depends on the initial and final positions. The work done in a round trip is zero. Learn these three points. Learn these three points. Let me say it again. It does not depend on the path. It only depends on the initial and final positions. The work done in a round trip is zero. That is, if you bring it back to where it started, the work done is zero. Ready, right, children? Everyone understood? Everyone understood? Okay, right? Okay, right? Children, let's take the case of gravity. The mobile phone is here. If I take the mobile phone upwards, the work done by gravity is negative. If this mobile phone comes down, the work done by gravity is positive. So, negative and positive, total work is zero. So, isn't the work done by gravity zero? Look at the examples. Electrostatic force. Gravitational force. Magnetic force. Lorentz force. And elastic force. Elastic force. The force of a spring. The force of a spring is also a conservative force. The force of a spring is also a conservative force. Okay, right, children? So, if a conservative force does work, that work done will always be constant. Or, the total mechanical energy remains constant. Just remember this. We are learning this. When we move forward a little, we will learn it. Now, on the other hand, you tell me. What will be a non-conservative force? A non-conservative force is a force that depends on the path. For example, frictional force. We know that frictional force depends on the path. Energy is lost there in the form of heat and sound. Right? We know that when I move a body from here to here, not all the work I do will be converted into energy. Some energy will be lost due to friction. It will be lost as heat. Understood? So, a force that depends on the path is a non-conservative force. There, the total energy will not be constant. Some energy will be lost. For example, frictional force, viscous force. These are all what, children? Non-conservative forces. So, will the work done in a round trip be zero? No. If it is non-conservative, the work done in a round trip will never be zero. Ready, right, children? That is, if a body is simply moved. If I move a body forward, the work done by friction is negative. If I move this body backward, is the work done by friction negative? Friction always does negative work, right? Friction always does negative work. But the work done by the gravitational force is negative when moving upwards and positive when moving downwards. But the work done by friction is always negative. So, the work done will never be zero. It will never be zero. So, in a round trip, in a round trip, the work done will not be zero. Ready, right, children? That is, the work done by frictional force and viscous force are examples. Okay, right? The opposite of the previous one. Pay attention to the question that comes from here. Can everyone answer this question? Let's see. Question from the 2022 model. Work done in the motion of a body over a closed loop is zero for a conservative force. True or false? Give the reason for your answer. Question from the 2025 model. This model seems to really like conservative and non-conservative forces. Because there is a question about it in this model question paper. Yes. The statement is true. Give the reason. I need the reason. Okay. The statement is true. Okay, I agree. This statement is true. I need the reason for it. Why is it zero? Why is it zero in a round trip? Why? We know that in a round trip, the displacement is zero. Therefore, the work done is what, children? Zero. Write down these simple things as we said before. Okay. Work done by a conservative force on a closed loop is what? Zero. Work depends on the initial and final positions, not on the path. It does not depend on the path. So, it only depends on the initial and final positions. Therefore, the displacement is zero. Therefore, the work is zero. Okay, ready, children? Yes. I believe everyone understood. Then, the question repeated in the 2025 model and March 2024 is the same question, without changing a single letter. Tell me, children. Classify the following into conservative and non-conservative forces. Elastic force. Tell me, children. Elastic force is conservative. No, no. Yes. Frictional force is non-conservative. Magnetic force is conservative. Viscous force is non-conservative. Ready, ready, ready, right, children? Shall we move on to the next one? Question from the 2023 model. What is a conservative force? A force that does not depend on the path. What will be the work done in a round trip? It will be zero. If you write this much, you will get marks. This is a question from the March 2023 exam. Okay. Next, let's see. Let's move on to the next topic. Let's move on to the next topic. Pay attention, children. Law of conservation of mechanical energy.

Now, within this chapter, there are three derivations. One is the law of conservation of mechanical energy. Then there is collision. And then there is the spring. There are only three derivations, okay? Let's finish quickly. So, look. Law of conservation of mechanical energy. What does the law of conservation of mechanical energy say? For a freely falling body, the total mechanical energy of the body remains constant. Or, a conservative force, like the force of gravity, is a conservative force. The energy or work done by a conservative force is always constant. Ready, children? If a conservative force acts, the total mechanical energy of that body will be constant. Pay attention, children. Look. Look. So, we are going to find out that a body is in free fall. So, if this body is in free fall, no matter which point we take, the total mechanical energy there will be constant. For a freely falling body, for what, children? Yes, for a free fall body. For a free fall body, what can we say? The total mechanical energy remains constant. That is, if a conservative force does positive work, the total mechanical energy remains constant. Everyone remember this. We are going to prove that at point A, at point B, at point C, the total mechanical energy is constant. It is a freely falling body. It is in free fall. At the very top, we know that the total energy, the total mechanical energy, is what, children? Kinetic energy plus potential energy. No doubt, right? We know that at the top-most point, the body is at rest. Therefore, the kinetic energy is zero. Now, when we look at the potential energy at A, it is very simple. When we look at the potential energy, the body is at a height H. Therefore, the potential energy is mgh. It is mgh. So, what did we get for the total energy, children? We got mgh. Give it equation number A. Yes, A. Now, next. Next, total energy at B. So, when we look at B, what energies are there? This thing is falling down. So, we took a screenshot at B. So, at B, the body is in motion, and the body also has height. So, at point B, let's look. The total energy is potential energy, or kinetic energy plus what, children? Potential energy. Shall we find them one by one? First, we find the kinetic energy. We all know that kinetic energy is half mv squared, right, children? Call this equation number 2. Here, we don't need the velocity because we should get it as mgh. So, we don't need velocity here. Let's change the velocity. We know the equation of motion: v squared equals u squared plus 2as. Okay, children? In this case, the initial velocity is zero. The initial velocity is zero. And also, and also, let's call this equation number 3. And also, the acceleration is the acceleration due to gravity. The displacement, we can say, is this much distance it has come. Let's give the displacement as s, or let's call it x. Equation number 4. Equation number 5. Equation number 6. Put it into 3. Put equation 5 and equation number 6 into equation number 3. So, what did we get, children? What did we get, oh? v squared equals. We can write v squared equals 2gh. We can give 2gh. Now, if you want negative, you can put negative here. You can put negative here. If the total negative goes away, you will get 2gh. So, don't worry. Okay? So, we got v squared, right? Now, we put it in here. We put it into the second equation. So, we got kinetic energy equals half into m into v squared. If we cancel out 2gh, what do we get? We get mgh. Kinetic energy is mgh, children. Give it equation number A. So, we found the kinetic energy. Now, it's potential energy, right? We know that potential energy. When we look at this point, for potential, this is the height the body has. Right? The body has only this much height. This height is total height H. We took this much distance as x. This much distance has been covered. The remaining, we can say, is H minus x. So, potential energy is, we can say, m into g into instead of height, let's put H minus x. Ready? Ready? So, we can say like this: potential energy equals. Potential energy is what, children? m into g into H minus m into g into x. We can say mgh minus mgx. We multiplied mg inside. Ready? Let's call this equation number. Let's call this equation number 9. Put equation 8 and equation 9 into equation number 1. We put kinetic energy and potential energy into equation 1. So, what will we get? Total energy equals mgx. This is kinetic energy. Plus potential energy, mgh minus mgx. Children, children, cancel it out. Cancel it out. What did we get for the total energy of the body? We got mgh. Mgh. Did everyone understand? Did you understand it wonderfully? The total energy here is also mgh. Let's call this equation number B. Let's call this equation number B. Now, finally, at point C. At point C, let's find the total energy. At point C, let's find the total energy. So, it comes and comes and comes and comes and comes and comes and hits here. It's about to hit. If it hits the surface, energy will be lost. So, we took the point just before hitting. So, we write the same here. The first equation is: total energy equals kinetic energy plus potential energy. Here, we find the kinetic energy. Kinetic energy is, we all know. Kinetic energy equals half mv squared. Here, we find the velocity. To find the velocity, we take the equation of motion: v squared equals u squared plus 2as. We take s. Equation number 3. Okay, children? We know, children, pay attention. We know. We know. We know, children. Tell me. We know. Yes, everyone tell me. We know. What is it? What do we know? Yes, yes. Initial velocity is zero. Initial velocity is zero. Acceleration due to gravity is g. Displacement. This much distance has been covered, right? This much distance has been covered, right? If this much distance has been covered, then we can say the displacement is H. Yes, displacement is H. H. Yes, yes, yes, yes. Okay, children? Okay, children? Okay, oh children? Ready? Ready? Ready? Okay. So, let's call this equation number 4. Let's call it 5. Let's call it 6. Put this into 3. Put 4, 5, and 6 into equation number 3. So, what will we get? v squared equals u squared plus 2gh. We got v squared equals u squared plus 2gh. Give it equation number 7. Put 7 into 2. Put equation number 7 into equation number 2, children. So, what will we get? Kinetic energy equals half into m into 2gh. If we cancel out, we can write here too. Kinetic energy is mgh. Give it equation number 8. Now, what will be the potential energy? We know potential energy is zero. Because it has reached the very bottom. If it reaches the very bottom, what will be the potential energy, children? What will be the potential energy, children? Tell me. Tell me. Tell me. Tell me. Tell me. What will be the potential energy if it reaches the very bottom? Can you tell me, children? Yes, yes, yes. Okay, children? Okay, yes, yes, yes. Children, there's one thing. Our magnetic force is non-conservative. I think I said it wrong. Magnetic force is a non-conservative force. It depends on velocity and path. It is actually non-conservative. Non-conservative. I said conservative, sorry. Electrostatic force is conservative. Others are non-conservative. Okay. So, what is the potential, children? Here too, what can we give for potential, children? Potential energy, we can give. 9. So, let's put 8 and 9 into 1. Let's put equation number 8 and equation number 9 into equation number, what can we call it, children? 1. So, what will we get? Yes, total energy equals. Total energy, again, we can write, is mgh. It is mgh. Let's call this equation number C. So, look. At point A, and also at point B, we got total energy as mgh. Similarly, total energy at B is mgh. Similarly, total energy at C is mgh. So, what can we say, children? What can we say? The total mechanical energy remains constant throughout the motion. Throughout that motion, the total mechanical energy is constant. Okay? Okay? Ready, children? Ready, oh? Ready, oh? Ready, oh? Yes, yes, yes, yes. Okay, ready, ready. Children, regarding the magnetic force case, we would have to explain the forces in more detail, like qv cross b. That actually comes in the second year. For now, just remember it's non-conservative. Children, everyone pay attention. Let's move to the next question. Prove the principle of conservation of mechanical energy in the case of free fall. This is a question that everyone knows, asked repeatedly, repeatedly, repeatedly. Ready, ready, ready, ready. Asked many times, many times. It's a good question. It's a good graph question. Pay attention. It's a graph question. Yes, here we can put energy. Here we can put energy. Here we can put the distance s. Okay? At the very top, at the very top, at the very top, we know x is H. At the very top, x is H. When it reaches the bottom, it becomes zero. Okay? We are going to take it in that way. So, pay attention. If this is, let's give it a height. Or let's give it a height. Let's give it H. That's better. Let's give it height. Let's give it H. That's what was said. Let's give it H. Okay. If the height is zero, then tell me, children. If the height is zero, who will be who will be? If the height is zero, the kinetic energy is maximum. This is kinetic energy. Kinetic energy is maximum. At the top-most point, at the top-most point, here H is zero. At the top-most point, let H be maximum. Let it be maximum. At the top-most point, who will be maximum? We know it's potential energy. Potential energy, we are giving it blue color. Potential energy, I will give it blue color. In blue color, we know if it's zero, then the kinetic energy will be maximum. Potential energy will be zero. Now, if we take the maximum height, potential energy will be maximum. Kinetic energy will be zero. Kinetic is red color. I gave it red color. Join both. Yes, we will get a graph like this. This is kinetic energy, and this is potential energy. Now, I am going to draw one more. This is total energy. This is the graph. Isn't it a simple graph? This is total energy. If height is zero, at maximum, kinetic energy will be zero. This is kinetic energy, and this is potential energy. Why is this straight line coming? Everything is a linear function. We know kinetic energy can be generally written as mgh, right? It can be written as mgh. Potential energy can be written as mgh. Both are linear functions. Because they are linear functions, we will only get a straight line. That's all. Okay? This same graph, when we study it in springs, will look like this. What is that thing? The potential energy of the spring is half kx squared. It is half kx squared. The potential energy of the spring is half kx squared. It is a parabolic curve. That's why it looks like this and like this. Similarly, if we look at kinetic energy, kinetic energy is half kx squared minus x squared. This is kinetic energy. Both of these are squared, so they are parabolic functions. That's why it looks like this. Don't get confused. This is a straight line, and that is a parabola. So, that derivation is finished. March. The total mechanical energy of free fall is constant. Repeated, repeated question. Now, let's move to the next question. 2022 improvement. The same question. State the law of conservation of energy. Prove the law of conservation of mechanical energy in the case of a freely falling body. Now, let's go to the spring. Spring. A derivation of work done or energy in a spring. Then the collision chapter is finished. Ready, children? Shall we look? Let's look. So, we are going to study the spring. What is a spring? It's an example of a variable force. Right? Which is conservative. Here, the force is a variable force. What is a variable force? It means the force changes according to the displacement. Now, I am going to stretch this spring. As the displacement of this spring increases, I will have to apply more and more force. Okay? So, in the case of a spring, we can say that a force will develop inside the spring. When we apply a force here, a force will develop inside the spring. The name of that force is restoring force. Because I have stretched this spring. I have stretched this spring. When I remove the force, if the spring is to go back, won't it need a force inside the spring? The force I apply will be stored in it. That's why it's called restoring force. To go back, the force I applied will be stored in it. So, look, children. So, in the case of a spring, the restoring force will be what? We know that as displacement increases, we will have to apply more and more force. So, can't we say the restoring force is proportional to what, children? Displacement. Let's put a negative sign. The negative sign is because the restoring force is in this direction, and the displacement is in that direction. That's why. So, we can say restoring force equals minus kx. K is a constant. The name of that is spring constant. Spring constant. What is K, children? Spring constant. Spring constant. Ready? Okay? Everyone understood, I hope. What we need here is potential energy. We are stretching this spring, stretching it, stretching it, stretching it. I am doing work by working, working, working to stretch the spring. The work I do will be stored inside that spring. In the form of potential energy. So, we are going to find the potential energy inside the spring. Come with me. Come with me. Ready? So, everyone look. Everyone look. So, to move this spring by a small displacement, let's call it dx, the work done by me, I am finding it. It's a small work done, right, dw? Yes or no? Yes, sir. Yes, sir. Sir, I am drawing the spring here. Just the spring. I am drawing it from here. Yes, spring. Initially, the spring was here. Initially, the spring was where? At this point. So, to move this spring by a small displacement, displacement dx, by a small displacement dx, the work done by me, I am finding it. It is dw. What is the equation of that work? It's me applying the force. My force is in this direction. The restoring force is in the opposite direction. So, work done equals, it's my applied force, right? It's my applied force, right? So, work done equals f into dx. Is there any doubt for anyone in this? No. No one has any doubt in this. We need to find the total work done, children. What will we do, children? We found the work done for a small displacement. It's f dx. To find the total work done, what will we do? Integrate it. Who will do it? Integrate it. So, the small work becomes big work. Integral. From x equals zero, if I call this distance x, up to where am I taking it? Up to capital X. This is the initial position. This is the final position. So, I have to integrate from zero to x. I have to integrate from zero to f. I have to integrate from zero to capital X. Up to there, I have to do work. f dx. f dx. We know that the force we apply and the direction of the force we apply are opposite to the restoring force. Right? Yes or no? The force I apply, if it's in this direction, the restoring force will be in the opposite direction. So, we can say F equals minus Fr. Like Newton's third law. Like Newton's third law. Now, we know the equation for Fr, right? What is the equation for Fr, children? What is the restoring force? It is minus minus kx. If we cancel it out, we get kx. So, the force we apply is what, children? It is kx. Put kx instead of force. Or, simply, it's okay to write this. Just put kx. Ready, children? So, work equals. Work equals integral from zero to x. Instead of force, let's put kx. kx. And then what is there? kx dx. Children, instead of force, sir put kx. Put kx. Now, what is it? We can take k outside, right? We took k outside the integration. Integral from zero to x dx. Now, look. When we integrate this, what will we get? K into, children, what will we get? x squared by 2. If we integrate x, the answer we get is x squared by 2. We have to put the limits. Finally, we have to put the limits from where to where we integrated. So, upper limit minus lower limit. What will we get? dw equals k into. Upper limit capital X squared by 2. Minus lower limit. So, the answer is what? dw equals kx squared by 2. This work. This work is converted into potential energy, right? So, this work is converted into potential energy. Potential energy of the spring is potential energy of the spring equals half kx squared. It is half kx squared. Ready, children? Ready, oh? Now, if this is found in terms of spring force, then here it will be negative. It will be negative. If it is found in terms of the force we applied, it is half kx squared. If it is found in terms of spring force, it will be negative. That's all. It's half kx squared. The potential energy there is an equation that all children should remember. And also, everyone look at this graph. We know the restoring force, the spring force, is called the restoring force. Here, the spring force is minus kx. Based on that, the graph will be a linear graph. Since it's negative, we draw it in this quadrant. Ready? Look. Look. This is how we get the graph, children. The slope of this graph. If we take the slope of this graph, we get Fs by what? We can write x. So, Fs is, if you look, what is it? It's kx. kx divided by x. So, we get the spring constant. Remember the slope of this graph. It is the spring constant. The area will be half kx squared by 2. Its magnitude is half kx squared by 2. So, we get energy. Remember, the area of this graph is what, children? The area of the graph is energy. Half kx squared. What will it be? It will be the potential energy of the spring. Just remember this graph. It will be a straight line. The slope is the spring constant. The area is the potential energy. Then, the question from the 2025 model. Everyone tell me the answer. I expect all children to tell the answer. I expect all children to tell the answer. Quickly, quickly, quickly, tell the answer. Everyone tell the answer, children. Everyone tell the answer. What is it? What is it? What is it? State whether the potential energy in the following case increases or decreases. Spring is stretched. We stretched the spring. We compressed the spring. Okay. When we compress the spring, when we stretch the spring, does the potential energy increase or decrease? Tell me, children. Everyone tell me. When we compress the spring, and when we stretch the spring, what happens? In both cases, pay attention. In both cases, the energy increases. In both cases, when we stretch it, it has potential energy. When we stretch it, there is potential. When will the potential energy be zero? When the spring is just resting, the potential energy is zero. When we stretch it, or compress it, the potential energy will be maximum. So, in both cases, the potential energy, children, increases. Potential energy increases. Ready, ready. Next question. Expression for energy of a spring. We know that. We did it. 2021 model. 2021 March questions. Is the force conservative or non-conservative? Tell me, children. Spring force, we know, is a conservative force. 2020 model question. Consider a spring applied with force within the elastic limit. Draw the variation of restoring force with displacement. The question is asking for the variation of restoring force with displacement from the mean position. The question is asking for the graph. Draw that graph. This is restoring force. This is the displacement. Okay, children? You can draw this graph in this direction and in this direction. This is our graph. Ready? Ready? Ready? Ready? Ready? What does the area of the graph represent? What will be the area of the graph, children? The area is half kx squared. Or minus half kx squared. The area will be energy. Energy stored. Ready, ready, ready. 2020 model question. Next question. Kinetic energy of an oscillating body. This will be studied in detail in oscillations. This will be studied in detail where? In oscillations. I will just show you that graph. I will just show you that graph. You will study this graph wonderfully in oscillations. Yes, this is our graph. This is our graph. Okay. We need to put a lid on top of this. This is our graph. Yes. Let me explain each part. Let me explain each part. I will give this here. So, this will be zero. At the mean position, the spring is oscillating. x equals zero. When the spring goes this way, we call it minus x. When the spring goes this way, we call it plus x. This distance is called x. Okay? At the mean position, we know the spring has zero potential energy. So, what graph will this be? This is the graph of potential energy. At the mean position, kinetic energy will be maximum. This is the graph of kinetic energy. This is total energy. We will study this in detail where? In oscillations. Okay? Will uncle teach you how this graph came? It comes in oscillations. The graph comes in oscillations. Tomorrow, uncle will teach you. Ready, children? Now, look at this question. Calculate the work done by the force from the given force. You have to find the work done. Force displacement graph. What will be the work done, children? Tell me. What will be the work done? Work done is the area, right? This is our force displacement graph. I told you, if they ask for the area of any graph, just multiply y into x. Right? So, look. The area is force into displacement. That is work. So, we just need to find the area. Shall we find it? We can find it simply. There are two parts. This is the first part. This is the second part. Okay, children? So, the area is total area equals A1 plus A2. What will be the area of the first part, you tell me? The area will be. Area equals. Here, the length is 10. The breadth is 100. So, 10 times 100. Plus. What about the second part? The second part is. We all know this is a trapezium. It's a trapezium. There are two parallel sides. So, to find the area of a trapezium, we all know. Half into. Half into A plus B. A is this side. It is 10. B is this side. It is 50. 100 plus 50. A plus B into height. The distance between these two parallel sides is the height. That is 10. So, let's cancel it out. What do we get, children? Here, we get 5. So, it is 1000. Plus. What is it? 150. 150 times 5. What do we get, children? 1000 plus 0. 5 times 5 is 25. Carry over 2. 5 times 15 is 75. Plus 2 is 77. So, we get 750. So, 1000 plus 750 is 1750 joules. That is the work done there. So, the area under the force displacement graph is work done. We are going to move to the final section, collision. Before that, let's study power and then go to collision. Power. What is this power? Power is the rate of work done. The work done by us in a unit of time is called power. What work is done in one second is called power. So, power is work by time. Work by time is called power. Ready, children? Next, an equation for power. Force into dash. Let's see. We are going to tell another equation for power here. Work is force into displacement. Displacement by. So, let's see. Here, we are going to tell another equation for power. Work is force into displacement. Displacement by.

