Transcription
If you remember, Nora was puzzled over how to find the instantaneous speed of the ball. She wanted to know the speed of the ball at the INSTANT it was at position ‘B’.
In this video, we will see that the answer to this question gives rise to one of the central ideas of calculus called differentiation. And this idea is built upon the idea of limits that we discussed earlier.
Anyway, do you remember the difficulty we faced in finding the instantaneous speed? If an object is travelling with a constant speed, we can easily determine its speed at any instant of time. At every instant, the speed will be the same. It will be equal to the total distance travelled by the ball divided by the total time taken to cover that distance. But we know that in reality, as we drop the ball, its speed keeps on increasing. So in our case, this formula only tells us the average speed of the ball. If we blindly apply this formula to find the instantaneous speed at position ‘B’, we will get the answer to be zero over zero; which doesn’t make any sense. So how should we approach the problem?
At the moment, we only know how to find the average speed of an object. So let’s play around with this concept and see if we can find any headway. Nora had earlier figured out the average speed of the ball after it started its motion and reached midway at position ‘B’. It turns out to be ‘fifty centimetres per second’. Can you tell me what does this average speed tell us? It means that on an AVERAGE, the speed of the ball is ‘fifty’ as it travels this distance. We could think of it as a ballpark figure that gives us an idea of the speed in this duration!
Now let me ask you an easy question! We know that the speed of the ball initially is zero. And we know that the average speed in this time duration is 50. So what can we tell about the instantaneous speed of the ball at position B? Which of these options will be correct? Intuitively, we can say that it’ll be greater than 50! Why? We know that as the ball falls, its speed keeps on increasing from zero till it reaches the floor. So for some initial part, it will be less than the average speed. Hence, after some point, the ball has to speed up and travel with a speed GREATER than the average speed. Hence, its speed as it reaches ‘point B’ will definitely be greater than ‘fifty’. To understand this point, you could look at a set of horizontal bars like this with increasing lengths. If I tell you that the average length of these bars is 50, and the length of the first bar is 10, then you’d be sure that the length of the last bar will definitely be more than 50. Only then will you get the average as 50.
Let’s come back to our problem now! Now let’s say we find where the ball is half a second after we drop it. The information we have is that it covers a distance of twelve and a half centimetres and reaches this position ‘C’. You would be wondering from where we get these values. Don’t worry about them for now. Ideally, we can experimentally determine them, but we have calculated them using some equation. And for ease of calculation, I have taken the acceleration due to gravity to be ‘one’ instead of ‘nine point eight’ metres per second squared.
Now let’s get back and calculate the average speed of the ball between the starting position A and ‘C’ and also between positions ‘C’ and ‘B’. Here are the formulae for average speed. Can you calculate these values? The values turn out to be ‘twenty five’ and ‘seventy five’ respectively. So now the picture is a little clearer. Now the average speed between these two points is 75. So what can we tell about the ball’s speed at position B? Based on the concept we saw earlier, the ball’s speed at ‘B’ will actually be GREATER than ‘seventy five’. How? Because we know that as the ball falls, its speed increases continuously. So the ball’s speed at ‘B’ will be greater than that at ‘C’. And therefore to cover this distance in half a second, its speed will cross this average at some point between ‘B’ and ‘C’.
With the information given, we’ve concluded that the instantaneous speed of the ball at B will be definitely greater than 75. We still haven’t got the answer to the instantaneous speed of the ball at position B. How do we get that answer? For that, let us ask ourselves a question… What happens after the ball crosses position B? It’s obvious that at any point below B, the ball’s speed will be greater than its speed at ‘B’. And that tells us something even more interesting! If we find the average speed between position ‘B’ and some point D below it, we will know that the instantaneous speed at ‘B’ will be less than this value. So if the average speed of the ball between B and D is say one hundred and twelve point five, then we can say that the instantaneous speed of the ball at B will be definitely LESSER than this value. What we’re trying to do here is get a RANGE. We will continue this in the next part. Our goal is to find the instantaneous speed at B, and we know that it is greater than 75!
Now we’re given some more information about what happens after the ball crosses position B. At time ‘t’ equals one point two five seconds the ball is at the position ‘D’, which is twenty eight point one two five centimetres below position ‘B’. Now let’s find the average speed between ‘B’ and ‘D’. Can you try? It turns out to be one hundred and twelve point five. And what does this tell us? As we saw before, it means the instantaneous speed of the ball at ‘B’ will be definitely less than this value. So we get the speed of the ball at ‘B’ to be somewhere between ‘seventy five’ and one hundred and twelve point five. We’re getting closer to the answer of finding the instantaneous speed of the ball at position B. It will be between these two numbers.
Let’s proceed further and find the average speed of the ball between point B and points closer to it. We’re given some more information now. We see that the ball is at positions E and F at zero point nine five seconds and one point zero five seconds respectively. Let us try calculating the average speed in these two regions. The calculations give us these two values. Ninety seven point five and one hundred and two point five! The instantaneous speed at B will be between these two values! Now we see that we have an even better estimate of the instantaneous speed at ‘B’. Did you get the idea? What should be our next step? Yes, we should find the average speed values between point B and the positions closer to B. So, as we calculate the average speed values closer and closer to position ‘B’, we get closer to the actual speed of the ball at ‘B’. We see that these numbers approach the number ‘100’. From the top as well as from the bottom! If we continue to calculate the average speed in even smaller intervals, we will get even closer to the number hundred. So we can conclude that the Instantaneous speed of the ball at ‘B’ will be hundred centimetres per second.
This is the basic idea. We get closer and closer to the answer with this approach. We see that the average speed approaches a number, which will be the speed of the ball at position ‘B’. This process is called the LIMIT PROCESS. We say that in the limit this time interval gets closer and closer to zero; the average speed of the ball approaches the instantaneous speed of the ball. Nora’s problem is now resolved. She knows