Time means velocity. So, force into velocity is another equation for power. Thus, we can say power equals force into velocity. Ready? Ready, man? Ready? Ready, man? Ready, man? Come on, come on. Let's go to the next question. Are we going to the next question? 2020 model question. Kilowatt-hour is the unit of dash. Kilowatt-hour. What is kilowatt? Watt is the unit of power. Hour is the unit of time. Power into time, what is it, children? Power into time, what is it? Power into time is work, isn't it? Isn't it work? Power into time. So, kilowatt means power. Power into time means work. So, what is this the unit of? It is the unit of energy or work. It is the unit of energy or work. The KSEB bill that comes to our house is in kilowatt-hours. The energy we consume is in kilowatt-hours. We say it is the unit of energy, kilowatt-hour. Then we have reached the last topic. Collision. Collision. Collision. There are three types of collision: elastic collision, inelastic collision, perfectly elastic collision. What are they? Elastic, inelastic, perfectly elastic. Shall we look at them one by one? First, we are going to look at elastic collision. If it is an elastic collision, remember, children, momentum is conserved. Momentum is conserved. Kinetic energy is conserved. Everything is conserved. If it is an elastic collision, okay? For example, collision between subatomic particles. That is an elastic collision. Now, if it is an inelastic collision, momentum is conserved, but kinetic energy is not conserved. In any collision, momentum is conserved. Momentum is conserved. Kinetic energy is not conserved. Total energy is also conserved. Total energy is also conserved. Okay, children? But kinetic energy is not conserved. The collision between two vehicles, the collision between us, all are collisions. Collision when playing carrom board is an inelastic collision. Now, another type of collision is perfectly elastic collision. That is, after the collision, the two colliding particles stick together. After the collision, they stick together. They will stick together. So, if we throw mud at the wall, won't the mud stick to the wall? What collision is that? It is an inelastic collision. Okay, children? In an elastic collision, momentum is conserved, kinetic energy is not conserved, total energy is conserved. Okay. What is conserved in all collisions? Momentum. Momentum is conserved in all collisions. Ready, ready. 2023 model question. 2023. Next is elastic collision in one dimension. It's a small derivation. Let's finish it quickly. A small derivation. Elastic collision in one dimension. In one-dimensional motion, it is an elastic collision, children. There are two bodies. Their masses are m1 and m2. There are two bodies. Their masses are m1 and m2. Their velocities are u1 and u2. Okay? They collide. When they collide, the velocity of m1 becomes v1, and the velocity of m2 becomes v2. This collision is elastic. It is an elastic collision. So, look. They have two bodies. They have initial velocities. After collision, they have final velocities. So, we know the law of conservation of mass. Not the law of conservation of mass, but the law of conservation of momentum. Momentum before collision equals momentum after collision. Momentum before collision, find it. What do you get? m1u1 + m2u2. After collision, what is it, children? m1v1 + m2v2. Let's bring the mass terms together. Let's bring m1 and m2 together. So, let's bring m1 terms together. m1u1. This one will come here. Sorry, this one will come here. So, m1v1 will come here. Minus m1v1. Equals. Similarly, what is here? m2v2 is here. Who will come here? m2u2. Let's take m1 as common. So, what do we get? u1 minus v1. Equals. If we take m2 as common, children, v2 minus what do we get? We get u2. Call this equation number one. So, using conservation of momentum, we have formed an equation. Similarly, the law of conservation of kinetic energy. Kinetic energy before collision equals kinetic energy after collision. Similarly, we can form an equation. It's just that a square will appear everywhere. A square will appear everywhere. So, kinetic energy before collision is half into half into m1u1 squared plus half into m2u2 squared equals half into m1v1 squared plus half into m2v2 squared. Let's cancel everything. Let's cancel everything. Half is there everywhere. Half is there everywhere. Let's cancel it. Let's bring the m1 terms together. Let's bring the m1 terms together, children. Pay attention, children. m1u1 squared minus m1v1 squared equals m2v2 squared minus m2u2 squared. Take m1 as common, children. If we take m1 as common, u1 squared minus v1 squared. Equals. If we take m2 as common, children, u2 squared minus Sorry, there is a small change here. Here it is v2. Here it is v2. This one is here. This one is coming here. So, here it is u2. Here it is u2. So, u2 squared minus What do we get, children? Sorry, v2 squared minus u2 squared. v2 squared minus u2 squared. Put this as equation number two. We need to find the relative velocity before collision and the relative velocity after collision. For that, we don't need mass. To cancel the mass from both sides, we can divide this by this. If we do equation 2 divided by equation number 1, we get the answer. I'll borrow equation 1. I'll copy it. I have copied it. I am going to borrow equation 2 as well. Pay attention. We wrote equation number one. We are copying equation number two. Come on. Come here. Yes. We are dividing both. We are dividing both equations. When dividing, who can we cancel? m1, m2, everything can be cancelled. u1 squared minus v1 squared. We know that a squared minus b squared is (a plus b) into (a minus b). So, we can write u1 minus v1 into u1 plus v1. v1 divided by u1 minus v1 equals v2 minus u2 into v2 plus u2. Divided by v2 minus u2. Cancel, cancel. Don't look at anything. Just cancel it. Just cancel it. If you write that last thing, you get the answer. What is the last answer? u1 plus v1 equals v2 plus u2. Now, to bring the relative velocity before collision, let's bring this here. Yes. u1 minus u2 equals v2 minus v1. We can take a minus common if needed. If we take a minus common, v1 will become plus, and v2 will become minus. Isn't it great? Didn't we get it perfectly? What is this called? This is called relative velocity. What would it be before collision? What is this called? This is called relative velocity. Relative velocity after collision. It will be asked in the exam. If asked to prove that relative velocity before collision equals relative velocity after collision, children, write this. Let me teach you one more small thing. What about two dimensions? We must teach that. Yes, two-dimensional collision. There is no figure, don't worry. I am here, so why worry about the figure? Isn't it? We need to draw the figure. It would be better if the figure was imported. We won't have to waste time drawing the figure. Do I have it? Yes, I have it. What is there without a figure? Isn't it? It would be good to have the figure. Only then will we feel like doing it. So, I will import the figure. Everyone, keep listening. Drink water, children. Go, you have taken so many classes, you are tired. Go and drink water. I will find the figure. Figure, figure, figure, figure. I think it's with me. Find it. I think I am drawing in class. That's the thing. I have only this heading in all the slides. Guys, I am drawing everything while sitting in class. Yes, yes. Brother will give it now. Guys, brother will give it now. I will have to draw the figure at the end. I can draw it. There is a modified one with me. Did you get it? Turn it on. Come, come, come, come. The figure has come. Now we are going to rock it. Yes, come on, come on. Come on. Empadodocument. Yes, what a beautiful figure. Only the heading is missing. The same heading, exactly like that. Only the figure is missing. Look, children. So, we have a mass called m1. I didn't draw this figure. We have a mass called m1. Its velocity is u1. Similarly, we have a mass called m2. Its initial velocity is u2. They collide. Since it is two-dimensional, one goes here with velocity v1, and another goes here with velocity v2. What we need to look at here is that since it is two-dimensional, initially there is no problem. Both are moving in the x-direction. Finally, if you look, this one and this one are in the x-y. If we resolve this, one component will be in the x-direction, and one component will be in the y-direction. That is, there is momentum in the x-direction for this body, and there is momentum in the y-direction. Similarly, this body's momentum has momentum in the x-direction and momentum in the y-direction. So, what happens when we look here? We know that according to the law of conservation of momentum, momentum before collision equals momentum after collision. We need to split this into two. What is momentum before collision in the x-direction equals momentum after collision in the x-direction? Similarly, we need to do it in the y-direction. That's what we are going to do. So, all that is written. Conservation of momentum parallel to the x-axis. So, can we find the momentum in the x-axis? So, if we look, we know that before collision, the momentum is m1u1 + m2u2. So, write the momentum before collision: m1u1 + m2u2. Now, after collision, we only need to write the momentum in the x-direction. Look. The velocity of this one in the x-direction is v1 cos theta1. The velocity of the second body in the x-direction is v2 cos theta2. So, after collision, we can say the momentum is m1v1 cos theta1 + m2v2 cos theta2. Writing this much is enough. Yes. What is it, children? Momentum. Momentum before collision in the x-direction equals momentum after collision in the x-direction. Ready? Now, let's find it in the y-direction. Momentum before collision in the y-direction equals momentum after collision in the y-direction. Shall we find it? If you look, initially there is no momentum in the y-direction. Both are moving in the x-direction. So, initially, there is no momentum in the y-direction, so we can call it zero here. Now, after collision? After collision, there is momentum upwards: m1v1 sin theta. There is momentum downwards: m2v2 sin theta. They are in opposite directions. So, we can write m1v1 sin theta1 minus what comes, children? m2v2 sin theta2. Why is there a minus here? One momentum is upwards, and the other momentum is downwards. So, one is plus, and one is minus. Ready? If needed, we can bring this one here. Simple. Just writing this equation is enough. That's our task. Now, the last and final: conservation of kinetic energy. Since kinetic energy has no direction, don't worry, don't worry. Kinetic energy before collision equals kinetic energy after collision. What will be the kinetic energy before collision? Half m1u1 squared plus half m2u2 squared. Now, after collision? Simply, you know the velocity. m1v1 squared plus half into m2v2 squared. Here, we don't need to find the x and y components separately because kinetic energy is a scalar quantity. It is because velocity is a vector quantity that we found x and y separately. This is a scalar quantity, so we don't need to do anything like that. If you know the velocity, you can find the kinetic energy of that body. No matter which direction it is, it doesn't matter to us. So, that's all for this chapter. We have finished the chapter. Is there a last question? Quantity conserved in elastic collision. Tell me, children. What quantity is conserved in elastic collision? We know that in an elastic collision, total linear momentum is conserved. Total kinetic energy is not conserved. Ready? Finished. Is there anything left in this? No slide. Yes, there is. Laws of motion. A blank slide has come. Next is laws of motion. Brother will come with that. Brother will teach two chapters and leave. Which ones are they? Laws of motion and gravitation. When will it come? After two chapters, it will be around 6 o'clock. Okay. Okay. It's around 6 o'clock. Wait a minute. Okay, fine. The questions being asked are about collision. I will tell you about collision. In collision, collision in one dimension is asked the most in exams. Two-dimensional is very, very, very rarely seen. It has only been seen in an improvement exam in some year. Questions from it will not be repeated. But you must understand the different types of collisions. That is, our, what to say, elastic collision, inelastic collision. Don't we have types of collisions? You must learn the properties of each precisely. Also, in collision in one dimension, the derivation that relative velocity before collision is numerically equal to relative velocity after collision is very important. Jaseel sir has taught you. Learn it precisely. So, children, shall we move on to our next chapter? Let's move on to the most important chapter called Laws of Motion. If you are ready, all the younger brothers and sisters in the chat box, as always, if you put a muscle, we will start our performances. Are you ready? If you are ready, respond. We are going to learn two very interesting chapters. Not just Laws of Motion, but also Gravitation. I will teach you and then wind up. We are going to learn the two most important and crucial chapters excellently. Are you ready, my treasures? Yes, if you are ready, respond. Nivetha, actually, I didn't go into those derivations because the questions related to them are not seen in exams in such depth. Nivetha, don't think otherwise, dear. Ready. So, guys, we are moving forward. Shall we start, dear? Respond. Why are you not responding? Everyone turn on. I am on. Why are you not on? Is it the tiredness after eating food? If you are tired after eating food, it won't work. You have to stand by me. Everyone stand by me. We will learn excellently. We will finish Laws of Motion excellently. You already know this, right? Laws of Motion has questions worth about eight marks. Gravitation also has a possibility of questions worth about eight marks. We are going to secure about 16 marks, or the passing marks, in the next two and a half or three hours. You all have to stand by me, brother, firmly. To make everything set, we are going to start. We are going to start the chapter called Laws of Motion. In the previous chapters, we learned precisely about the parameters of a moving body, whether it was motion in a straight line or motion in a plane. Velocity, acceleration, displacement, distance, time of flight, horizontal range, maximum height. All these were parameters of a moving object. Their properties. But now we are going to learn about what causes that motion. That is, we are done. Now we are going to move into dynamics. In this chapter, we are going to deal with dynamics. That is, mainly about force and the laws related to that force. We are going to learn about force and the laws related to it. That is, Newton's first law, second law, third law. We are going to learn all of them excellently. We are going to understand them precisely. Let's not delay. First, force. What is force? Force is an external agent that changes the state of a body. Force. If we apply a force to a stationary object, that object will start moving. If we apply force to a moving object, that object will come to rest. The external agent that we apply to change the state of an object is called force. No doubt, right? No one will have any doubt. Don't you see two types of forces here? What are those two types of forces? First, Jaseel brother is pulling me. What is the name of the force of pulling? It is called pull. Pull. The force of pulling is called pull. What is pushing called? Jaseel brother pushed me for so long. The name of the force of pushing is called push. So, these are two types of forces. We can classify them as internal and external. Here, at the most basic level, we can classify force in two ways: pull and push. Pull means pulling. Push means pushing. Don't forget, okay? Next, what is force? Force is what causes acceleration in a body. Similarly, force is a vector quantity. It has magnitude and direction. Understand this. Now we are getting to the most important part. What is the unit and dimension of force? Everyone, comment the unit. You all must know. What is the unit of force, children? There should be no doubt. Come on. There should be no doubt. The unit of force is Newton. Newton is the unit of force. It is written in lowercase letters, okay? Newton is the unit of force. It is symbolized by capital N. When writing the name, you should use lowercase letters. Let me tell you another unit. If force equals mass into acceleration, how can we write it? Kilogram meter per second squared. We can write it like this. One kilogram meter per second squared is equivalent to one Newton. One kilogram meter per second squared is equivalent to one Newton. Understand that too. Let me tell you another thing. You have learned about CGS system of units, right? In the CGS system, in what unit do we express force? In the CGS system, in what unit do we express force? Nivetha is saying precisely. What unit is it? It is dyne. So, how many dynes is one Newton? Tell me, children. How many dynes is one Newton? Nivetha is saying excellently. One Newton is equal to 10 to the power of five dynes. It might be asked in the exam. It might be possible to ask in the exam. It is very good to learn it. Learn the CGS system of units too. Newton is in SI. Force is expressed in dynes in the CGS system of units. One Newton is equal to 10 raised to the power of 5 dynes. Understand this unit conversion precisely. Now, let's move on to the dimension. The dimension is very simple. We all know it. m a, right? It is an equation we have learned. Even though we are learning about it again, so the dimension of force. The dimension of mass is capital M. The dimension of acceleration is LT raised to the power of minus 2. What do we get? MLT raised to the power of minus 2. This is the dimension of our force. Also, very important. You learn this in the chapter Units and Measurements. However, understand it here too, okay? Okay, ready. Shall we move forward? Next is a simple question. But that question is very important because it has been asked in the last two years' question papers. That is its importance. It is a one-mark question. Comment. What will be the net force acting on a body in equilibrium condition? What will be the net force acting on a body in equilibrium condition? Everyone, comment. All my younger brothers and sisters, comment. What will be the net force acting on a body in equilibrium condition? There should be no doubt. It will be equal to zero. In equilibrium condition, the net force will be zero, children. You don't need to have any doubt. It will be equal to a big zero. Understand that. Next, linear momentum. Momentum means the product of mass and velocity. You all know this, right? Simply, that's enough for us. So, momentum equals mass into velocity. Momentum is a vector quantity. Its direction will be the same as the direction of velocity. Momentum is the product of mass and velocity. Momentum is a vector quantity. Understand that its direction will be the same as the direction of velocity. Fine. Next, unit and dimension of momentum. Let's move on to the unit. Momentum equals mass into velocity, right? So, what will be the unit? The unit of mass is kilogram. The unit of velocity is meter per second. So, we get kilogram meter per second. Now, coming to the dimension. The dimension of mass is capital M. The dimension of velocity is LT raised to the power of minus 1. So, what do we get? MLT raised to the power of minus 1. Didn't we get it precisely? Didn't we get the dimension of momentum? What is another physical quantity with the same dimension as momentum? What is another physical quantity with the same dimension? I have taught you. Comment. What is the physical quantity that has the same dimension as momentum? I believe you will remember. Yes, great. Awesome. Impulse. It is impulse. Momentum and impulse have the same dimension. Understand that momentum and impulse have the same dimension, all my children. Important, right? Moving on to the next. We are moving on to the important topics of this chapter. Newton's first law of motion. What does Newton's first law of motion say? Every body continues in its state of rest or uniform motion until an external force acting on it. Every object will continue in its state. Until when will it continue? Until an external force acts on it. If it is a moving object, it will keep moving. If it is a stationary object, it will remain stationary. Until an external force acts on it. If an external force acts, the state will change. So, every body continues its state of rest or uniform motion until an external force acting on it acts externally. If an external force acts on it, the state will change. So, Newton's first law of motion says that this Newton's first law of motion introduces a concept. What is that concept? Tell me. Yes, Minha is saying excellently. It is inertia. Newton's first law of motion provides us with the idea about inertia. What is inertia? It is actually an inability. It is the inability of a body to change its state by itself. Inertia is the inability of a body to change its state by itself. The example I always give is the example I always give you. Let's take two people. Let's assume this is our Jaseel sir. Let's assume this is our Jaseel sir. And we always take someone else. Who is it? Yes, let's take an elephant. Right? Let's take an elephant. See, the elephant doesn't have ears. Let's add ears. Okay, ready. A trunk and no trunk. A tail. Ready. So, Jaseel sir is here, and an elephant is here. Okay? Do I need to write it down? JMK and elephant. These two are sleeping. Both are sleeping. I go and say, "Wake up and run," I am the elephant's keeper. Who will wake up and run first? Jaseel sir is sleeping. The elephant is sleeping. If I go and say, "Wake up and run," who will wake up and run first? Everyone, tell me. Who will change from the sleeping state to running first? There should be no doubt. It will be our JMK who will wake up and run first. Right? JMK is sleeping. If told to wake up and run, Jaseel sir will wake up and run. But will the elephant run like that? The elephant cannot run. The elephant will wake up, stretch, adjust its trunk, adjust its tail, and then run. So, the elephant will take a little more time. What is special there? Yes, Jaseel sir's inertia is less. The elephant's inertia is more. The elephant cannot change from rest to motion quickly. But Jaseel sir can change from rest to motion quickly. What is the reason for that? What is the reason? Inertia is the inability to change its state by itself. Or the inability to change from rest to motion. It is less for Jaseel sir. Jaseel sir will change quickly. But the elephant has more inertia. The elephant cannot switch from rest to motion quickly. What is the reason for that? The question is, why does the elephant have more inertia? What is the reason for that? Yes, the elephant has more mass. Because the elephant has more mass, it has more inertia. Therefore, the elephant cannot change from rest to motion quickly. What is the most important parameter that inertia depends on? It is mass. We can say that mass is the measure of inertia. Right? If mass increases, inertia increases. If mass decreases, inertia decreases. Understand that. This is very, very, very important. Everyone must learn this, okay? So, there are three types of inertia. First, inertia of rest. Second, inertia of motion. Third, inertia of direction. These are the three types of inertia. These are the three types of inertia. Inertia of rest, inertia of motion, inertia of direction. I will tell you the definition from here itself. Inertia of rest means the inability of an object to change its state of rest.

Inertia of rest, meaning the inability of a body to change its state of rest, is inertia of rest. What would inertia of motion be? It is the inability of a body to change its state of motion. It is the inability of a body to change its state of motion. What would inertia of direction be? It is the inability of a body to change its direction. Inertia means inability. If it is the inability to change the state of rest, it is inertia of rest. If it is the inability to change its moving condition, it is inertia of motion. If it is the inability to change direction, it is inertia of direction. That's all, it's simple. First, inertia of rest. It is the inability of a body to change its state of rest by itself. The inability to change its state of rest is called rest. Shall I give an example? Yes, I am standing in a bus. The bus driver suddenly started the bus. Aren't we falling backward? We are falling backward. What is the reason for that? The reason for that is inertia of rest. The reason for that is inertia of rest. Let me go into the details of it precisely. Pay attention, I am standing in the bus. When the bus suddenly starts forward and moves forward, our legs are in contact with the bus, right? Our legs are in contact with the bus. So, when the bus moves forward, since our legs are in contact, our legs also move forward. But our upper body, you know, the upper body has more inertia. They prefer to remain in a state of rest. The upper body does not want to move forward with the bus. So, even as the legs move forward, the upper body, due to its inertia, remains still. Because it remains still, we fall backward. So, when the bus suddenly starts, what is the reason we fall backward? It is inertia of rest. What is the reason we fall backward, children? Understand that it is inertia of rest. It is important. Next, let's move on to inertia of motion. It is the inability of a body to change its state of motion. Let me give a similar example. Suppose I am standing in a moving bus. If the bus driver suddenly applies the brakes, what will happen? I will fall forward. What will happen? I will fall forward. What is the reason for that? Let me explain it similarly. The legs are in contact with the bus. So, when the bus suddenly stops due to braking, the legs, being in contact, also come to rest suddenly. But our upper body has more inertia. It prefers to remain in a state of motion. It does not want to suddenly change to a state of rest. The upper body prefers to keep moving. Due to its inertia, it will tend to move forward. That is why we fall forward. That is why we fall forward. Did you understand? So, when a moving bus suddenly stops, the reason we fall forward is inertia of motion. It is because we prefer to continue in the moving state that we fall forward. Did you understand it clearly? You must study. Let me give another example. Athlete Usain Bolt. Usain Bolt is running 100 meters. He runs and runs and reaches the finish line. He reaches the finish line, and will he stop immediately? He will never stop. Usain Bolt will run another 50 meters. Why is he running like that? He cannot change from motion to rest as if a switch is turned off. Even after crossing the finish line, there is inertia of motion. Therefore, only after running a little further will he be able to change from motion to rest. Did you understand? He has to continue in that moving state for some more time. Only then can he gradually change to a state of rest. A body cannot change from rest to motion or from motion to rest as if a switch is turned on. Fine. Last one. Let's move on to that, which is inertia of direction. It is the inability of a body to change its direction. Pay attention, children. I tie a stone to a rope and am spinning it. We all know that if a body exhibits circular motion, in which direction will the direction of its velocity be? Without any doubt, the direction of its velocity will be tangential. Right? The velocity will be tangential to the circular path. When the rope breaks, how will this stone move? It will move tangentially. Did you understand? Because the stone prefers to continue in its direction. So, even if the string or rope breaks, the stone will move tangentially. Because that body prefers to continue in that direction. It does not want to change its direction. It is the inability of a body to change its direction by itself that is called inertia of direction. Understand it precisely. You must know this concept. So, inertia of motion means the inability to change the state of motion. Inertia of rest means the inability to change the state of rest. Inertia of direction means the inability to change direction. Understand that it is all an inability. Now, shall we work out some questions related to this? There are not many questions, just one or two questions to help me understand if the concept has reached you. What is the measure of inertia? Tell me. What is the measure of inertia? All brothers and sisters, can you tell me? What is the parameter that we can call the measure of inertia? Without any doubt, it is mass. Mass is the measure of inertia. Mass is the measure of inertia. Are you all ready? Comment, guys. Why are you just watching? Be active in the chat box, leave the trivialities, and comment. Comment, children. Yes, mass is the measure of inertia. Next question, a possible question asked in the 2020 improvement exam. When a moving bus suddenly stops, passengers fall forward. This is due to what? Can you comment? When a moving bus suddenly stops, what is the reason passengers fall forward? What is the reason? Without any doubt, the reason is inertia of what? Inertia of what? I taught you, right? Yes, inertia of motion is the reason. The reason is inertia of motion. When a moving bus suddenly comes to rest, the reason passengers inside fall forward is inertia of motion. Understand that. Yes, this is inertia of motion. Ready? Great. Everyone is responding wonderfully. Shall we move on to the next question asked in all these years? What question would that be? What question would have been asked in all these years? Don't you all know? It is Newton's second law of motion. It was asked in the March 2025 exam as well: Newton's second law of motion. What does Newton's second law of motion state? The rate of change of momentum is directly proportional to the applied external force. The rate of change of momentum is directly proportional to the external force we apply. The rate refers to how much it changes with respect to time, how much change in momentum occurs over time. That is the rate of change of momentum. Understand that. External force is directly proportional to the rate of change of momentum, or we can write dp/dt. It is okay to represent it with delta, and it is okay to write dp. dp means change in momentum, and dt means time interval. So, F is proportional to dp/dt. To remove proportionality and make it equal, whom do we bring in? Tell me. We bring in a proportionality constant, right? Here, the proportionality constant we bring in is called k. So, F is proportional to k * dp/dt. Equation number two. Here, the proportionality constant we introduce is k. So, we can say F = k * dp/dt. Understand that. What is the value of k? You don't need to have any doubt. The value of k is 1. So, we can say F = dp/dt. Equation number three. Now, each of my children must tell me. I believe you will tell me. Let's replace momentum. Tell me, what is the equation for momentum? I taught you, didn't I, children? Tell me, what is the equation for momentum? Everyone comment. What is the equation for momentum? The equation for momentum is momentum = mass * velocity, right? Yes. So, F = d * (mass * velocity) / dt. Equation number four. Mass * velocity / dt. Equation number four. Shall I take the mass here commonly out? Mass is constant, I am going to take it out commonly. So, F = m * dv/dt. Equation number five. My children, tell me, what is d/dt? Change in velocity by time means what? It is acceleration. So, Force = mass * acceleration. This is equation number six. Force = mass * acceleration. This is our sixth equation, equation number six. Ready? Did you understand it precisely? Is it set? Yes, yes, Maria is commenting with precise and detailed answers. I believe everyone understood this. Okay, children? No doubts? Okay, children? No doubts? If it is clear, put a fire emoji in the chat box. Then, shall we work out just one question to support this? We will look at just one question. We will only look at one important question. A constant retarding force of 50 Newtons is applied to a body of mass 20 kilograms. A force of 50 Newtons is exerted on a body of mass 20 kilograms. The initial velocity of the body is 15 meters per second. How long will it take to stop? The question is, how long will it take for the body to come to rest? Everyone, read the question carefully one more time. A body is moving. Its initial velocity is 15 meters per second. An external force is applied to the body. That force is called a retarding force. A retarding force means a force applied in the opposite direction, opposing the motion. So, that retarding force is 50 Newtons. If so, how long will it take for the body to come to rest? Shall we look at it? I will explain it easily. Pay attention. Imagine this is a body. Okay? This is the body. What is the mass of this body? Mass is 20 kilograms. So, m = 20 kilograms. What is the velocity at which the body is traveling? It is traveling at a speed of 15 meters per second. Okay? In which direction is it traveling? Suppose we assume the body is traveling in this direction. So, a force acts in the opposite direction, opposing its motion. A force acts in the opposite direction. What is that force? The force is 50 Newtons, acting in the opposite direction. Understand that. These are what are mentioned in the question. There is a body, its mass is 20 kilograms, and it is traveling at 15 meters per second. So, a force of 50 Newtons acts in the opposite direction. Fine. Okay. Shall we write down the given data? What is the given data? Let's write the initial mass. Mass = 20 kilograms. Here, you must understand that this is a retarding force, meaning a force acting in the opposite direction, opposing the motion. So, how should we write it? We should write it as -50 Newtons. Don't forget the minus. Also, the initial velocity u is 15 meters per second. The final velocity v is 0. What do we need to find? We need to find the time. How much time will it take for the body to come to rest? We have to find the time. Shall we look? Children, many are commenting with answers. Let's see if they are correct. So, the formula we will definitely use to find time is v = u + at. To substitute in this formula, we need acceleration. So, let's find acceleration first. F = mass * acceleration. So, acceleration = force / mass. Force is -50, mass is 20. If we cancel, what will we get? 0, 0 goes away. -5/2 = -2.5. Meters per second squared. We got acceleration as -2.5 meters per second squared. Now, can we find the time? Look, v = u + at, right? From this, we are going to find time. So, time = or look, v - u = at. If so, time = v - u / a, right? v is 0, u is -15. Acceleration is -2.5. Equal to -15 / -2.5. The minus and minus go away. 15 / 2.5 is 6. We get 6 seconds. The answer is 6 seconds. How much time is it, children? It is 6 seconds. Understand that. It is very important. You must study. Is it clear? Yes, Maria, the answer is correct. Yes, Rejimol, the acceleration is correct. Did everyone get the final answer? Ajil, the answer is correct. Yes, the final answer is -15 divided by -2.5. The answer is 6 seconds. That is our correct answer. I believe everyone understood. Then, shall we move on to the next topic? Next topic is impulsive force. Impulsive force means it is a large force acting for a short time period. A large force acting for a short time is called impulsive force. For example, hitting a ball with a bat and sending it away is an impulsive force. Hitting a nail with a hammer is an impulsive force. But the characteristic of impulsive force is that it is time-varying. It is not a force with a constant magnitude. Therefore, measuring impulsive force is not an easy task. So, what do we measure? Yes, therefore, what we measure is the total effect of the force called impulse. So, the total effect of impulsive force is called impulse. So, the name given to the effect of impulsive force is impulse. Did you understand? Impulsive force is time-varying. It is not easy to measure. So, we measure the effect of that impulsive force. What is the name given to the effect of that impulsive force? It is called impulse. So, impulse = force * time. Understand that. Impulse = force * time. So, we can say I = F * t. That's simple. Ready? F * t. So, what will be its unit? Tell me quickly. What will be the unit of impulse? Impulse = force * time. What will be its unit? The unit of force is Newton, and the unit of time is second. So, what will we get? Impulse = Newton-second. What will we get? Newton-second. Yes, it is Newton-second. The answer is correct. You are all commenting with wonderful answers. Let's move forward. Shall we move on to the next very important, very significant small topic, children? Let's move on to a small topic. That is the relation between impulse and momentum. We are going to derive the relation between impulse and momentum. Stay with me. We are going to look at the relation between impulse and momentum. Understand that. We know the equation from Newton's second law: F = dp/dt. Let's multiply dt here. So, F * dt = dp. Children, what is force * time? Everyone comment. What is force * time? Without any doubt, force * time is impulse. Force * time is impulse. So, we can write impulse = dp. That is, impulse = change in momentum. Yes, the answer is correct. Change in momentum is equal to impulse. Understand that change in momentum is equivalent to impulse. That is very, very, very important. Shall we move forward? Yes, guys, we are moving forward. We are moving on to the next very important topic, that is Newton's third law of motion. It is a very small topic, children. It was asked in the 2021 improvement exam. It is not asked very often. The second law is the law that is repeated the most. The third law is asked very rarely. So, what comes according to Newton's third law? Roman, come with your face washed, there will be no problem, everything will be fine. Ready, ready. What does Newton's third law state? For every action, there is an equal and opposite reaction. Isn't it the law that is most convenient for us to say and learn? The third law states that for every action, there is an equal and opposite reaction. If we consider firing a bullet, the gun applies a force to the bullet, and that is why the bullet moves forward. That force is called action. Similarly, the bullet applies a force to the gun, and that is the reaction. That is why the gun moves backward. The backward motion of the gun is called the recoil of the gun. We are about to study that. Is it clear? So, the force the gun applies to the bullet is called action, and the force the bullet applies to the gun is called reaction. Action and reaction are equal and opposite. Is it clear? For every action, there is an equal and opposite reaction. That is what Newton's third law states. Now, there is a small question, a small doubt. Even though action and reaction are equal and opposite, why do they not cancel each other out? Even though action and reaction are equal and opposite, why do they not cancel each other out? Can you comment? What is the reason for that? Why would action and reaction not cancel each other out? The answer is simple. Action and reaction act on different bodies. If action is on the bullet, reaction is on the gun. They act on different bodies. Action and reaction are equal and opposite. But they do not cancel each other. But they do not cancel each other since they act on different bodies. Action and reaction act on different bodies. That is why they do not cancel. Fine. Moving on to the next question. Can you guess what the most important question asked in all these years is? Can you guess what this most important question asked in all these years is? If you can guess, please guess. What is this question? What is the question asked in the March 2024 exam, including the March 2024 model? Without any doubt, it is the law of conservation of linear momentum. That is the law of conservation of linear momentum. I will explain it very simply. Just stay with me. We know from Newton's second law that F = dp/dt. Suppose our external force is zero. Let's assume we are not applying any external force. Let's assume the external force is zero. So, let's put zero for force. Zero = dp/dt. If we multiply dt here, it also becomes zero, right? So, dp = 0. Let's call this equation number one. Let's call this equation number two. This is equation number three. Let me explain it once more. This is the formula from Newton's second law. If no external force is applied, F = 0. So, 0 = dp/dt. If we multiply dt here, it will also become zero. Children, tell me, change in momentum is zero. Then what will be the momentum? dp means change in momentum. If the change in momentum is zero, what will be the momentum? Comment, everyone. If the change in momentum is zero, what will be the momentum? You don't need to have any doubt. Momentum will be a constant. P = constant. P = constant. This is taken as equation number four. Momentum will be a constant. Understand that momentum will be a constant. This is the question asked in all these question papers. It is a very simple two-mark question. If no external force acts, the total momentum will be constant. That's all. But this is the question asked in all these question papers. It is an important question to study. Shall we move on to the next topic? Next topic is recoil of a gun. We are going to find the recoil speed of a gun. It is the recoil of the gun that we are going to study next week. Stay with me. Everyone, stay with me. Recoil of the gun. Pay attention. This diagram alone is enough. This is our gun. The mass of the gun is capital M. The recoil speed of the gun is capital V. Recoil speed means when firing, the gun moves backward, right? That backward motion is called recoil. So, that recoil speed is capital V. Okay? Now, in the case of the bullet, the mass of the bullet is small m, and the speed of the bullet can be considered as small v. Is it clear? Did you understand it precisely? No doubts? No. Now, we are going to proceed to our equation. We are going to find the recoil speed. According to our law of conservation of linear momentum, the external force is zero. If the external force is zero, what will be the momentum? Without any doubt, momentum = a constant. Right? Momentum will be a constant. If momentum is constant, pay attention. Total momentum before firing = Total momentum after firing. Equation number one, equation number two, equation number three. Total momentum before firing = Total momentum after firing. Right? What is the momentum of the gun and bullet before firing? Before firing, what will be the momentum of the gun and bullet, children? Aren't both at rest? In a state of rest, definitely both are at rest. Therefore, before firing, the momentum of both will be zero. Because both have no velocity. If there is no velocity, is momentum not zero? Definitely. Therefore, before firing, momentum will be equal to what? It will be zero. What is the momentum after firing? The momentum of that bullet is small m * small v. The momentum of the gun is capital M * capital V. Equation number four. This is our fourth equation. Ready? The momentum of the bullet is small m * small v. For the gun, it is capital M * capital V. Everyone must study this. It is very important. Let me rearrange this. Pay attention. If m * v = what will come? -M * V = what will come? -M * V. If so, V = -m * v / capital M. This is equation number five. This is equation number six. So, capital V = -m * v / M. Everyone must study this. This is the final answer for recoil speed. Capital V = -m * v / M. Ready? Did you understand it precisely? Is everything set? Is it clear? If it is clear, let's move forward. If you are ready, let's move forward. Shall we work out questions related to this? Yes, Niveda, everyone is commenting with the right answer. Gopi, Gopika, I think it is Gopika, or Gopi? Whoever it is, they are commenting with the correct answer. Let's move on to two things related to this. Two things that can never be avoided. The first is a numerical question. A numerical question asked in the March 2025 exam, March 22 exam. There is a bullet. The mass of the bullet is 15 grams. The mass of the bullet is 15 grams. The velocity of the bullet is 100 meters per second. The mass of the gun is two kilograms. We need to find the recoil speed of the gun. Shall we look? Mass of the bullet, small m = 15 grams. Converting to kilograms, 15 / 1000 = 0.015 kilograms. Right? 0.015 kilograms. Converted to kilograms. The speed of the bullet is small v, which is 100 meters per second. The mass of the gun is two kilograms. We need to find the recoil speed of the gun. Ready? Let's find it. What, children? Are you going to find the recoil speed equation? We know it is -m * v / capital M. So, -0.015 * 100 / 2. What will that be? -1.5 / 2. What will that be? -0.75. -0.75 meters per second. -0.75 meters per second. Study this. The reason for the negative sign is that the gun moves in the opposite direction to the bullet's movement. That is why there is a negative sign. -0.75. Appu, you did not put the minus sign. You must definitely put the minus sign because the negative sign is important here. The gun is moving in the opposite direction to the direction the bullet is moving. Therefore, you must put the negative sign anyway. Don't have any doubt about it. Put the negative sign. There is one more question related to this. Pay attention. Which of the following statements is correct? You must study questions at the top level. But you must study these concepts and derivations. Derivations will not be repeated there. So, all the derivations you...

You need to study the concept, and the rest will be only question discussion. Which is the correct statement? Speed of gun and bullet are equal. Incorrect, right? The gun and bullet will never have the same speed. Momentum of bullet and gun are equal in magnitude but opposite direction. Correct, right? That is, minus mv equals capital mv, you learned that. So the second statement is correct. Momentum of gun and bullet are equal in magnitude and in the same direction. Never in the same direction. Velocity of gun and bullet are not equal. So the second option is the right one. The second one is the correct option, you should learn this. Shall we move to the next one? Next one, apparent weight of a body in a lift. We are going to look at the apparent weight of a body in a lift, and the actual weight. Everyone please pay attention. Let me explain it simply. Everyone please pay attention. I will explain it easily. Suppose this is a lift. Inside the lift, this is a weighing machine. The lift and everything should be there. This is a weighing machine. It's a weighing machine that calculates weight. Who is standing on top of this weighing machine? Let's assume our Jaseel Annan is standing on top of this weighing machine. This is our Jaseel Annan. Jaseel Annan is standing on top of the weighing machine. Okay? Isn't that something everyone knows? Yes. In the situation where Jaseel Annan is standing on top of this weighing machine, a force will act downwards. Please comment what is the force acting downwards. Everyone should comment. In this situation, please comment what is the force acting downwards. In the situation where Jaseel Annan is standing on top of the weighing machine, what is the force acting in the downward direction? Isn't this something everyone should know? What acts in the downward direction? You don't need to have any doubt. Our weight acts in the downward direction. Weight acts in the downward direction, right? Weight, or mg, acts downwards. Correct? mg, or weight, acts downwards. No doubt. Now, won't there be a reaction here? There will be a reaction. In which direction will the reaction be? The reaction is upwards. Reaction acts in the upward direction. R, or reaction. In which direction will the reaction act? It will act upwards. Clear, right? So, weight acts downwards. In which direction will the reaction act? Yes, the reaction acts upwards. It will act in the upward direction. Who will act? Our reaction will act. Reaction will act in the upward direction, and weight will act in the downward direction. Fine. Does anyone have any doubt? No one has any doubt, right? Did you understand it clearly and precisely? Now, there is one more thing you need to understand. What is the actual weight here? The actual weight is mg. You should recognize this. Actual weight means actual weight equals mg. Actual weight means mg. So, what is the apparent weight? Apparent weight means the reaction, or R. Apparent weight means the weight we feel. If I am moving in a lift on a weighing machine, my weight, the reading on the scale, will keep changing according to my motion. Did you understand the reason? I am sitting on a weighing machine inside a lift, and the lift is moving upwards and downwards, accelerated, and at constant speed. So, according to my motion, the weight shown on the weighing machine will keep changing depending on how the lift is moving. But will my actual weight change? My actual weight will not change. It will always be constant. My actual weight is mg. But the weight shown on my weighing machine is the apparent weight, or R. This apparent weight will keep changing depending on how it is moving. But the actual weight will not change. Apparent weight is a reaction. It will keep changing according to our motion. But recognize that the actual weight, or mg, will not change. Now, let's move to the case of our lift. First case. How is the lift? Suppose the lift is at rest, or moving upwards or downwards with uniform velocity. Let's assume the lift is at rest, or moving upwards or downwards with uniform velocity. Ready? Shall we look? Let's look. Pay attention. Suppose we move the lift upwards. Suppose the lift is moving in the upward direction. This is me. My mass is m. I am standing on a weighing machine. So, the force acting on me. I am considering uniform velocity. Moving with uniform velocity is all the same. So, suppose I am considering that I am moving with a uniform velocity. So, what will be the equation for net force, F equals? Tell me, everyone. Comment. What will be F equals? Quickly, everyone. What is the equation for F? Can't it be explained most simply? Calculate the total forces acting here. Which are they? R, or reaction, acts upwards. mg acts downwards. Correct? R, or reaction, acts upwards. mg acts downwards. Since we are moving upwards, how will we write the equation? We will write it as r - mg. What is it? r - mg. This is termed as Equation Number One. This is our first equation, okay? F = r - mg. What is it, children? You should write r - mg. Because we are moving upwards, that's why we gave r - mg. Right? Fine. Next. Instead of force, according to Newton's second law, can we put mass times acceleration? So, mass times acceleration equals r - mg. This is taken as Equation Number Two. This is our second equation. Mass times acceleration equals r - mg. Equation Number Two. Children, tell me. In the situation of moving with uniform velocity, what will be the acceleration? Everyone comment. Everyone comment what the acceleration will be when moving with uniform velocity. Quickly comment. When moving with uniform velocity, acceleration is zero. If acceleration is zero, what will this entire term become? Yes, this entire term will become zero because acceleration is zero. So, how will we get it? We will get zero equals r - mg. So, what will r be equal to? It will be mg. This is Equation Number Three. This is Equation Number Four. r equals mg. Did you understand it precisely? That is, the apparent weight and the actual weight will be equal. When a lift is at rest, or moving upwards or downwards with uniform velocity, the apparent weight and the actual weight will be equal. Actual weight is equal to apparent weight. Understand that the actual weight and apparent weight will be equal. That's very, very important. It's something to be learned. It's an important thing. Set, right? Are you happy? Are you confident? Then let's move to the next thing. Case Two. Moving to the second case. Everyone pay attention. The lift is moving downwards. The lift is moving downwards with uniform acceleration. The lift is moving downwards, accelerated. The lift is moving downwards with acceleration. So, let me mark it. Sorry, moving upwards. Not downwards. So, upwards with uniform acceleration. I am marking it. It's already marked here. I will mark it darker again. The lift is moving upwards, accelerated. The lift is moving upwards, accelerated. Right? Fine. Now, pay attention here. Everyone pay attention. Here, as we have already done, what is the equation for force? F equals, what is it, children? Here, since the motion is upwards, the force on the upward side is R minus the downward force, so we can give r - mg. This is taken as Equation Number One. Instead of force, we can give mass times acceleration. So, mass times acceleration equals r - mg. Equation Number Two. Now, pay attention to the next thing. In this case, it is moving with acceleration. No doubt. In this case, the lift is moving accelerated. There is acceleration here. If so, what will be the equation for R? r equals m. r = g + m. Is that correct? r = m + m. This minus mg will go to the other side and become plus mg. mg + m. That is, r = g + a. This is taken as Equation Number Three. This is Equation Number Four. m into g + a. Equation Number Four. Ready? Did you understand it precisely? Does anyone have any doubt? No one has any doubt, right? Is it clear to everyone? Do my children have even a sliver of doubt? If so, pay attention here. Everyone pay attention here. There is another very important thing I want to tell you all. Normally, normal weight, or actual weight, is what? W equals mg. But here, the apparent weight obtained is m into g + a. Not mg. It's m into g + a. So, what can we say? You don't need to have any doubt. In this case, the apparent weight will be greater than the actual weight. Because the actual weight is just mg. But here, the apparent weight obtained is m into g + a. You should understand. In this case, apparent weight is greater than actual weight. Understand that the apparent weight is greater than the actual weight. This is what you need to learn according to the second case. This is what each of you needs to learn according to the second case. This is important. This is very important. Understand and learn it. Shall we move to the next case? Let's move to the next case. Each of you needs to answer. This is the case where each of you needs to answer. Case Number Three. Can you answer? Look. Can you answer Case Number Three? The lift is moving downwards. The lift is moving downwards with uniform acceleration. The lift is moving downwards, with uniform acceleration. Can you find out? Yes. The lift is moving downwards with uniform acceleration. The lift is moving downwards, accelerated. How to find out? It's easy. Simple. F equals. Since it's moving downwards, first we should write mg. mg - r. This is Equation Number One. Here too, instead of force, we can give mass times acceleration. Equals mg - r. Plus r will come to this side. So, r equals mg - ma. So, r equals m common out, g - a. This is Equation Number Two. This is Equation Number Three. This is Equation Number Four. So, r equals m into g - a. Did you understand it confidently? So, we all know that the actual weight W is mg. Here, the apparent weight obtained is m into g - a. Not mg. It's m into g - a. So, who will be greater? The actual weight will be greater. The apparent weight will be greater. There is no doubt. In this case, we can say that the actual weight is greater than the apparent weight. That is, in the situation of moving downwards with acceleration, what is the actual weight? It is greater than the apparent weight. Here, the actual weight is greater than the apparent weight. Everyone needs to learn this. This is very, very, very important. This is a question that is constantly asked in exams. Understand and learn it. Next one. Free fall. The lift's mechanism has failed, and it is falling downwards. It is in free fall. If an object is in free fall, the acceleration it possesses will be g. Correct? So, in this case, the acceleration will be equal to the acceleration due to gravity. This case, the acceleration is equal to the acceleration due to gravity. So, since it is falling downwards, what will be the equation for F? It will be mg - r. Correct? Yes. Instead of F, we give ma. ma equals mg - r. So, what will r be equal to? mg - ma. So, r equals mg. Since we can give g instead of a. So, mg. So, r, or apparent weight, will be equal to what? It will be zero. Or, no weight will be felt. It will be a state of weightlessness, a state of no weight. Weight loss will be felt. There will be no weight of any kind. In the situation of free fall, that object will have no weight. So, r, or apparent weight, will be felt as zero. Okay? Okay? Yes, yes, yes. Chronic, stellar. Okay, ready. When in free fall, its acceleration will be constant, g will be its acceleration. Is it crystal clear to all my younger brothers and sisters? If this is set, please send a "set" message in the chatbox. I am confident about this. I don't have even a sliver of doubt. I am very much confident about this. If you say so, send a "set" message. Then we can do one question and move directly to friction. We can do just one question and then move directly to friction. Okay, ready. Respond. Your lack of response is bad. Everyone respond, children. Then, just one question. There is a child. The child's mass is 30 kilograms. The child is standing inside a lift. The lift is moving upwards with uniform acceleration. What will happen to the child's weight? Isn't this something everyone knows? In the situation of moving upwards with acceleration, the child's weight will increase. Apparent weight will increase. No doubt. It will increase. Right. The expression for the apparent weight of the child. What will be the expression for apparent weight? r equals m into g minus a. Sorry, g plus a. It will be g plus a because it is accelerating upwards. Next. If the lift is moving downwards with constant acceleration of 5 meters per second. In the situation where the lift is moving downwards with constant acceleration, what will be our apparent weight? We can say the same thing. If moving downwards with constant acceleration, r equals what? It will be m into g minus a. Because it is downwards. So, r equals m. They have said the mass is 30 kilograms. Let's take g as 10. 10 minus 5. Is that correct? Yes. r equals 30 into 5. So, r equals 150 Newtons. r will be 150 Newtons. Apparent weight equals 150 Newtons. Is it clear, children? Did you understand it confidently? Are all my younger brothers and sisters confident? If this is set, can all of you comment a "set" message in the chatbox? If you say this is confident, this is clear, I don't have even a sliver of doubt, then comment a "set" message in the chatbox. Here, the thing you need to recognize is the unit. It's not apparent weight, children. Weight should be expressed in Newtons. Or you can give kilogram-weight. Just giving kilograms is wrong. Because it is weight. Since it is weight, expressing it in Newtons is the safest. Everyone should try to express it in Newtons. Okay. Moving to the next thing. Friction. What is friction? The duration of the live session will be approximately seven thirty to eight o'clock. Okay. Friction. This chapter will be almost until five o'clock. With gravitation, it will be until about 6:30. I will definitely take the class. Then there is one more chapter for you to take. Jaseel Sir. So, it will be almost until 7:30. Hopefully. Ready. Next. Friction. This is a question asked in March 2024 and Christmas 2023. Friction. What is it, children? Frictional force means, suppose we are pulling this box. We are pulling this box. So, the force that opposes that motion, acting in its opposite direction, is frictional force. It is a force which opposes the motion or the impending motion. That is, whether an object is moving or has a tendency to move, the force that resists that motion, that opposes that motion, acts in the opposite direction. Understand that frictional force acts in the opposite direction. Okay. The first friction is static friction. What is the first friction? Static friction. It's simple. Let these points remain there. I will explain it simply. I took a table. I took whom? I took a table. This is a table. Pay attention. This is our table. Pay attention. This is our table. On top of this table, I have placed whom? I have placed a box on top of this table. A big box. Okay? I have placed a big box on top of the table. Right. I am applying a force to this box. The box is not moving. But it has a tendency to move, right? When I apply an external force, even if the body is at rest, it has a tendency to move. Therefore, there is frictional force. So, the frictional force when a body is at rest is static friction. What friction? Static friction. So, when I apply a force, there is static friction. I increased the force I applied. So, static friction also increased. I increased the force I applied to the maximum. Then static friction also reached its maximum value. The maximum value of static friction is called fs maximum. That is called limiting friction. That is called limiting friction. Clear, right? Let me say it once more. We have placed a box on top of a table. I applied an external force to that box. So, the static friction there will act. So, as I increased the force I applied, the static friction also increased. So, I increased the external force I applied to the maximum. Then static friction also reached its maximum value. The maximum value of static friction is called limiting friction. If the external force I apply is greater than this limiting friction, then that object will change from rest to motion. It will change to motion. Clear, right? Did you understand it precisely? So, the friction that exists when an object is at rest is called static friction. As we increase the external force we apply, static friction also increases. When the external force is maximum, static friction also attains its maximum value. The maximum value of static friction is called limiting friction. If the external force we apply is greater than the limiting friction, then the body will start to move from rest. The body will change from rest to motion. Did you understand it precisely? So, these are the points I have given here. Please learn them precisely. Just understand this concept. The maximum value of static friction is called limiting friction. That is important. Next, let's move to kinetic friction. What is kinetic friction? It is the friction that occurs when a body is moving. The frictional force that occurs when a body is in a moving condition is called kinetic friction. What is it called? Kinetic friction. This kinetic friction has a specialty. Kinetic friction is always less than the limiting friction, or static friction. Okay. Kinetic friction is always less than static friction. The kinetic friction is always less than the static friction. Because the frictional force acting on a moving body is less than the frictional force on a stationary body. Understand that. Kinetic friction will always be less than static friction. Okay. Can you guess which question has been asked in the recent question papers from the 2022 model to the 2020 model? If you can predict or guess, please tell me, children. Which question is this? Which question is repeatedly asked in so many question papers? You can easily say. It is the laws of static friction and the laws of kinetic friction. That law is asked. First, the law of static friction. What is it? Limiting friction depends upon the nature of the surfaces in contact. Limiting friction depends on the nature of the surface. What is this nature of the surface? That is, as I always say, the tile in our hall is not the same as the tile in the bathroom, right? In the hall, we put a smoother tile. In the bathroom, we put a rougher, more textured tile. Because there is a possibility of slipping there. It's a place where water is always present. So, this limiting friction depends on the nature of the surface. It depends on what kind of surface it is. The value of limiting friction is independent of the area of contact. It is independent of the area. Limiting friction is directly proportional to the normal reaction. Limiting friction is directly proportional to the normal reaction. Shall we look? So, I am going to start writing. Pay attention. So, fs maximum. fs maximum is directly proportional to the normal reaction. Or, fs maximum equals. To change the proportionality to equality, whom do we introduce, children? To change the proportionality to equality, we need to introduce a proportionality constant. Correct? We need to introduce a proportionality constant. Whom do we introduce here as the proportionality constant? Tell me. Yes. The proportionality constant we introduce here is mu s. So, we can write it as mu s into n. This is Equation Number One. This is Equation Number Two. So, fs maximum equals mu s into n. Clear, right? So, what is mu s? mu s is called the coefficient of static friction. mu s is called the coefficient of static friction. We can write the equation for mu s as fs maximum by n. Can we write it as mu s equals fs maximum by n? Did you understand the matter perfectly? No doubt, children. Please learn this. You must learn this. It is important. That is, our limiting friction depends on the nature of the surface. It does not depend on the area. It is directly proportional to the normal reaction. So, fs maximum is directly proportional to normal. So, to change the proportionality to equality, the proportionality constant we introduce is mu s. mu s means coefficient of static friction. mu s is called the coefficient of static friction. We can write mu s equals fs maximum by n. Did you understand the matter perfectly? If it's okay, can you comment a heart emoji? If you say this is okay for me, no doubt, then comment a heart emoji. Then we can move to the next one. Let's move to the laws of kinetic friction. Let's learn it in the same way. Pay attention. Let's move to the laws of kinetic friction. Let's learn it in the same way. Pay attention. What will kinetic friction depend on? Definitely, it will depend on the nature. It is independent of the area. It is independent of the relative speed. Kinetic friction. It is the same as what is there. It depends on the nature. It is independent of the area. It is independent of the relative speed. Because here, since it is kinetic friction, speed plays an important role, right? Because that body is moving. But kinetic friction is independent of speed, independent of relative speed. Understand that. So, what does it depend on? Kinetic friction is directly proportional to the normal reaction. So, how can we write it? fk is directly proportional to normal reaction. Or, fk equals, what can we write? mu k into n. This is Equation Number One. This is Equation Number Two. So, what is mu k? mu k means the coefficient of kinetic friction. What is mu k, children? That is the coefficient of kinetic friction. You must learn it. You must learn the equation for mu k. Did you understand the equation perfectly? These questions are constantly asked in exams. They are very important questions. They are questions that must be learned. This is very, very, very important. Next, kinetic friction actually has two types. What are they? Sliding friction and rolling friction. Kinetic friction has been divided into two types. There are two types of our kinetic friction: sliding friction and rolling friction. You should understand. When a body is rolling on another body, the friction that occurs in the rolling situation is called rolling friction. If it is sliding, that is, if it is slipping and rolling, the friction that occurs there is called sliding friction. Which one will be less? I always give the example. Is it easier to pull a gas cylinder or to roll it? We know that rolling a gas cylinder is very easy. That is, our sliding friction is less than our rolling friction. Sorry. Rolling friction is less than sliding friction. Rolling friction is less than sliding friction. Because rolling is very easy compared to pulling. So, understand that rolling friction is less than sliding friction. That is important. One question is finished. Friction is almost over. What is the name of the maximum value of static friction? Everyone should comment. All children should comment. Then I will go and get a glass of hot water. By then, comment. If it is maximum static friction, yes, water is important. Tell me. Great. All are awesome. What is it, children? That is our own fs maximum, or limiting friction. That is it. That is our limiting friction, or fs maximum. Learn it. That is limiting friction, or fs maximum. Next, angle of friction. Very simply. Angle of friction is very easy. Just understand the concept. Everyone pay attention. Suppose we take a surface. We take a surface. Suppose a box is sitting on this surface. A box is sitting on this surface. This is our box. Suppose a box is sitting on this surface. Everyone pay attention. A box is sitting on our surface. Ready. What are the forces acting here? Can you tell me? You can definitely tell me. What is the force acting downwards? Downwards, definitely mg, or weight, will act. In which direction is the normal reaction?

Let's act. The normal reaction is the usual reaction. We can replace it in any way. That is, in the previous case, we gave it in the case of the lift. Here, I am giving confusion, okay? Here, I am giving the normal reaction, both are the same. We can give 'A' or 'B', no issue. Okay. So, the weight of that body is acting downwards as mg. The normal reaction is acting upwards as N. Right? Fine. Now, pay attention here. I am applying a force on that body. Everyone, pay attention to the direction of the force I am applying. The direction of the force I am applying is this way. Even though the body is not moving, I am applying a force. So, there will definitely be friction. You have to comment on what kind of friction it is in a situation where it is at rest. What will be the friction force in a situation where it is at rest? The body is at rest, but because I am applying a force, a frictional force will act there. Which frictional force will act? Yes, even though the body is at rest, a frictional force will act there. What friction is that, children? You don't need to have any doubt, that is our static friction, or Fs maximum. That is our static friction, or Fs maximum. It is in the opposite direction to the applied force. If we apply force in that direction, static friction will act in the opposite direction. Static friction will act in its opposite direction. Now, pay attention here. This is an important thing that everyone should pay attention to. Shall we draw the resultant of this normal reaction and static friction? I am drawing a resultant of this normal reaction and static friction. Pay attention. This is the resultant. This is the resultant of the normal reaction and static friction that I am drawing. This is the resultant for normal reaction and static friction. I have given it as 'R'. This is the resultant. What is the angle of friction? Everyone, understand this. The angle that this resultant makes with the normal is theta. This theta is called our angle of friction. Let me tell you once again. Understand this. That is, a body is standing on a table. Its weight is acting downwards as mg. The normal reaction is acting upwards. I am applying an external force on it. The direction of the force is marked. Static friction, or Fs maximum, will act in the opposite direction of that force. So, R is the resultant of this static friction and normal reaction. R is the resultant of this normal reaction and static friction. So, the angle that this resultant makes with the normal is theta. The angle that this resultant makes with the normal is called theta. Theta is the angle between the resultant and the normal reaction. This angle is called the angle of friction. Okay? If it is clear, send a "set" message in the chat box. If you understand this, send a "set" message in the chat box. Then we can move on to the last thing related to friction. The last topic related to friction is the angle of repose. We are moving on to the last topic related to friction, that is, the angle of repose. If it is set, respond a little more. Ready, set, set. Okay, fine. Next, we are moving on to the angle of repose. What is the angle of repose? Simply put, this is an inclined surface. I have placed something here on this inclined surface. I have placed something here. Can you see the angle? Now, can you see it? I can place the thing in another color. I have got the thing. Ready. Can you see it? Okay. This is an inclined surface. I have placed something here. Can you see it? Everyone, can you see it? I think you can see it. This thing is rolling down. Is it falling down? No. If I lift it a little more, is it rolling down? No. If I lift it a little more, is it rolling down? No. When I lifted it a tiny bit more, it rolled down. Right? When I lifted it a tiny bit more, it rolled down. So, the minimum angle of inclination required for this thing to roll down is called the angle of repose. So, if I place it like this, this thing will roll down. If I lift it, will it roll down? No. If I lift it a little more, will it roll down? No. If I lift it a little more, it will roll down. No. When I lifted it a tiny bit more, it rolled down. Right? When I lifted it a tiny bit more, it rolled down. So, the minimum angle of inclination required for it to roll down is called the angle of repose. Understand this. Suppose a box or a box is placed on an inclined plane, and the angle of inclination there is theta. We all know that the weight mg is acting downwards. We can resolve that mg into two components. Yes, the horizontal component is mg cos theta, and the vertical component is mg sin theta. You need to understand one thing. That box tends to fall downwards, right? So, doesn't friction oppose it? You need to understand that clearly. Therefore, in which direction is the frictional force acting? Look, the direction of the frictional force is this way, because that box tends to fall downwards. So, the frictional force acts in the opposite direction, opposing that motion. Fine. Clear. So, weight acts downwards as mg, two components mg cos theta, mg sin theta. Frictional force acts in the opposite direction to oppose the downward fall. Normal reaction acts perpendicular to the surface. Clear, right? Have you understood the parameters within it clearly? No doubt, right? I believe so. Now, children, everyone pay attention. It is stated here. Now, the angle theta is the angle of repose. I have given the definition of what it is here. Everyone, pay attention. The minimum angle of inclination at which a body placed on an inclined plane begins to slide down is called the angle of repose. It is called the angle of repose. Just understand it. Shall we move forward? Yes. Pay attention. Everyone pay attention. Just before sliding down. That is, before rolling down, the box is in equilibrium. Before sliding down, the box is in equilibrium. We all know. In equilibrium, what is the net force acting on it? Quickly comment, children. What is the net force acting on it in equilibrium? Can you comment quickly? Everyone, please comment. What is the net force acting on the body in equilibrium? Quickly comment. We all know. The net force acting on the body in equilibrium will be zero. There is no doubt. It will be equal to zero. If so, pay attention here. This normal reaction will be equivalent to the mg cos theta below. Similarly, Fs maximum will be equivalent to mg sin theta. They will be equal and opposite, right? Normal reaction will be equivalent to mg cos theta. Fs maximum will be equivalent to mg sin theta. Right? So, all those forces will be equal and opposite. That is what is given here. mg sin theta equals Fs maximum. mg cos theta equals normal reaction. Ready. Shall we divide them? We are not thinking of anything else. We are going to divide them with our eyes closed. So, mg sin theta divided by mg cos theta equals Fs maximum divided by N. Pay attention, children. Here, mg and mg cancel out. What is sin theta by cos theta? Tan theta. Tan theta equals Fs maximum by N. We were taught that. What is Fs maximum by N? Tell me, children. That is, Fs maximum divided by normal reaction. We know it is the coefficient of static friction. It is μs, right? That is, μs. Coefficient of static friction. This is the derivation for the angle of repose. Learn this. This is the derivation for the angle of repose. Just learn it. It is important. It is asked in exams. It is very important. This is the equation for the angle of repose. Tan theta equals μs. Are you confident? Did you understand it perfectly? This is the angle of repose. That is, you are just concluding. The tangent of the angle of repose depends only on the coefficient of static friction. Understand that the angle of repose depends only on the coefficient of static friction. Next, advantages of friction. Friction helps us walk in all ways. The braking system of vehicles works through friction. There are many such advantages of friction. There are more disadvantages than that. Wear and tear will occur. Energy loss, power loss. Friction is the cause of such things. So, understand that friction has disadvantages as well as advantages. How can friction be minimized? We can polish. We can use lubrication. We can use ball bearings. Understand lubrication, children. That is, oil and grease. We call these lubricants. So, we can use lubricants. We can polish. Also, we can use ball bearings, steel balls. If you look at the pedals of a bicycle, you can see steel balls. These are all methods we use to reduce friction. Okay? Ready? Now, we are moving on to the last topic in this chapter. We are only teaching two things. We can discuss more numericals and top-level questions related to it. Only two derivations. We are moving on to the last two derivations. Come on, children. Are you ready? What is this question? Tell me, children. Comment on what question was asked in March 2025. What is the question? Comment everyone. Comment on what question was asked in March 2025. Can you comment? Tell me, what is that question? Yes, motion of a car on a level circular road. That's it. Motion of a car on a level circular road is shown here. Okay, right. This car actually has a tendency to skid outwards. It has a tendency to skid outwards. We all know that when we watch races and see them skidding like this, they tend to skid outwards. So, what prevents it from that motion? What acts towards the center? What prevents it from such a motion? There is no doubt, it is our own, it is our own static friction, or Fc. It is our own static friction. Static friction is what prevents the object from going outwards. Understand that. Okay? The static friction that prevents it from going outwards. Fine. Let's move on to the derivation. So, for a car to move on a circular track, it definitely needs centripetal force. Here, who acts as the centripetal force? Who acts as the centripetal force? It is the static friction. Understand that the static friction acts as the centripetal force. The static friction is what acts as the centripetal force. I told you, static friction. Write it down. Fs maximum. Centripetal force is provided by static friction. Here, the static friction acting as the centripetal force is what keeps the car on the circular track. So, Fc = Fs maximum can be said. Centripetal force will be equivalent to static friction. This is taken as equation number one. The equation for centripetal force is mv^2/r, right? We can give capital R, because the radius of the circular track is capital R, right? Yes, it is capital R. We can give it as capital R. Pay attention everyone. What is the equation for Fs maximum? It is μs * N. This is taken as equation number two. Why did I write v maximum here? Because we are going to find the maximum speed. That's why I replaced v^2 with v maximum^2. It can be written as equal to μs * N. Ready? Now, pay attention here. Everyone pay attention. The normal reaction here is equivalent to mg. Normal reaction = mg. Let me replace it. I am going to replace it. Everyone pay attention. mv maximum^2 / r = μs * N. This is equation number three. Let me cancel it. I will cancel m. So, v maximum^2 / r = μs * g. Equation number four. If so, what will v maximum^2 be? v maximum^2 = μs * g * r. So, what will v maximum be? v maximum = sqrt(μs * g * r). This is very, very, very important. You all need to learn this. It is a very important equation. This is important. Equation number five. Equation number six. Are you confident? Are all the younger brothers and sisters confident? Is it set? Okay, if so, put a heart emoji in the chat box. If you understand this, put a heart emoji in the chat box. Yes, Niveth says he understood. If everyone understood, comment with a heart emoji. Then we can move on to the last one. We can move on to the last one. This question has been asked in so many question papers. We are going to move on to our last derivation. We are going to move on to the most crucial derivation in this chapter. Even if you don't learn anything else, you must learn this question. We are going to move on to a question that can be said so. Guess what it is. Come on, guys. What is that question? Without any doubt, you can say it. That is our own, that is our own banking of roads. We are going to move on to banking of roads. Motion of a car on a banked road. We are going to derive the motion of a car on a banked road. It is easy. We all know that when we go to hilly areas like Ooty, Kodaikanal, Munnar, Wayanad, etc., we see hairpin bends. In such bends, the outer edge of the road appears to be higher than the inner edge. The outer edge of the road is slightly increased, or its height is slightly more than our inner edge. The phenomenon is called banking. Okay. The outer edge of the road is slightly raised above the inner edge. It will be slightly higher. That phenomenon is called banking of roads. Because of this banking, vehicles do not skid off the road. So, if the outer edge is raised higher than the inner edge, then only vehicles can continue on the road properly, otherwise, there will be a tendency to skid outwards. Understand the reason, okay? This theta is what we call the angle of banking. Theta is known as the angle of banking. Learn it. Theta is the angle of banking. Okay? Shall we move forward, children? Now, we are going to proceed to the derivation. Stay with me, everyone. Let's derive it together. Okay? Just watching like a movie and understanding is not possible. As Lalettan's dialogue says, "It's overambition, my friend. It won't work." We cannot learn this by just watching like a movie. Because this is a high-level thing. This is not a derivation that can be easily cracked under normal circumstances. Not one, not two, even those who have written it many times may get confused at the last moment. Stay with me. Everyone stay with me. Consider this as our mass. Okay? This is our car. Okay? This is the car on the road. This thing is the car. So, we are going to resolve the forces on the car. Everyone stay with me. Stay with me. This is my request. I am drawing the first force downwards. Quickly tell me what is acting downwards. No doubt, who is acting downwards is our own weight. Weight acts downwards, mg. In which direction does the normal reaction act? It is perpendicular to the surface, right? Here, I am drawing the normal reaction. Drawing the normal reaction perpendicular to the surface. Are these two things okay? Yes. The angle of banking is theta. Theta is the angle of banking. I believe everyone has understood this much. I expect no doubt. So, the car tends to skid outwards. So, the frictional force will act in a way that opposes it. So, shall I draw the frictional force, children? This is our frictional force. That frictional force acts in a way that opposes the outward motion of the car. So, this is the direction of the frictional force. This is the direction of our frictional force. It acts in a way that opposes that motion. So, the frictional force, or Fs maximum, will act in this way. Fs maximum will act in this way. Fine. Next, I am going to resolve this Fs maximum into two components. Horizontal and vertical. Stay with me, everyone. Stay with me. I am going to resolve Fs maximum into two components. Horizontal and vertical. I am going to split Fs maximum into horizontal and vertical components. I am going to resolve it. Stay with me, guys. Okay. I am going to split Fs maximum into two components. Pay attention, children. Everyone pay attention. Let's take this as horizontal and this as vertical. So, if this angle is theta, what will this angle be? Alternate angles, right? This will also be theta. If this is theta, you can write this value. You can write it very simply. This will be Fs maximum cos theta. This will be Fs maximum sin theta. It will be Fs maximum cos theta and Fs maximum sin theta. Clear, right? No doubt, right? No one has any doubt, right? Crystal clear? Next, we are going to resolve this normal reaction into two components. I am going to resolve this normal reaction into two components. Stay with me, children. Tell me. I am going to resolve the normal reaction into two components. It is the same way. Pay attention, everyone. I am going to divide the normal reaction into two components as well. Come on, guys. It is very easy to resolve the normal reaction into two components. But many will get confused. This angle is 90 minus theta. This angle is theta. Just think about that. Don't think too much. This angle is also theta. If it is theta, what will be the vertical component of the normal reaction? This is N cos theta. This will be N sin theta. This is N sin theta. That's all. We have resolved the static friction into two components. We have resolved the normal reaction into two components. That's all the work. Nothing else. Our derivation is finished. Finished. The most important part of our derivation is finished. We have resolved N cos theta and N sin theta. Similarly, we have resolved Fs maximum into Fs maximum sin theta and Fs maximum cos theta. The work is done. Now, shall we start? Okay. We are starting the derivation. Stay with me. So, here we need to see who provides the centripetal force. The forces in this direction provide the centripetal force. So, Fs maximum cos theta and N sin theta are providing the centripetal force here. Right? So, shall I write it down? Centripetal force is provided by Fs maximum sin theta + sorry, Fs maximum cos theta + N sin theta. These two are providing the centripetal force here. Can we give this as equation number one? Given as equation number one. Next thing. Let me replace Fs maximum. Look, Fs maximum is μs * N. I am going to replace it. So, pay attention. Centripetal force = μs * N cos theta + N sin theta. Equation number two. Have you understood it clearly? No doubt, right? No. Next, let's move on to the expansion of centripetal force. It is mv^2/r. Here too, we are finding the maximum speed, so it is mv maximum^2 / r. = μs * N cos theta + N sin theta. This is equation number three. This is taken as equation number three. Did you understand it perfectly? No doubt, right? No doubt for anyone. Next, so the matter of centripetal force is over. Next, are these two forces equal? N cos theta and Fs maximum sin theta are equivalent to mg, right? Yes. The upward force and the downward force are equal. So, shall we equate them? So, mg + Fs maximum sin theta = N cos theta. I am going to write it. Pay attention. mg + Fs maximum sin theta = N cos theta. Equation number? This is equation number four. Ready. Here too, we can replace. Let's keep it only for mg. mg = N cos theta - Fs maximum sin theta. Equation number? This is equation number five. Clear, right? Here, we can replace Fs maximum. mg = N cos theta - μs * N sin theta. Equation number? This is equation number six. Ready? Next, let's move on. Everyone pay attention. We are going to the next step. The next step is very simple. We are going to divide the two equations. We are going to divide equation number three by equation number six. Shall I divide them? Let me take this up. That is the best thing. I am copying this. Copy. Paste here. Divided by. I am going to divide this. Let me take the next person. Let me also copy this and paste it below. Set. I have placed him below in the same way. We don't need the equal to here. We have put an equal to. That's enough. Next work. This mg will go upwards, of course, to the numerator. mg will go upwards. So, there will be a small change in the equation. Look. mv maximum^2 / r * mg = This term has no change. I am writing it as it is. Because there is no change. Fine. Let me put the equation number. This is equation number seven. This is equation number eight. Ready. Next step, pay attention. I am canceling m here and m here. v maximum^2 / r * g. Right? v maximum^2 / r * g. So, what will this term be? I am copying this term again. Copy. Write it below. Because there is no change. Fine. Let me put the equation number. This is equation number nine. Now is the time. Now is the game. This whole equation, I am going to divide by N cos theta. That is, I am going to divide the numerator and denominator by N cos theta. So, pay attention. v maximum^2 / r * g = μs * N cos theta / N cos theta + N sin theta / N cos theta. Divided by. N cos theta / N cos theta - μs * N sin theta / N cos theta. I have divided the whole equation by N cos theta. Now, just cancel the common terms. Our work is done. Look. Here, N cos theta and N cos theta. Cancel. Here, N and N. Cancel. N cos theta, N cos theta. Cancel. m, m. Cancel. So, in the first term, what remains? Only μs. So, I am writing it down. Pay attention, everyone. v maximum^2 / r * g = Only μs will remain. + sin theta / cos theta is tan theta, right? μs + tan theta. Divided by. What is below? It is one. So, 1 - μs tan theta. Because sin theta by cos theta is tan theta, right? So, μs tan theta. Equation number? This is equation number eleven. This is taken as equation number eleven. Did you understand it clearly? No doubt, right? No doubt for anyone, children. Next, we can find v maximum^2. So, v maximum^2 = r * g will go here. So, r * g * (μs + tan theta) / (1 - μs tan theta). So, what will v maximum be? v maximum = sqrt(inside the root, write these terms). r * g * (μs + tan theta) / (1 - μs tan theta). This is equation number twelve. This is equation number thirteen. Did you understand it crystal clear? Did all the younger brothers and sisters understand it? This is the maximum permissible speed on a banked road. The maximum speed a vehicle can travel on a banked road is this. Now, I will also tell you the optimum speed along with this. Pay attention. Optimum speed is very simple. To find the optimum speed, look. Optimum speed, v optimum. There will be no friction there. So, μs.

When zero is given to zero, this term in the denominator will completely become zero. So, I will write it down if needed. V maximum is equal to sorry, V optimum, V optimum is equal to root of R G tan theta divided by one minus zero. Zero plus tan theta is tan theta. Tan theta by one means only tan theta. So, what is the optimum speed obtained? Optimum speed is root of R G tan theta. Optimum speed is root of R G tan theta. Please learn this as well. Everyone should learn this. This is very important. This is asked in exams. This is very, very, very important. Questions based on this are asked. This is equation number 14. We have completed the chapter called Laws of Motion, 15, 16. If you are confident, put a heart emoji in the chat box, and we can move to the next chapter. If you are confident, put a heart emoji in the chat box, and we can move to the next chapter. Are you ready? Are you ready to move to the next chapter, Gravitation? If you are ready, let's put a muscle emoji in the chat box. We can finish Gravitation in a short time. Ready, sir. Need rest. Hey, why do you need rest when I don't have it? What order is this, Nirmala? Don't play. We need to finish quickly. Because you also need to revise. After today's live ends, you need to go through at least the slides again. If we take more time, it won't be good. Because we need to wind up the complete live by 9 o'clock. If we give you half an hour rest, it's just a waste of time. So, I'm not stopping. I'm moving forward. Sir, drink some water. You look exhausted. Okay, thank you. I'll go and get some water. I'll go and get some water. I'm back. I'm back. I'm back. Ready, ready. Don't say anything. I'm not replying to one of your comments. You wait. I have given you time for that. Go. Run away. So, Gravitation. Gravitation. Gravitation. First is Universal Law of Gravitation. There is nothing special to say. What is Universal Law of Gravitation? Gravitation is asked for about eight marks. What, my dear students, is Universal Law of Gravitation? It means that all bodies in the universe attract each other. The gravitational force of attraction is directly proportional to the product of their masses and inversely proportional to the square of the distance. Every body in the universe attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This is something everyone knows, right? Everyone knows this. I have given the statement here. Shall we write the equation? The equation for gravitation is Equation Number 1. The force is inversely proportional to the square of the distance between them. Equation Number 2. That is, if the mass increases, the force increases. If the distance increases, the force decreases. Because the gravitational force is inversely proportional to the square of the distance. Let me combine these two equations. So, F is proportional to M1 M2 by R squared. This is taken as Equation Number 3. Students, to change the proportionality and make it equal, whom shall we introduce? Everyone comment. To change the proportionality and make it equal, whom shall we introduce? Everyone comment. Whom shall we introduce? Introduce the proportionality constant. F is equal to G M1 M2 by R squared. This is taken as Equation Number 4. This is the fourth equation. The work is done. This is called the Universal Law of Gravitation. Everyone must learn this. This is the Universal Law of Gravitation. That's all. Everyone knows the value of G, right? What is the value of G? 6.67 into 10 to the power of minus 11 Newton meter squared per kilogram squared. Next, we have already found the dimensions of G. I won't go into that. Next, we move to the most important part. The question "Write any two properties of gravitational force" can be asked. Shall we look at the properties of gravitational force? It is always an attractive force. Gravitational force will not repel. It is always attractive. It is always an attractive force. It will never repel like magnetic force or electrostatic force. Also, it is independent of the medium. It is independent of the medium. Suppose I consider two masses. The gravitational force between them when placed in air will be the same as when placed in water. It will be the same force. If I place it in petrol, the gravitational force will not change, no matter in which medium I place it. Understand that gravitational force is independent of the medium. It is the weakest force in nature. Because there is gravitational force between all bodies around us. There is gravitational force between me and the camera in front of me. There is gravitational force between me and the monitor in front of me. Even then, we don't experience it significantly. We don't experience it much. So, the reason is that it is the weakest force in nature. It is the weakest force in nature. It forms an action-reaction pair. It forms an action-reaction pair. What is this action-reaction pair? Suppose I stop Nivedan in the chat box. I stop Nivedan here. Then I say, "Nivedan, I attract you." The body called me will exert a gravitational force on Nivedan. Right? My gravitational force will pull him towards me. Similarly, he will also exert a gravitational force on me. So, in which direction will it pull? It will pull towards me. So, the gravitational force between us acts in opposite directions. So, an action-reaction pair is formed. If I consider the gravitational force I exert as action, then the gravitational force exerted by Nivedan will be the reaction. So, an action-reaction pair is formed there. Understand that. It is a long-range force. It doesn't act only over short distances; gravitational force acts over long distances. Because there is gravitational force between planets. There is gravitational force between the Sun and Earth, and all planets. Understand that. Therefore, it is a long-range force. It doesn't act over short distances; gravitational force acts over long distances. Another thing to mention is that many students watching this live are starting to send screenshots of lecture notes when I mentioned a student's screenshot of Theresa. Many people who are writing lecture notes, many people with carbon IDs, many people have forgotten. Many people have sent them to me. Everyone study well. Write lecture notes. It will be very effective for your revision. Because this slide, you might not be able to revise quickly at your convenience before entering the exam hall. The mobile might not permit it. But with this, you can mark very simply. If we study, we can tick. It will be very convenient. That's why I say write lecture notes. Okay. Next is acceleration due to gravity. What is acceleration due to gravity? Without going into the law, I usually teach Kepler's laws at the very end. I will teach it this time as well. What is acceleration due to gravity? We all know. I always give this example. Suppose I have a crane. Using the crane, I lift an elephant and Jaseel sir. If I drop both at the same time, who will fall first? Comment. I lift an elephant and Jaseel sir to a certain height. Then I drop both at the same time. Who will fall first? Definitely the elephant will fall first. Because the elephant has more mass. The elephant will fall first. Right? Yes. But if I remove all the air here and create a vacuum condition here. What is here? A vacuum condition. If I remove all the air and lift both and drop them again, who will fall first? If I remove all the air and lift both and drop them, both will fall at the same time. Right? In a vacuum condition, if we drop an object towards the Earth's surface, it will fall with a constant acceleration. That acceleration is called acceleration due to gravity. So, the elephant and Jaseel sir will have the same acceleration. That's why both fell at the same time. The acceleration is called acceleration due to gravity. In a vacuum condition, if we drop a body towards the Earth's surface, its motion is free fall. So, if it is in free fall, the acceleration of that body is called acceleration due to gravity. So, Jaseel sir and the elephant will have the same acceleration in free fall. That is acceleration due to gravity. That's what we are going to find. Okay? So, you can see something above the Earth. What is it? This is a small mass. A small mass. This is the mass of our Earth, capital M. Let the radius of the Earth be capital R. So, won't there be a gravitational force between this Earth and that mass? Of course, there will be. There will be a gravitational force between the Earth and that mass. You know the equation, right? F is equal to G M M by R squared. How did this come? Do you have any doubts? I will clear them. Pay attention. Generally, it's G M1 M2 by R squared, right? Instead of M1, I put the mass of the Earth, capital M. Instead of M, I replaced it with the mass of the body, small m. The distance between them is what? It is equivalent to capital R, or the radius of the Earth. It is equivalent to capital R, or the radius of the Earth. Right? Isn't this distance equivalent to the radius? Distance from center to center. Yes, fine. Next, instead of gravitational force, what can we put? We can put Mg. Let me replace it with Mg. Mg is equal to G M M by R squared. Did you understand clearly? This is taken as Equation Number 2. Let me cancel the constant terms here. Let me cancel this mass and this mass. We get the equation for acceleration due to gravity, G. What do we get? G = GM by R squared. This is the expression for acceleration due to gravity. If a body is in free fall on the Earth's surface, the acceleration it possesses is called acceleration due to gravity. Learn this equation. This is very, very, very important. What does acceleration due to gravity depend on? It only depends on the universal gravitational constant, the mass of the Earth, and the radius of the Earth. Its value is 9.8 meters per second squared on the Earth's surface. Learn that too. If this is set, send a "set" message in the chat box. If this is set, all my younger brothers and sisters, send a "set" message. Then we can move on to the next things. There are only two questions related to acceleration due to gravity waiting for you. I will definitely make you do both questions. No doubt. The first question was asked in March 2025. We cannot avoid it at all. Because it is such an important question, that's why I'm teaching it. Acceleration due to gravity is independent of ____. Acceleration due to gravity is independent of what? What is acceleration due to gravity independent of? There is no doubt. It is, just remember its equation. What is the equation? G = GM by R squared. It only depends on the universal gravitational constant, mass, and radius. So, what is it independent of? It is independent of the mass of the body. It is independent of the mass of the body. Are you ready, my dear ones? Did everyone understand? Definitely independent of the mass of the body. What is the maximum value of G on the surface of the Earth? 2024 model, 25 model, 22 model. So, it is a question asked in model exams. We can expect a question for the upcoming 2026 model as well, right? Definitely, we can expect it. So, where is acceleration due to gravity maximum? Then we need to draw a diagram to understand whether it is at the poles or the equator. It's simple, pay attention. We all know G is equal to what? Sorry guys. G is equal to what? G = GM by R squared. That is, acceleration due to gravity increases when the radius decreases. So, look, students, the distance from the center to the equator is very large. It is less towards the poles. So, when considering the poles, R is less. Right? At the poles, R is less. So, G will be more. Where will acceleration due to gravity be more? Understand that it is more at the poles. Acceleration due to gravity will be more at the poles. Is it clear? Shall we move to the next topic? There is nothing special in the next topic. You all know it. Can you guess what the most important question from this chapter asked in all these years would be? Can you guess? If you can guess, please tell me. Yes, set. World of, I can't read your names because I have short sight. It's time to change my glasses. I can read many names now. Yes, yes. I can read names now. Okay. World of. What a name? Jovial. World of Jovial. Okay. Variation with height. Ajile, you are a gem, Aj. Aj is answering correctly. It is variation with height. That's all. Variation of G with altitude or height. What change happens to acceleration due to gravity when height increases? We are going to derive that. Don't Lalettan say that the taste of tea increases with height? Similarly, what change happens to acceleration due to gravity when height increases? We are going to do that. Yes, guys, we are going to derive. So, students, understand. The mass of the Earth is capital M. The radius of the Earth is capital R. The acceleration due to gravity on the Earth's surface is G. The acceleration due to gravity on the Earth's surface is G. The acceleration due to gravity at a height H is G dash. So, what change happens to acceleration due to gravity when height increases? Let's look. Okay, let's look at everything. Come on, pay attention, students. We can do it easily. First, acceleration due to gravity on the surface of the Earth. What is the acceleration due to gravity on the surface, students? Everyone understand. On the surface, G is equal to GM by R squared. G = GM by R squared. I've given it as Equation Number 1. That's on the surface. What is the acceleration due to gravity at a height? What is the acceleration due to gravity at a height? Comment. What will be the acceleration due to gravity at a particular height? Come on, guys. What will it be? Here, we just need to make a small change. What is G equal to? G M divided by, instead of R, what will come? In the case of being on the surface, the distance from the center to the surface is R. We all know that. But in the case of being at that height, what is the distance from the center? Don't we need to add the height to the radius? So, in the denominator, what will come? R + H whole squared. This is Equation Number 2. This is our second equation. R + H whole squared. Ready? Did you understand clearly? Did everyone understand clearly? R + H. This is Equation Number. Next, next one. Pay attention, everyone. Pay attention. What we are going to do now is divide the two equations. Divide Equation Number 2 by Equation Number 1. Divide them. Pay attention. Let's call it G dash. Sorry guys, just a minute. Let's call this G dash. It's not wrong to call the acceleration due to gravity at a height G dash. Let's call it G dash. So, G dash by G is equal to what, students? G dash is GM by (R + H) squared. G is GM by R squared. Let's call this Equation Number 3. When dividing the two, the term in the denominator will come to the numerator, right? It will come reciprocally. Of course, when the term in the denominator comes to the numerator, it will come upside down. So, let me bring it reciprocally. Pay attention. G dash by G is equal to what will come? GM divided by (R + H) squared into R squared by GM. Into R squared. GM. Equation Number 4. Ready? Did you understand clearly? No doubts here. Let me cancel the common terms here. Look, this GM and this GM cancel out. We will get a fifth equation. This is an important equation. G dash by G is equal to what, students? R squared divided by (R + H) squared. This is Equation Number 5. This is Equation Number 5. This is an important equation, that's why I'm painting it yellow. This is one of the most important equations. This is an equation that you should give great importance to in your numerical questions. Many numerical questions are solved using this. This is one of the most important equations. Shall we move on to the rest of the things? Shall we move on to the rest of the things? Pay attention here. I am going to take R out of this denominator. I am going to take R out commonly from the denominator. Pay attention. So, G dash by G is equal to what will come? The R squared above remains as it is. I am taking R out below. But you should understand that there is an R squared here and a whole squared here. So, when R comes out, it will come as R squared. If we remove it from the place where R is, what will we get? One. Right? So, in 1 + H, we cannot take R out. So, it will become H by R. The whole squared is kept as it is. Equation Number 6. Did you understand clearly? No doubts, right? So, G dash by G is equal to R squared divided by, when R is taken out from below, since there is a whole squared, it will come out as R squared. When R is removed from its place, it becomes one. We cannot take R out of H. So, we get H by R. So, 1 + H by R whole squared. Did you understand clearly? No doubts, right? So, G dash by G = R squared divided by, when R is taken out from below, since there is a whole squared, it will come out as R squared. When R is removed from its place, it becomes one. We cannot take R out of H. So, we get H by R. So, 1 + H by R whole squared. The next step is trivial. This R squared and this R squared cancel each other out. So, let me write down the remaining terms. Pay attention. Step 7. That is, G dash by G is equal to what will we get? 1 divided by 1 + H by R whole squared. Equation Number 7. Step 7. Let's take this (1 + H/R) whole squared to the numerator. Let's take 1 + H/R to the numerator. So, G dash by G is equal to what will we get? 1 + H by R raised to the power of -2. This is Equation Number 8. 1 + H by R raised to the power of -2. Equation Number 8. Did you understand clearly? Now, pay attention, everyone. We are going to apply a condition of the binomial theorem. You all know that 1 + x raised to the power of n is approximately equal to 1 + nx. Right? 1 + x raised to the power of n is approximately equivalent to 1 + nx. When x is very small. When x is very small, we can write 1 + x raised to the power of n is approximately equal to 1 + nx. Ready? Fine. So, in place of x here, it is H by R. So, if H by R is very small, we can put this power inside, right? Here, the power n was put inside, and it became 1 + nx. Similarly, here, what is in place of x is H by R. So, if H by R is very small, we can put the power inside. Understand that. Okay. Now, pay attention. H by R must be very small. When will H by R be small? Yes, the numerator must be large and the denominator must be small. That is, H is greater than, sorry, sorry, I went the opposite way. The numerator must be small and the denominator must be large. Only then will H by R be very small. Do you understand? Right? When the numerator is small and the denominator is large, the answer will be small. It's the same here. The numerator must be small and the denominator must be large. That is, H must be very, very, very small compared to R. Only then can we consider H by R to be small. Fine. Clear. Is it okay? Let me give an example. Suppose it is 100 by 5. Here, the numerator is large and the denominator is small. What is the answer? The answer is 20. But if it is 50 by 100? Here, the numerator is small and the denominator is large. What is the answer? It is 0.5. The answer is small. Did you understand? So, when will the answer be small? The numerator must be small and the denominator must be large. Only then will the answer be small. It's the same here. The numerator must be small and the denominator must be large. That is, H must be very, very less than R. Only then can we consider H by R to be small. Fine. Clear. Now, if H by R is very small, we can put the power inside, right? We know that. Let's put the power inside. Pay attention. G dash by G is equal to, if the power goes inside, what will happen? 1 + H by R raised to the power of -2. The -2 comes inside. So, 1 - 2H by R. This is Equation Number 9. Let's multiply this G to the right side. G dash is equal to G into (1 - 2H by R). This is Equation Number 10. This is considered as Equation Number 10. This is our final equation. G dash is equal to G into (1 - 2H by R). This is our final equation. Everyone must learn this very important equation. This is an equation that must be learned and understood. This is very important. If this is clear, put a heart emoji in the chat box. Nivedan will give the entire derivation. If this is clear, put a heart emoji in the chat box. All my younger brothers and sisters, put a heart emoji. This is clear to me. There is not a single doubt. If all the steps are crystal clear, put a heart emoji in the chat box. Come on, guys. Again, I remind you that this equation can only be used under this special condition. You should realize that this equation can only be used under this special condition. Another thing I want to tell everyone is that we must also write a conclusion for this. It's no use just bringing it to the final answer. We must definitely write a conclusion. That is, what change happens to acceleration due to gravity when height increases? We must conclude that. Of course, we can conclude using this equation. But there is an easier equation than this. Pay attention. That easy equation is this. Pay attention. If the height increases, when height increases, we know that the value of R + H will increase. If the height increases, the denominator, that is, (R + H) squared, will increase. If (R + H) squared increases, G dash will decrease, right? Aren't they inversely proportional? G dash will decrease. Definitely. When height increases, acceleration due to gravity will decrease. Is it clear? Did you understand clearly? When height increases, acceleration due to gravity will decrease. Understand that. Everyone must learn this. This is very important. When height increases, acceleration due to gravity will decrease. Okay, fine. You must understand this condition. This final equation is applicable only under this condition. Shall we do a question? There are not many questions. Only one question. A question: At what height above the surface of the Earth is the acceleration due to gravity half of that on the surface of the Earth? Only one question. At what height is the acceleration due to gravity half of that on the surface of the Earth? It can be cracked easily. Stay with me, everyone. Stay with me. It can be cracked easily. What is our equation? G dash by G is equal to R squared divided by (R + H) squared. Do you remember this equation? You should remember. I taught you. The first equation I highlighted. That equation. Let's replace that equation. Multiply G to the right side. So, G dash is equal to G into R squared divided by (R + H) squared. This is taken as Equation Number 2. Students, tell me. What is G dash? The question is, at what height is the acceleration due to gravity half of that on the surface of the Earth? So, G dash is half of the acceleration due to gravity on the Earth's surface, right? G dash is half of the acceleration due to gravity on the Earth's surface, right? Definitely half. So, instead of G dash, we can put G by 2. Instead of G dash, let's replace it with G by 2. So, G by 2 is equal to what will come? G into R squared divided by (R + H) squared. This is Equation Number 3. Let's cancel the common terms. G and G cancel out. So, what will come? 1 by 2 is equal to R squared divided by (R + H) squared. Equation Number 4. Let's cross-multiply. So, (R + H) squared is equal to what will come? 2R squared. Cross-multiply. If we take the square root, R + H is equal to what will come? Root 2 R. So, H is equal to what will come? Root 2 R minus R. Okay. So, pay attention, everyone. Pay attention. Root 2 is 1.414. R minus R is just R. For simplification, I'll just put 1R. For ease of simplification. If you take R from 1.414 R, what is left? So, 0.414 R. This is the final answer. This is Equation Number 5. This is Equation Number 7. This is our final equation. Equation Number 8. The height is 0.414 R. Did you understand clearly? Does anyone have any doubts? If so, I can explain it again. At what height is the acceleration due to gravity half of that on the Earth's surface? This is the question. So, the equation we can use is this. The equation I highlighted first. That equation itself. I replaced G by multiplying it to the right. So, G dash is equal to G into R squared by (R + H) squared. Now, pay attention here. What can we replace G dash with? At what height is the acceleration due to gravity half of that on the Earth's surface? So, the acceleration due to gravity will be half of that on the Earth's surface. So, G by 2. Instead of G dash, put G by 2. So, G by 2 is equal to R squared divided by (R + H) squared. G and G cancel out. So, 1 by 2 is equal to R squared divided by (R + H) squared. Cross-multiply. This one to here and.

This is translated from Malayalam to English.

He multiplied this here and A + SR squared, took the root on both sides. What is the root of A + A squared? It is A + A. What is the root of A squared? It is A. A equals root A minus A. When R goes here, it becomes minus R. So, root means 1.414. So, you get 1.414 A minus 1 R. Is 0.414 A, which is 1.414 minus 1 R, correct? So, 0.414 A. If this is okay, can everyone comment with a heart emoji in the chat box? If you understand this, have no arguments, no doubts, and are confident, can everyone comment with a heart emoji? Then we can move on to the next one. You can guess the next one. Can you guess which derivation has been asked in all these years? Can you guess which important derivation has been asked in all these years? This is the most important, most possible derivation. What is it, children? Come on guys, respond! What has been asked in all these years? What was asked even in Christmas 2023? Variation of G with Yes, variation of G with depth, or what change happens to acceleration due to gravity when depth increases. What change happens to acceleration due to gravity when depth increases? That's what we are going to look at. So, understand, dear ones. The acceleration due to gravity on the surface is G. Let's assume the one at depth is G dash. So, the acceleration due to gravity at depth D is G dash, and the one on the surface is G. Fine. What is the acceleration due to gravity on the surface? Let's assume it is G. Shall we start? We are going to start the derivation. Pay attention. Let's start with the equation that everyone is familiar with. Mass equals density times volume. Isn't that known to everyone? That is, density equals mass by volume, right? Density equals mass by volume. So, mass equals density times volume. No doubts, right? So, mass of the Earth equals density rho. What is the volume? It is 4/3 pi R cubed. It is capital R, right? The radius is capital R, right? So, we wrote 4/3 pi R cubed, capital R cubed. We gave this equation number one and this equation number two. What is the acceleration due to gravity on the surface of the Earth? G, G. G equals GM by R squared. We gave this equation, which everyone knows, as equation number three. My children, understand here. Instead of this mass, let's replace this mass. Instead of this mass, let's replace this mass. Then G equals. Then Gs equals what will come? G by R squared. G by R squared, or G into. Instead of mass, I am directly replacing it. Rho into 4/3 pi. 4/3 pi R cubed divided by R squared. Equation number four. Pay attention here. We are making an important replacement. Not a replacement, but a cancellation. This R squared and R cubed. R squared cancels with R squared. We are going to rearrange the remaining terms. Okay? Rearrange the remaining terms. So, G equals. Let's take all the constant terms first. 4/3 pi. Shall we take 4/3 pi out? Yes. 4/3 G rho pi. What is left? Only R, right? 4/3 G rho pi. Only R will be left. No doubts, right? No one has any doubts. This is equation number five. What will be the next G dash, children? Tell me quickly. G dash equals. The acceleration due to gravity on the surface is 4/3 G rho pi R. What will be the acceleration due to gravity at a certain depth? What will G dash be? Can you comment? What will G dash be, children? You can do it. What is G dash? Children, recognize this. In the case of the surface, the distance from the center to the surface is R. Okay? When we come to this depth, what is the distance from the center? Is it not reducing the depth from the radius? Or is it R minus D? Definitely, it is R minus D. Replacing. That is, 4/3 G rho pi. Instead of R, it is R minus D. Equation number six. Did you understand clearly? Now, we are going to divide these two equations. Okay. Divide equation 6 by 5. Then what will equation be? G dash by G equals 4/3 G rho pi R minus D divided by 4/3 G rho pi R. This is equation number seven. Equation number seven. Let's cancel the constant terms. Let's cancel all the constant terms. This 4/3 G rho pi and this 4/3 G rho pi. I have cancelled them. Let's write down the remaining terms. Pay attention. The remaining terms are G dash by G equals what will we get? R minus D divided by R. Right? R minus D by R. Equation number eight. Now, my children, pay attention. This is a situation where many people will get confused. The correct answer is Sinan, right. Right answer. Dear. Then pay attention here. Let's multiply G there. Before that, I am taking R out commonly. So, G dash G equals. If we take R out commonly from the numerator, if we remove it from where R is, we get 1. We cannot take D out. D by R. So, R into 1 minus D by R. All divided by R. Equation number nine. Let's cancel the common terms. Let's cancel R and R. So, G dash by G equals what will we get? 1 minus D by R. Equation number ten. So, what will G dash be? Everyone pay attention. What will G dash be? G dash equals. If we multiply G there, G into 1 minus D by R. G into 1 minus D by R. This is the final answer for acceleration due to gravity. This is the final answer for acceleration due to gravity variation with depth. What change happens to acceleration due to gravity when depth increases? This is the final answer based on that. Step eleven. Finished. Now, as we said for height, what change happens to acceleration due to gravity when depth increases? We need that conclusion. Definitely, we need that conclusion. So, that's what we are going to. Shall I tell you an easier equation? Shall I tell you an equation that you can conclude most conveniently? Pay attention. This depth, right? Assume depth increases. Assume depth increases. What will happen to R minus D? If depth increases, will R minus D be less? Let me give an example. 100 minus 25 is 75. What is 100 minus 60? It is 40. That is, this term, when the value of the second term increases, the answer decreases. Right? Here, the value of the second term is less, the answer is more. When the value of the second term increases, the answer decreases. That is, if depth increases, R minus D will definitely decrease. Understand that R minus D decreases. If R minus D decreases, acceleration due to gravity also decreases. Understand clearly. If depth increases, the value of R minus D decreases. If R minus D decreases, acceleration due to gravity also decreases. Because everything else is constant. So, R minus D and acceleration due to gravity are directly proportional. If R minus D decreases, acceleration due to gravity also decreases. Understand one thing. Increases. Acceleration due to gravity is like height. When height increases, acceleration due to gravity decreases. Similarly, when depth increases, acceleration due to gravity also decreases. Shall we look at a single question related to it? Just one question. We can discuss the remaining questions at the top level. Shall we look at just one question here? One question. What is the question? Weight of the body is determined by acceleration due to gravity. What will be the weight of the body at the center of the Earth? Tell me quickly. What will be the weight of a body at the center of the Earth? All brothers and sisters, tell me quickly. What will it be? You don't need to have a shred of doubt. What will be the weight of a body at the center of the Earth? Yes. So simple. The equation we all know is Weight equals MG. So, since it is at the center, it is not wrong to give it as MG dash. So, what is the formula for G dash? G dash equals G into 1 minus D by R. So, G dash equals G into 1 minus. Tell me. To reach the center of the Earth, how deep do we need to dig? Assume I want to dig to the center of the Earth. How deep do I need to dig? We need to dig to a depth equal to the radius of the Earth. That is, depth is equivalent to the radius. If so, what will D by R be if depth equals radius? D by R will be equal to 1. Zero. G dash will be equal to zero. If G dash is zero, what will the weight be? The weight will definitely be zero. So, what will the weight be at the center of the Earth? Understand that the weight will be zero. Clear? If D equals R, then D by R will be 1. Fine. Did you understand clearly? This is a possible question in exams. Shall I move on to the next derivation? Can you predict which important derivation has been asked in all these years? Can you guess which important derivation has been asked in all these years? Can you predict? If you can, please predict. What will that question be? What will that question be? The one asked in all these years, even in 2025? There is no doubt. This is known as Escape Speed. What is escape speed? If I throw an object from here with a single push, it will come back to Earth after some time. Right? It will come back to Earth. But when an object is projected from the Earth, it never comes back to Earth. The minimum speed required to overcome the Earth's gravitational field or the Earth's gravitational force and escape is called escape speed. If we throw an object from the surface of the Earth, it will come back to Earth. The reason for this is the gravitational force exerted by the Earth on it. However, the minimum speed required to overcome this gravitational force and escape to infinity is called escape speed. In some cases, it is also called escape velocity. The correct word is escape speed. Okay. Right. Escape speed is the correct word. Shall we look at it? What is the escape speed of the Earth? What is the escape speed of the Moon? These are very repeated questions. So, the escape speed of the Earth is 11. Sorry, guys. Let's write a bit lower. The escape speed of the Earth is 11.2 kilometers per second. The escape speed of the Moon is 2.38 kilometers per second. Learn this. Both are asked repeatedly in exams. The escape speed of the Earth and the Moon are asked repeatedly in many exams. Learn it. Let's move on to the derivation of escape speed. So, what is the work done to take the object from the surface of the Earth to infinity? Just write that. We don't need to start from integration method here. Just start from here. What is the work done to take an object from the surface of the Earth to infinity? Work equals. Learn this. You only need to learn this. Work equals GMm by R. This is the work equation. Equation number one. That is, the work done to bring an object from the surface of the Earth to infinity is GMm by R. This work is actually equivalent to what? The body is in motion, right? So, this work is equivalent to what energy? The energy possessed by a moving body is kinetic energy, right? Therefore, this work is equal to kinetic energy. No doubt. Kinetic energy equals. We can write work. So, 1/2 Mv squared equals GMm by R. Can we write it more neatly? It looks untidy. So, GMm divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape speed. It is because it is escape speed that I have given 'e' as a subscript. Okay. Ready. The board is starting to hang a lot. Guys, pay attention. G M M divided by R. This is known as equation number two. This is equation number three. You need to pay attention here. It is not the usual speed, but escape speed. Therefore, I am giving a subscript 'e' to v because we are going to find the escape

It is difficult to find the total work to bring it in. So that's why we take two points in between, P and Q. So initially, we find the work to bring the mass called M from P to Q. Then, by integrating it, we find the total work. Then, by integrating it, we find the total work. This is the plan. So, where is the mass called M now? It is at P. So, we find the work to bring it from P to Q and then integrate it to find the total work. Let me explain the diagram once more. This is our Earth. The mass of the Earth is capital M and the radius of the Earth is capital R. So, this is infinity. We need to bring it from infinity to A. Point A is at a distance R from the center. Next point. What about point P? It is at a distance x from the center. Point Q is at a distance dx from P. It would be very good if you draw this accurately. Let's draw it. Fine, okay. Shall we start the derivation? We are going to derive. The gravitational force on a body at the point P is given by, that is, what is the gravitational force on the object when it is at P? The mass is at point P. So, what is the gravitational force in that situation? The distance is x. So, what will be the gravitational force, children? Isn't it something everyone knows? Gravitational force F is equal to GMM divided by x. So, x squared. This is equation number one. Okay, right. Next, if the body is displaced from P to Q, what is the work to be done to displace the body from P to Q? We know it's a small displacement, so it's small work. The small distance is dx. So, it's small work. What is small work? DW. DW is equal to what? Force into displacement, right? What is the force? Force into displacement dx. What is the force? GMM by x squared into dx. Equation number two. We got the work, GMM by x squared into dx. It's small work, but we all know we don't need to bring it from P to Q. To bring the mass from infinity to A. So, to find the total work, we need to integrate this small work. We are going to integrate it. Stay with us, guys. Integrate both sides. Integral of DW is equal to integral of GMM by x squared into dx. When integrating the right side, where are we bringing it from and to? From infinity to A, right? Point A is at a distance R. So, understand the limits. It is from infinity to R. Equation number three. Understand the limits accurately. Infinity to R. Equation number three. Let's take the constants out of the integration. So, we are going to take GMM out of the integration. If we integrate DW, we get total work W. Which is equal to GMM integral from infinity to R of 1 by x squared dx. Equation number four. Children, I am taking 1 by x up. That is, I am taking x squared to the numerator. x squared, that is, 1 by x can be written as x raised to the power of -2. When taken to the numerator, the power becomes negative. So, pay attention. W is equal to GMM integral from infinity to R of x raised to the power of -2 dx. Equation number five. Everyone, pay attention. The integral of x raised to the power of n dx is x raised to the power of n+1 divided by n+1. That is, the integral of x squared dx is x raised to the power of 2+1 divided by 2+1. The integral of x cubed dx is x raised to the power of 3+1 divided by 3+1. So, what will be the integral of x raised to the power of -2 dx? Can't you say it easily? DW is equal to GMM x raised to the power of -2+1 divided by -2+1. The limit is from infinity to R. Equation number six. -2+1 is -1, right? Definitely -1. So, W is equal to GMM x raised to the power of -1 divided by -1. Limit from infinity to R. Equation number seven. I am taking this -1 in the denominator out. I am taking this minus sign straight out. So, we can put a minus sign in front, right? Definitely, we can put a minus sign in front. So, W is equal to what? Minus GMM. Inside the bracket is x raised to the power of -1. x raised to the power of -1. Limit from infinity to R. Equation number eight. x raised to the power of -1 can be written as 1 by x. So, W is equal to -GMM into what? 1 by x. Limit from infinity to R. Equation number nine. First, let's substitute the upper limit. W is equal to -GMM. The upper limit is R. So, 1 by R minus 1 by infinity. This is equation number ten. What is 1 by infinity, children? 1 by infinity is zero. So, what do we get? W is equal to -GMM into 1 by R. Equation number eleven. Shall we write it together? It's the final answer. So, W is equal to -GMM by R. This is the work. Potential energy is stored. U is equal to -GMM by R. This is our final equation. Gravitational potential energy U is equal to -GMM by R. Equation number twelve. This is thirteen. This is equation number. If you are confident, put a heart emoji in the chat box. I understand. I understood it perfectly. If you say there is no doubt, put a heart emoji in the chat box. Then we can move on to the next. The next thing is very simple. That is gravitational potential. After gravitational potential energy, the next task is gravitational potential. It can be said easily. What is gravitational potential? It is the work done to bring not any mass, but a unit mass. The work done to bring a unit mass is called gravitational potential. Gravitational potential energy is the amount of work done to bring a mass from infinity to a particular point in Earth's gravitational field. Gravitational potential means the mass we introduce will be a unit mass. Pay attention. In gravitational potential energy, it was -GMM by R. Understand here accurately. Mass means one. Unit mass. So, what will the equation be? V is equal to -GM by R. -GM by R. That's all. This is the expression for gravitational potential. That's all. Rajimol, you are right. Last derivation. The end of this chapter. That is, the energy of an orbiting satellite. We are going to look at the energy of an orbiting satellite. It will have two energies because it is moving. It will definitely have kinetic energy. The satellite is at a certain height from Earth. It will also have potential energy. So, everyone understand. Total energy will be the sum of what energies? Kinetic energy and potential energy. No doubt. Equation number one. Let's find each energy first. Let's find kinetic energy first. What is the equation for kinetic energy? Half M V squared. That is equal to half M into V, which is orbital velocity, right? Orbital velocity is known. Root of GM divided by R+H. Squared. This is equation number two. This is equation number three. Half M into root of GM by R+H. Squared. Put the square inside, kids. I am going to put this square inside. Pay attention, everyone. What do we get? Kinetic energy is equal to one by two M. When the square is put inside, the root goes away, right? So, GM divided by R+H. Understood perfectly. I'll make it neat. I'm going to make it neat. That is, kinetic energy can be written as half GM M divided by R+H. This is the expression for kinetic energy. This is equation number four. This is equation number five. Understood perfectly, right? Kinetic energy is obtained. One by two GM M by R+H. Potential energy can be found very simply. There is no difficulty. Because you know the general equation for potential energy. U is equal to what? Minus GM M by R. So, this is the equation for potential energy here. So, what will come here? There is only a small change. No big change. Potential energy will be minus GM M. No change in that. Instead of R, what will be the radius? Instead of R, what will come? R+H will be the radius. R+H. There are no other changes. This is equation number six. This is our sixth equation. Potential energy is equal to -GM M by R+H. Understood crystal clear, right? No doubt at all? Confident? Now, shall we add the two energies? Let's add the two energies. Let's go. Add both energies. Ready, children? Pay attention, everyone. So, total energy is equal to kinetic energy, which is one by two GM M by R+H. Potential energy is minus GM M by R+H. Equation number seven. From this, if we want, we can take out GM by R+H. So, total energy is equal to GM M by R+H. If we take it out, what is left, children? One by two minus one. One by two minus one is minus one by two, right? Total energy is equal to GM M by R+H into minus one by two. What is the answer? Total energy is equal to minus one by two GM M by R+H. This is our final answer. This is our expression for total energy. And this is the final equation. Everyone must learn this. Very important. The equation for total energy. This is very, very, very important. Questions related to this are frequently asked in exams. You must learn it. Minus one by two GM by R+H. Here, you need to understand some relations. Some relations. If you pay attention, you will understand. Total energy is equal to the kinetic energy equation with a minus sign put in front. So, minus of kinetic energy can definitely be said. That is the first thing. Next, let's look at the relationship between total energy and potential energy. This is potential energy. If you multiply this potential energy by one by two, what do we get? We get total energy. So, total energy is equal to one by two times of potential energy. We can write it like that. No doubt. Next, what is the relation between potential energy and kinetic energy? Look, it's simple. We can tell the relation between the two. This potential energy, right? How can we find potential energy? Isn't it enough to multiply this kinetic energy by minus two? If we multiply kinetic energy by minus two, what do we get? We get potential energy. Do you understand? If we multiply this kinetic energy by minus two, and cancel the two twos, what do we get? We get potential energy. So, we will also write that relation if needed. That is, potential energy is equal to minus two times of kinetic energy. Potential energy is equal to minus two times of kinetic energy. Just learn these. They might be asked in exams. One-mark questions related to this are frequently asked in exams. This is important. Let's move forward. We are moving to the next. Pay attention. Next. We are moving to the next. Everyone stay with us. What is the last topic in this chapter that has been asked in all these years? Tell me quickly. Thermodynamics will be taught tomorrow. Today, it's seven chapters, right? What are you talking about? Isn't it the first seven chapters? Aren't the remaining chapters taught tomorrow? Why are you like this? What is the question asked in all these years, including the model questions? No doubt, it is Kepler's laws. There are three laws of Kepler: Law of Orbits, Law of Areas, Law of Periods. We are going to learn all three. First, Law of Orbits. What is the Law of Orbits? We all know that all planets revolve around the Sun in elliptical orbits. All planets revolve around the Sun in elliptical orbits. The Sun is one of the foci of the ellipse. The Sun is one of the foci of the ellipse. That's all. All planets revolve around the Sun in elliptical orbits. An ellipse has two axes. The longer axis is called the major axis. The shorter axis is called the minor axis. Isn't this what you learn in Maths? So, the Law of Orbits states that all planets revolve around the Sun in elliptical orbits. The Sun is one of the foci of the ellipse in which they revolve. Understand this. So, guys, we are moving forward. Next is the Law of Areas. What does the Law of Areas say? The Law of Areas is very simply put, the area covered by the planet in a proper time interval around the Sun will be the same. The area covered by the planet in a proper time interval around the Sun will be the same. That's the point. Simply put, let's say a planet moves from P to P dash in a time called delta T. Let's say it takes three months. So, in three months, the planet traveled from P to P dash. The area it covered during that time is delta A. The area covered by the planet with the Sun during that time is the area of this shaded portion, let's say it is delta A. So, the planet traveled from P to P dash in a time called delta T. Let's say the area it covered in that time is delta A. The planet revolves and revolves and revolves and revolves and revolves and reaches here. Now, the planet travels from A dash to A in the same way. Let's assume it also travels in the same time called delta T. So, if the time is the same, the area covered here will be the same as the area covered here. The same area will be covered in that time. Because the time is the same. In the same time interval, the area covered by the planet with the Sun will be equal. That is, the area swept by the planet in equal intervals of time is equal. The area swept by the planet in equal intervals of time is equal. That is, the aerial velocity is constant. Delta A by delta T is equal to constant. Or aerial velocity. Aerial velocity is constant. Understand this accurately. Equal intervals of time, equal areas will be covered. Aerial velocity is constant. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to one by two into L divided by M. Equation number five. Children, understand. One by two is a constant. Mass is constant. There is no external torque acting here. Angular momentum is constant. You learned this in the previous chapter, right? In the seventh chapter, you learn that if there is no external torque, the total angular momentum is constant. Isn't that what you learned? Therefore, since there is no external torque here, momentum is constant. Aren't all the terms on the right side constant? One by two is constant, L is constant, M is constant. No doubt. Set, right? Therefore, we can write delta A by delta T is equal to a constant. That is, aerial velocity is constant. The planet covers equal areas in equal intervals of time. The planet covers equal areas in equal intervals of time around the Sun. That is, aerial velocity is constant. Understand this accurately. This is the Law of Areas. If you want the derivation, I will tell you. If you want the derivation, I will tell you. Pay attention. I am going to find the area of this portion. I am going to find the area of this portion. We will consider this as a triangle. Delta A is equal to half into base into height. Half BH. So, delta A is equal to half into what is the base? The base is equivalent to the position vector, right? This is equivalent to the position vector R from the Sun. So, instead of the base, we put R. And what is the height equivalent to? The height is equivalent to this displacement. Isn't it displacement? Displacement from P to P dash. Children, displacement is velocity into time. Displacement is velocity into time. So, instead of height, we can put velocity into time. V into delta T. I am taking delta T to this side. So, delta A by delta T is equal to what? Half into R cross V. This is equation number one. This is equation number two. This is equation number three. Children, tell me. Instead of velocity, what can we replace? Momentum is equal to mass into velocity. Velocity is equal to momentum divided by mass. It's a vector, children. Momentum by mass. I will replace it. Instead of velocity, let's put momentum by mass. Pay attention. Delta A by delta T is equal to half into R cross P by M. We all know that before that, let's put the equation number. This is equation number four. We all know that R cross P is what? It is angular momentum L. R cross P is angular momentum L. I will replace R cross P with angular momentum L. So, delta A by delta T is equal to

Every particle's same angular velocity BD linear velocity. All particles have the same angular velocity and different linear velocities, or the same linear velocity and different angular velocities. Same angular velocity and same linear velocity. Different angular velocity and different linear velocity. What is it, children? Tell me the answer. The rotation of the second hand of a clock is rotational motion. In pure rotational motion, as we all know, the angular velocity of all particles will be the same, and the linear velocity will be different. Option A is the right answer. Option A is the correct answer. Shall we move on to the next one? We are moving on to the next one. In pure rotational motion, every particle of the body has the same angular velocity. If a body is in rotational motion, the angular velocity of all particles within it will be the same. Is the statement true or false? We know the answer is true. All particles will have the same, children, the same angular velocity. Are you ready? Are you ready? Are you ready? I hope everyone understood. I believe it's okay. I believe it's okay. Is it okay, children? Yes, yes, yes, yes. Okay, set, set, set. Let's move on to the next one. Let's move on to the next one. Yes, the next is the center of mass. What is this center of mass? It is the imaginary point where the entire mass of a body is concentrated. We call it the center of mass. It is the name we call the imaginary point where the entire mass of a body is concentrated, the center of mass. So, with that center of mass, we can balance this body. You are seeing the center of mass of some bodies. If it is a cylinder, its center of mass will be at the center of its axis. The center of mass of a disc, sphere, or ring will be at its center. Similarly, the center of mass of a rectangle or a square will be where the diagonals meet. Similarly, the center of mass of a triangle will be where the centroid is. So, these are all bodies with a symmetric shape. If it is a symmetric shape, the center of mass will be at the geometrical center. The center of mass will be at the geometrical center itself. Are you ready? Okay? Okay? Then there is a calculation for this center of mass. There is an equation to find the center of mass. Okay, let's look at that equation. If the body is in one dimension, okay, we have two masses, m1 and m2. This is one dimension. If it is one dimension, don't worry about anything else. If we have two particles, or if all particles are on a straight line, we can use this equation. Look at the equation, children. x1 is the position of the particle called m1. x2 is the position of the particle called m2. Then, let's assume the center of mass is somewhere here. The distance to the center of mass is x. Then the simple formula to find it is something everyone should know, everyone should learn. x is equal to, children, the equation is m1, taking the first mass, its x-coordinate, plus taking the second mass, its x-coordinate, if there is a third mass, plus the third mass multiplied by its x-coordinate, divided by the total mass, m1 plus, what is it, children, m2. This is our equation. Okay, what about two dimensions? In two dimensions, there will be a y as well, right? So, in two dimensions, we already know x. What is special about the dimension? Children, there will be a y-coordinate as well. There will be a y-coordinate. Yes, let's see. I have two particles. I have placed two particles in an xy plane. Yes, it is the xy plane, two dimensions. Placed in the xy plane. There are two particles. The x-coordinate of the first particle is x1, its y-coordinate is y1. Taking the next particle, its x-coordinate is x2, its y-coordinate is y2. If there is a third particle, its x-coordinate is x3, its y-coordinate is y3. Then, let's assume the center of mass of these two particles is somewhere here. This is the first mass. This is the second mass. The x-coordinate of the center of mass is x. The y-coordinate of the center of mass is y. Then the equation to find it. The x-coordinate can be found as before. The equation to find the y-coordinate, you all know, right? y is equal to, children, what is the equation? If it is the y-coordinate, y1 will come. Plus, since it is the y-coordinate, y2 came. Divided by m1 plus, what is it, children, m2. Shall we draw this in a box? We have drawn this beautifully in a box. Ready? Ready? Did everyone understand? Did everyone understand? Now, if it is three dimensions, what else will come, children? A z-coordinate will also come. Okay? Now, in terms of position vector. Look, children, in terms of position vector. Position vector. Position vector. What about in terms of position vector? All children, look. What about in terms of position vector? Let's see. Yes, this is our plane. So, here is our first particle. Its mass is m1. Here is our second particle. Its mass is m2. So, we are drawing its position vector. Look carefully, children. This is the position vector of the first body. We called it r1. This is the position vector of the second body. This is the position vector of the second body. We called it r2. Now, I assume the center of mass is somewhere here. So, I am drawing the position vector of the center of mass. Pay attention, children. This is the center of the center of mass. Yes, this is the position vector of the center of mass. This is the position vector of the center of mass. I called it r. I called the position vector of the center of mass, what, children, r. Then, similarly, the equation to find the position vector of the center of mass is very easy. r is equal to. We take m1. Take its position vector. Plus, take m2. Take its position vector. Divided by the total mass, m1 plus m2. I hope everyone understood this. I hope the elder brother believes it. Did everyone understand this? If you understood the calculation of the center of mass, tell me you understood. If it is the x-coordinate, x1 plus m2x2 plus m3x3 plus m4x1 divided by m1 plus m2 plus m3, etc. Now, if we are finding the y-axis, m1, its y-coordinate, m2, its y-coordinate, m3, its y-coordinate, add all of them and divide by the total mass. Now, in terms of position vector? In terms of position vector, nothing. Take m1, its position vector, plus take m2, its position vector, take m3, its position vector, divided by m1 plus m2 plus, etc. Okay? This is how we find the center of mass. If everyone understood this, let's do a problem with an elder brother and sister related to this. A woman weighing 59 kg and a man weighing 71 kg are sitting at the two ends of a seesaw. The length of the seesaw is 3.5 meters. Where will the center of mass of both of them be? Shall we find out? Yes, when such a question comes, what you should do first is, if possible, draw an xy axis. Just draw an xy axis. So, let's place our first person, our elder sister, at the origin. We placed our elder sister at the origin. This is our first mass, m1, which is our elder sister. The mass of the elder sister is 59 kg. Now, our elder brother, children, is m2. A seesaw is a straight line. We know it's a straight line. So, if there are only two masses, you should prefer the x-axis. Okay? Now, if there are more than two masses, you can place the first two masses on the x-axis and the third mass anywhere. Ready? So, m2 is the mass of the elder brother. m2 is the mass of the elder brother. m2 is the mass of the elder brother. Okay. We know this distance, children, right? This distance is the length of the seesaw, isn't it? It is given as 3.5. It is 3.5. Ready? Ready? Ready? So, look here. This is one dimension. So, we only need to find x. What is the x-coordinate of m1, children? Let's call the x-coordinate of m1 as x1. Everyone comment what it is. What is the x-coordinate of the first body? Everyone comment. It is zero. It is at the origin. That's why we brought it to the origin. So, one of them became zero, right? What is the x-coordinate of the second mass? It is 3.5. Okay? We need to find the center of mass. Always try to draw the center of mass near the heavier mass. Where the heavier mass is, there the center of mass will be. So, this is our center of mass. We are going to find the x-coordinate of the center of mass. We called the distance to the center of mass as capital X. We need to find X. We are going to copy our equation. Everyone pay attention. What is our equation, children? The equation to find the center of mass is m1x1 + m2x2 divided by m1 plus, what will it be, children, m2. The value of m1 is 59 kg. x1 is zero, see? That's why we did that. Now, m2 is 71. x is 3.5. Divided by the total mass, which is 59 plus 71. If you do this, you will get the answer 1.91 meters. Who got the answer, children? Did everyone get the answer? What other activities are happening in the chat box? Did you get it, children? If you got the answer, say you got it. Everyone say you got it. Ready? Ready, sir. Okay, sir. Did you get it, children? Yes, yes, everyone got it. Yes, I got it. You didn't get it, right? Okay. Now, what will y be? Is it zero? There is no need to find it. Okay. So, we can say the center of mass will be at 1.91 meters away from the woman. You can write it down like that. If you draw the diagram, everything will be understood. This is a question from the March 2023 paper. The point at which the whole mass of the body is supposed to be concentrated is called what, children? What is the name of the point where the whole mass is concentrated? Quickly, quickly, quickly, tell me. Center of mass. CM is our center of mass. Next, we are going to study torque. Not torque, children, torque. Torque. What is torque? When we apply force to a door, doesn't the door rotate? The door doesn't go like this, like this, like this. The door goes like this. The door rotates. It turns around a fixed axis. This turning effect is actually what we call torque. Torque is the turning effect of a body. So, when we apply force to a body, if the body turns, we say there is torque. So, torque is the cause of rotation. If force is the cause of linear motion, torque is the cause of rotation. So, it is the rotational analog of force. In rotation, what we study instead of force is torque. Torque is the cause of rotation. If force is the cause of linear motion, torque is the cause of rotation. That is why it is the rotational analog of force. It is also called the moment of force. The thing related to this rotation is called moment. Not momentum, not momentum. Moment. Things related to rotation are called moment. Moment of force. Force that causes rotation. It is understood in that way. Torque is the moment of force. Now, what does torque depend on? Torque depends on three things: force, radius, and angle. You will understand if you look. If I apply force to this spanner, only then will it rotate. If I don't apply force, it won't rotate. If I apply good force, it will rotate well. If I apply small force, it will rotate a little. Okay? Does it depend only on force? Definitely not. If I apply force here, near the nut, the nut will rotate. It will never rotate. If I apply force here, it will rotate. If I apply force here, it will rotate a little better. If I apply force here, it will rotate well. We don't say, will it be easier to rotate by holding the spanner at the very end? Where do we place the handle of the door? Do we place it at the hinges? Definitely not at the hinges. Instead, it is placed very far from the hinges. Because as this distance increases, the turning effect increases. We can reduce our effort. Okay? So, the turning effect increases. If the radius increases, the turning effect increases. So, it depends on the radius or position vector. If the radius increases, the turning effect increases. Then there is one more thing, the angle. Torque also depends on the angle. Let's look at it in detail. Let's say I am applying force like this. I am applying force like this. This is our radius, right? I am extending the radius. I am extending the radius vector like this. This is the radius vector, and this is the force vector. The angle between this radius vector and this radius vector is what we call theta. What is theta? Theta is the angle between the radius vector and the force vector. Theta is the angle between the radius vector and the force vector. Then we can resolve the force into two components. Everyone pay attention. One component will come upwards. One component will come upwards. One component will come this way. What will this be, children? This is F cos theta. This is F cos theta. And this is F sin theta. Children, tell me, will F cos theta provide rotation? Will F cos theta provide rotation? No. It will never rotate if you apply force like this. If you apply force like this, it will rotate. Common sense. So, F cos theta is. Sorry. If you study now, will you get full A+? One thing I can tell you for sure. If you haven't started studying yet, and you still haven't started studying, you will not get full A+. I can tell you that for sure. If you don't study now, you will not get full A+. So, why not study? Whether you get full A+ or not is a different matter. If you don't study, you definitely won't get it. So, try studying. Whether you get it or not is another matter. If you don't study, you definitely won't get it. So, try studying. Okay? Okay? If you don't study anymore, you will get into trouble. Ready? Now you decide whether to study or not. Now, children, pay attention. Torque. What is torque? What is torque? What is torque? Torque depends on what? It depends on force, right? It depends on the angle, which is sin theta, right? It depends on sin theta. So, torque depends on the radius. It depends on the force. It depends on the angle. So, this is the magnitude of torque. The magnitude of torque is r sin theta. Ready, children? The magnitude of torque is r sin theta. Torque is a vector quantity. Torque has direction. The rule for determining the direction of torque is the right-hand thumb rule or the right-hand screw rule. That is, if our curled finger represents the rotation of a body, if the body is rotating this way, the thumb represents the direction of torque. If the body is rotating this way, this thumb represents the direction of torque. Look at the blades of your fan, children. See the fan rotating. See the fan rotating. Then tell me, in which direction will the torque generated by the fan's rotation be? Everyone comment. Another way to say it is, place your curled finger from the radius vector to the force vector. Place your curled finger from the radius vector to the force vector. From the radius vector to the force vector. From the radius vector to the force vector. Place your curled finger from the radius vector to the force vector. From the radius vector to the force vector. Place your curled finger from the radius vector to the force vector. The thumb represents the direction of torque. So, the unit vector going in the direction of torque, I call it n-cap. What is n-cap? N-cap is a unit vector along the direction of torque. The unit vector going in the direction of torque is what we call, children, what we call, n-cap. Torque is a vector quantity. What we have written so far is only the magnitude. So, we are going to represent torque as a vector. This is where we study the cross product. We study the vector product. Torque is a vector quantity. The radius is a position vector, which is a vector quantity. The force vector is a vector quantity. When we multiply the radius vector and the force vector, the person we get is torque. Torque is the multiplication of the radius vector and the force vector. The name we call this multiplication is vector multiplication or cross product. When we multiply two vectors and get another vector, the name for this is cross product or vector multiplication. We studied dot product, now we will study cross product. Ready? We are doing a problem for this. We are doing a simple problem, not a big problem like in the dot product where we find a large angle. Everyone look. So, look, we are going to represent this torque as a vector. We can all tell the equation for torque. The magnitude of torque is r sin theta. This is the magnitude, the measurement. Tell me now, 10 Newtons, 20 Newtons, these are magnitudes. 10 Newtons North becomes a vector. It becomes a vector only when direction is added. So, we got the magnitude of torque. Now, what else do we need? Its direction. We already know that the direction of torque is represented by n-cap. Direction of torque. It is found using the right-hand thumb rule. Okay? So, if we multiply this magnitude by n-cap, it has now become a vector. To change a magnitude into a vector, what should we do? Multiply by a unit vector. Okay? What is a unit vector? It is a vector with a magnitude of one unit. So, if we multiply this magnitude by n-cap, it becomes a vector. The direction of this vector will be the same as the direction of n-cap. The magnitude of n-cap is one, as we know. Okay. I got 10 units of torque. I got 10 units of torque. Okay? Then, if I multiply n-cap by this 10, won't its length increase? Yes, its length will increase. This is torque. Its direction will be the same as the direction of n-cap. Its length is now 10 units. So, to change a magnitude into a vector, learn this: if you multiply something by a unit vector, it becomes a vector. The direction of that vector will be the same as the direction of the unit vector. The direction of the unit vector is the direction of torque. That's all. We can also represent this symbolically. We can represent this symbolically. Look, everyone. Torque, tau, is equal to r cross F. Nothing else. We can represent rF sin theta with n-cap in this way symbolically. That's all. What is this? What is this? This is r sin theta multiplied by n-cap. We can represent it vectorially in this way. That's all. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This is. This. This is. This is. This is. This

B means we are taking the dot product of A and B. When we take the dot product of A and B, what do we get? We get a scalar quantity. A dot B means AB cos theta. A cross B is AB sin theta. For these to be equal, if their magnitudes are equal, the angle is 45 degrees. Sin 45 and cos 45 are equal. So, for the magnitude of A dot B and the magnitude of A cross B to be equal, the angle must be 45 degrees. If it is 45, A and B are equal. If it is 45, sin 45 and cos 45 are 1 by root 2. We can add that too, right? That is, the magnitude of A dot B and the magnitude of A cross B must be equal. We can only talk about the magnitude. Because A cross B is a vector, it cannot be equated to a scalar. So, we must mention the magnitude of A cross B. Even if we say A dot B, it's okay, but we must say the magnitude of A cross B, because A cross B is a vector quantity. A vector cannot be equated to a scalar. Okay? When is this equal? If theta is 45 degrees, if theta, the angle, is 45 degrees, A cross B and A dot B will be equal.

Next, let's look at the question. It is difficult to close an open door by pushing it from the hinge side. Why is it difficult to close and open a door by holding it at the hinges? We know it's difficult at the hinges, right? So, sir, hinges mean nothing. It's where we fix it, right? Hinges are what fix the door to the frame. Where is our grip? It's away from the hinges. We know that if the radius decreases, the turning effect decreases. If the radius decreases, the turning effect decreases, that's the reason. Didn't we do the 2021 model question with dignity, children? Next is the 2023 model question. We have the force vector. We have the radius vector. Find the torque. This is a vector multiplication. The 2021 model and the 2021 model and the 2021 model asked the same question. We can include this question in the topper series if we want, or you can do it as homework. I will show you one question. You do one as homework. Okay, let's do this. Seeing this, it is an important question. There is a possibility of such a question being asked. We have the force vector and the radius vector. Such questions are likely to be asked in the model exam. It was asked in 25, so maybe look at the pattern. It was asked in the 2021 model. It was asked in 2023. It might be asked in 25, but it hasn't been asked yet, so it might be asked in 24. The calculation is slightly different, but no one needs to know, only we need to know. Look, so we need to find the torque, right? Torque means we know it's R cross F. Torque means we all know it's R cross F. A shortcut method to find this torque is the determinant method, or else you will have to sit here tomorrow doing cross products. So, a shortcut method to find the cross product is the determinant method. Didn't we learn to find the dot product there? Finding the dot product is very simple. Multiply this with this, multiply this with this, multiply this with this, and add everything up. But for the cross product, there are some complexities. So, there is a very high chance of this being asked. So, draw two lines. This is called the determinant. In these lines, put I, put J, and also put K cap. Ready? Now, who comes first? R, right? So, write the I, J, and K of R below this. What should you write, children? Write three, write three, write three. Isn't it that simple? Now, who comes next? It's the force. So, write the I, J, and K of the force below. So, what is it? Write two, write minus five, and also write four. Include the sign here too, okay? Otherwise, it will be wrong. Now, what should be done? Now, the work to be done. Everyone pay attention. The main work is this. First, we take out the I cap. We take out the I cap. Can we take it out? Take out the I cap. Eliminate the row and column containing the I cap. Eliminate it, children. See all this? See all this? Multiply this with this. Multiply this with this. Put a minus in the middle. The work is done. So, look, multiply this with this. Multiply this with this. Put a minus in the middle. That is, 3 times 4 minus 6 times minus 5. Okay, children? Look, multiply this with this. Multiply this with this. Put a minus in the middle. Now, after I, children, what comes next? J comes. When J comes, remember one thing. Put a minus. What is it, children? Minus J cap. Minus J cap. Okay? Ready? Minus J cap. Minus J cap. Is it okay, children? Ready, children? Yes. Now, let's write this. I don't think we can write everything. The model exam won't budge from there. Nasty model exam. Okay. So, let's write J cap. When writing J cap, first it's plus, then it's minus. Okay? When writing J cap, eliminate its row and column. Just like before, multiply this with this. 3 times 4. 3 times 2. 3 times 4 minus 3 times 2. Now, next is plus. Now, after that, it's K, right? Plus, minus. Next is plus. Look, everyone, look. Next is K cap. Eliminate the row and column of K cap. Eliminate the row and column. 3 times minus 5. 3 times 2. 3 times minus 5 minus 3 times 2. Ready? Now, it's simple. Simple mathematics. I cap. 3 times 4 is 12. 12 minus 6 times minus 5 is minus 30. So, it becomes plus 30. Plus 30. Sir doesn't know how to add and subtract, okay? You all help me. J cap. J cap. 3 times 4 is 12. 3 times 2 is 6. So, 12 minus 6. Last one is K cap. K cap. Pay attention. K cap. What is here? Minus 15 and minus 6. Minus 15 and minus 6. Children, what is the answer? 27 I cap. Oh God. Did anyone get it? 27 I cap. Next, 12 minus 6 is 6. So, minus 6 J cap. Awesome. Now, minus 21. Minus 21 K cap. Did anyone get this? This is the torque here. And write one more thing: Newton meter. What is the unit of torque? Newton meter. Or Joule. Did everyone get it, children? Minus 11 I cap. Oh no, don't say that. Where did you get minus 11 I cap? It's 27 I cap, minus 6 J cap, 21 K cap. I hope everyone got it. You should do this as homework. Do it as homework. Just put the answer in the comment box. If you put the answer in the comment box, I will check and tell you. Sir, a motivation. If you don't study, children, you won't become anything. You will fail. Okay. This is the 2023 model question. If vector B is parallel or perpendicular. The cross product between them is zero. If the cross product is zero, are those two vectors parallel or perpendicular? We know that if they are perpendicular, the dot product is zero. If they are parallel, the cross product is zero. We all know when the cross product is zero. What is the answer, children? Parallel. What is the answer? Parallel. If they are perpendicular, we all know that the dot product is zero. Right? If they are perpendicular, we all know that the dot product is zero. Okay? Ready? Ready? This is the model question. Next question is the expression for torque. For a body in rotational motion, we all know it's tau sin theta or R cross F. You can write either. It's okay to write only the magnitude. Now, what is the method to change the magnitude of force? It's asking for a method to increase torque. If you increase the radius, what will increase, children? Torque will increase. Okay? Did everyone understand? The 2023 model question is also caught. The 2023 model question is caught. What is moment of force? What is moment of force? Moment is related to rotation. Moment of force is torque. Moment of inertia is moment of inertia. Moment of inertia is moment of inertia. There is nothing special about it. Moment of momentum is angular momentum. Okay? Sir, I don't know what happened to this physics. It was cut and deleted. I didn't see this physics. But ready, children, let's move on to the next one. Under what condition is the torque applied by a force zero? It's asking for the condition where we apply force, but the torque is zero. We are applying force, but the torque is zero. What is the condition? Either the radius must be zero, or theta must be 0 degrees or 180 degrees. Ready, children? Did everyone understand? Understand? Okay? Okay? Either the radius must be zero, or theta must be zero, or it must be 180 degrees. Yes, children. Let's move on to the next question. Let's move on to the next question. Yes. Moment of inertia is the next question. Let's finish quickly. Don't worry. Let's finish quickly. Moment of inertia. What is moment of inertia? We know what inertia is, right? Inertia is the inability of a body to change its state. That inability is known as inertia. So, here, if a body is rotating, it will try to continue in rotation. If we don't apply any torque, okay? Children, the Earth is revolving around the sun. It has been revolving for billions of years and is still revolving. Inability. The Earth decides one day, "I've been revolving for so long, why not take a break for a few days?" Right? Nothing will happen because it doesn't have the ability to change its own rotation. Someone else has to come and poke it with a pickaxe or a stick. As Archimedes said long ago, "Give me a place to stand and a lever, and I will move the Earth." We are learning that. The principle of moments comes at the end of this chapter. That is, we can move the Earth. If you give me a place to stand and a lever, I will move the Earth from its position. Torque comes there. It will change if torque is applied. It will continue to rotate even if torque is applied. So, we learned that the inertia of a body is related to its mass. If a body has more mass, it has more inertia. But in the case of moment of inertia, mass is not the only important factor. Radius is also an equally important factor. For example, the Earth revolves around the sun, and Pluto, the former Pluto, is no longer a planet, also revolves around the sun. Pluto has a larger radius than the Earth, right? Therefore, Pluto has a greater moment of inertia. Don't consider Pluto's mass. For ease of comparison, let's assume the mass of Earth and Pluto are equal. Who has a greater moment of inertia? If the radius increases, the moment of inertia increases. If the radius increases, the moment of inertia increases. To rotate an object, for example, I am taking this object here. I can rotate this object very easily by holding it here. See? Can you see? I can rotate it easily by holding it here. I can rotate it by holding it here. Because of the iPad and other things, I can rotate it. But if I try to rotate this same object by holding it here, oh my God, it's difficult. It's very difficult. No, no, no, no, no. So, it's a little difficult to rotate it by holding it here. But it's easy to rotate it by holding it here. What is the reason? If the radius increases, the moment of inertia of that object increases. When rotating like this, the radius is large. The radius is this much, right? This much radius. This much radius is visible. This much radius. But when rotating here, the radius is small. Okay. So, if the radius decreases, the moment of inertia decreases. It can be rotated quickly. Okay. So, we can say that moment of inertia depends on two factors: radius and mass. A body is rotating around an axis. You are looking at the axis. I am going to draw this body here. This is our body. Its mass is M. The distance from the axis is R. Then the moment of inertia of that body will be, what, children? MR squared. MR squared is the moment of inertia of the body. Children, what is it? MR squared is the moment of inertia of the body. Ready? Ready? Now, if there is more than one particle, if it is the moment of inertia of a system of particles, then don't worry. Add all the masses, add all the moments of inertia. Moment of inertia of the first body plus moment of inertia of the second body plus etc., etc., etc. If there are many particles, the total sum of the moments of inertia of all particles is the moment of inertia of that system. Shall we move on to the next one? We need to memorize the moment of inertia of some bodies. It's a direct hit question in the exam. You need to memorize the moment of inertia of some bodies. You will memorize it with me. You will memorize it right now. Ready? Let's see. Let's see the moment of inertia of each body. First, we look at the ring. The moment of inertia of a ring is MR squared. It's simple. If the radius of the ring is R, it's MR squared. The axis is passing through the center and perpendicular to the plane of the ring. Okay? Then the moment of inertia is MR squared. Now, about the diameter. The moment of inertia of the ring about this diameter. The ring is rotating with respect to this diameter, like Shaktimaan rotates. Then the moment of inertia of the ring is half of the previous one, MR squared by 2. Now, take a rod. This is a rod, right? This rod is rotating. If the length of the rod is L, if the length of the rod is L, then the moment of inertia of the rod is ML squared by 12. Similarly, take a disc. Remember the disc is half of the ring. It's easy to memorize. So, if the moment of inertia of the ring is MR squared, then the disc's is MR squared by 2. So, its half will be about the diameter. About the diameter, it will be half of that, that is, ML squared by 2 divided by 2, which is ML squared by 4. Right? Isn't half of half a quarter? Now, next is a hollow cylinder. If it's a hollow cylinder, look. When a hollow cylinder rotates, doesn't it feel like a ring rotating? When a hollow cylinder rotates, doesn't it feel like a ring rotating? It's MR squared. The moment of inertia of the ring. Now, if it's a solid cylinder? If it's a solid cylinder? Doesn't it feel like a disc rotating? If it's a solid cylinder, it feels like a disc rotating. That's MR squared by 2. Now, next is a solid sphere. Children, if it's a solid sphere? Children, if it's a solid sphere, it's 2 by 5 MR squared. If it's a solid sphere, don't worry, it's 2 by 5 MR squared. 2 by 5 MR squared. Ready? Ready? Okay, ready. How many minutes? Maximum by 8 o'clock, 8 o'clock, we can wind up. If it's a solid sphere, it's 2 by 5. If it's a hollow sphere, we know. Next, define moment of inertia. Define. 2025 model question. 2024 March question. Define moment of inertia. Moment of inertia. Define it. Rotation. Right? Next, what is its equation? The moment of inertia of a body depends on its mass and the distribution of mass about the axis of rotation. The axis of rotation. How far is the mass from the axis of rotation? It also depends on it. Write the equation for moment of inertia. Write these things. It's a one-mark question, but it's a repeated question asked in the model, March 2023, March 2021, and model 2020. What are the factors affecting moment of inertia? Write all this. Two marks. You don't need to write the other. Just write the definition and the equation here. What are the factors affecting moment of inertia? Mass and distribution of mass about the axis of rotation. Mass and distribution of mass about the axis of rotation. How far is the mass from the axis of rotation? If the mass is far from the axis of rotation, the moment of inertia is greater. Ready? Okay. What are you writing? D cubed. I don't know any of that. I don't understand what you mean. Yes. Next is model 2024. The rotation analog of mass in linear motion is the rotation analog of mass. Moment of inertia. Next question. There are two satellites. There are two satellites. Their masses are equal. They are rotating around the Earth. The satellites are rotating around the Earth. They are rotating. The satellites are rotating. They have different radii. Equal mass, but different radii. So, is the moment of inertia of the satellites the same? The question is. Absolutely not. The one with the larger radius will have a larger moment of inertia, right? Will the moment of inertia be the same or different? The moment of inertia will be different. Now, write the unit and dimension of moment of inertia. What is the unit of moment of inertia, children? If we take the unit of moment of inertia, we know it's kilogram. The unit of moment of inertia is kilogram meter squared. Kilogram meter squared is the unit of moment of inertia. Similarly, similarly, if we look at its dimension, if we look at the dimension of moment of inertia, we can say it's ML squared. Mass into radius squared. Ready, children? Ready, children? Okay. Yes. What is the measure of inertia? What is the measure of moment of inertia? That's what we said: mass and distribution of mass about the axis of rotation. Those are the factors that measure moment of inertia. Ready. Let's move on to the next one. Let's move on to the next question, children. Is everyone ready? If everyone is ready, here's the next question. What are the analogs of mass and force in rotational motion? Instead of mass, we study moment of inertia. Instead of force, we study torque. Next is radius of gyration. Children, radius of gyration means the moment of inertia of all bodies. We can represent the moment of inertia of any body like this. We can generally represent the moment of inertia of any body as MK squared. So, the table we studied, that is, each body has a K value. A ring has a K value. A disc has a K value. A sphere has a K value. Similarly, a rod has a K value. The K value of each body will be different. So, the name given to it is radius of gyration. So, we can find the radius of gyration. It is the square root of the ratio of the moment of inertia to the mass of the body. So, to find the radius of gyration of a body, divide the moment of inertia of that body by its mass and take the root. So, we can define radius of gyration like this. Radius of gyration of a body. The radius of gyration of a body. It is the square root of the ratio of the moment of inertia of the body to the mass of the body. Now, the physical definition is written. The radius of gyration of a body is the radius at which, if a point object of the same mass as the body rotates around an axis, the moment of inertia of that point mass is equal to the moment of inertia of the body. We can define it in that way. Whichever is easier for you, define it that way. I think writing the mathematical equation is a bit better. Or, if you are looking for full marks, you should memorize this definition. You can write it like this too. As I said, you can write it. Ready. Let's look at the radius of gyration of each body. Let's look at the radius of gyration of the disc and the ring first. So, we know the technique to do this is to find the radius of gyration. It's root I by M, right? I by M. So, if we look at the ring, if we look at the ring, if we look at the ring, the root of I is MR squared. MR squared divided by M. If we cancel M, what do we get, children? We get R. So, understand the technique. Cancel the mass in the moment of inertia and take the square root of the remaining part. So, you get R. Isn't it set up? Isn't it simple? Here, cancel M. Take the square root of the remaining part. The square root of R squared is R. The square root of 2 is root 2. Isn't it awesome? Next, look, children. Next, look, children. Here, cancel M. Take the square root of the remaining part. The square root of L squared is L. Root 12. Ready? Next, children, look. Here, cancel M. Take the square root. For R, it's R squared. For 2, it's root 2. So, R by root 2. Now, here, cancel M. Take the square root. You get R by 2. Now, next, here, cancel M. So, you get R. Here, cancel M. So, you get root 2. Here, cancel M. So, you get R into root 2 by root 5. You need to take the root of 2 and 5. This is how you find the radius of gyration. This is the question asked in the 2021 model, pay attention, children. Pay attention. The moment of inertia of a disc of mass M and radius R about an axis passing through its center and perpendicular to its plane is MR squared by 2. We didn't need all these big stories. We just needed to say the moment of inertia of the disc. Whatever it is. What is the radius of gyration? There's no need for all this. You will answer now. What do you get? I is MR squared by 2. Divided by M. If you cancel M, you get R by root 2. It will be a one-mark question. Okay? Ready, children? Ready? Let's move on to the next one. Let's move on to the next question. 2018 model question. Look, 2018. See? So, radius of gyration is important, right? If the moment of inertia of a rod is ML squared by 12, then the radius of gyration. Don't look at anything. Cancel M. The square root of L squared is L. The square root of 12 is root 12. If you want, you can simplify it. Root 12 can be written as 4 times 3, so 2 root 3. Correct? No compulsion, right? No compulsion. Yes. Yes. I hope you understood. Let's move on to the next topic. Angular momentum. Hey, the momentum of a rotating body due to rotation. We relate it to angular momentum. So, the momentum of a rotating body. The momentum of a rotating body is expressed as angular momentum, or moment of momentum. What is moment of momentum? Moment of force, we know, is torque. So, what will moment of momentum be? The name given to it is angular momentum. Okay. We know the equation for angular momentum. Remember the equation for angular momentum. L is equal to R cross P. This is the expression for angular momentum. The unit is kilogram meter squared per second. This is a vector quantity. Pay attention, children. This is a vector quantity. Ready? Everyone must learn angular momentum. We will do a derivation using this angular momentum. Before that, a 2024 question. Here. Which physical quantity is represented by the product of moment of inertia and angular velocity? Moment of inertia and moment of inertia. What is moment of inertia? It's I. If you multiply moment of inertia and angular velocity, which physical quantity do you get? You should know. You don't know. So, we know that instead of moment of inertia, we study mass in linear motion. Instead of angular velocity, we study velocity in linear motion. We all know the velocity of mass. What is it? Momentum. So, what is moment of inertia into angular momentum? What do you get here? Tell me. Angular momentum. Can you write it simply? Yes. Yes. Yes. Did all children understand? Children, I hope all children understood this. I omega. This is a 2024, 2021 question. Moment of linear momentum. Moment of momentum is called. We all know angular momentum. Angular momentum. Okay. Okay. Ready. Ready. Ready. Did everyone understand? Sir, light Fermi. What is this? I don't understand. Hey, let's move on to the next important question. The question you are most likely to see in your exam paper in this chapter is this derivation. The relation between angular momentum and torque. Children, we know angular momentum L is equal to what is the equation? R cross P. R cross P. Ready? We all know this. Take the derivative of this. If we take the derivative of this, dL.

DBT equal to DBT of children, what do you get, my dear? Let's write it as cross. Let's write it as cross. Is it set? We are taking the derivative. We know what the rule of derivative is. Just remember Nitin Sir and Madhav Sir. Remember the derivative of MN Nambiar. When we say derivative of N Nambiar, take M out and take the derivative of N. Take Madhav Sir out and take the derivative of Nitin Sir. Take Nitin Sir out and take the derivative of Madhav Sir. The program is finished. Remember this rule. This is a product rule. It is the rule of MN Nambiar. Ready, my dear? So, here, let's look at it, my dear. Here, we are applying that rule. So, DL by DT is first into first, isn't it, my dear? Instead of into, we should put a cross. Derivative of the second. Derivative of the second plus second into. We are putting a cross instead of into because it is a cross product. Derivative of the first. Derivative of the first. Is it set? Now, in the next step, a small disturbance. In the next step, we are going to do a small bit of a disturbance. Look, torque, torque equal to, oh no, oh no, DL by DT, no, no, here DP by DT. What is DP by DT? What is DP by DT? DP by DT means, we know, the rate of change of momentum. That, that is force. DR by DT, rate of change of position. Rate of change of position is velocity. DR by DT, rate of change of position is something we all should know. It is velocity. Put this in. Put this in the equation. Similarly, momentum. Momentum means, we know, mass into velocity. Put these three in our equation, my dear. So, what will we get? We can write DL by DT, DL by DT equal to R cross. My dear, you should pay attention. Instead of DP by DT, we can put F. Plus M into V cross. What is it, my dear? We can put V, right, my dear? Right, my dear? Right, my dear? P Q M into DR by DT is V. So, M V cross. Now, the equation that everyone should know. We all know A cross A. What is the angle between A and A, my dear? What is the angle between A and A? The angle between A and A is zero degrees. Zero degrees. So, A cross A will be zero. A cross A is zero. A cross A is zero. A cross V, what will it be? It will be zero. V cross V will be zero. So, look, my dear. Similarly, isn't V cross V zero? So, in the next step, we can write that. We know V cross V is zero. Similarly, we have seen this thing somewhere. What is R cross F? We know R cross F is torque. Put this in, and the answer is ready. What did we get? The answer is DL by DT. DL by DT equal to, my dear, what do we get? We can write torque. DL by DT, we can write torque here. Wonderful equation. Haven't we seen this equation somewhere? In linear motion, when DP by DT equals force, isn't it? DP by DT. Instead of that, the equation we are studying is set, isn't it? Did you understand it wonderfully, all my dear ones? Did you understand this? The rate of change of angular momentum is torque. The rate of change of linear momentum is force. Okay. Then, from here, in March 2023, the relation between the derivative and torque and angular momentum. 2025 model, 2024 model, 2022 model, March. The equation was asked to be derived. The next topic is the law of conservation of momentum. What is the law of conservation of angular momentum? Shall we look? Shall we look? We all know that torque is DL by DT. Shall I ask you, if we do not apply any torque to a body? If there is no torque. If we do not apply any torque to a body, then can we say that zero equals DL by DT? Or DL by DT is zero. If the derivative of something is zero, then that something must be constant. It is common sense. If DL by DT is zero, then that L must be constant. L is constant. The program is finished. So, this is what the law of conservation of angular momentum says. If there is no external torque, if we do not apply any torque to a body, then the angular momentum of that body will be conserved. Ready, my dear? Learn both. We need one more equation here. Pay attention. We already know that L is, what is it, my dear? L is I omega. We learned this here. L is I omega. Right? Angular momentum is I omega. Didn't we do the equation a little while ago? So, I omega is a constant. We can say from there. We can say omega is inversely proportional to I. This is very, very important. We will do application-type things with this. From here, we can say initial angular momentum equals final angular momentum. I1 omega1 equals. We will do problems with this. We will do application-type questions with this. These two equations are important. These two equations are important. So, it would be good to draw it here. I1 omega1 is the initial angular momentum. I1 omega1 is the final angular momentum. Just like the saying, "like father, like son," it is the same. A child is spinning like this. A child is spinning like this. When spinning like this, the moment of inertia is I2. When spinning like this, the angular velocity is omega2. Then omega1 equals omega2, or constant. If the moment of inertia of a body decreases, the angular velocity increases. Very, very important. With this, what is asked in the exam? Application-type questions are asked. In the 2025 model, we were asked about the law of conservation of angular momentum. Ready, my dear? Ready? Next is the application related to the conservation of angular momentum. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the next thing. Let's look at the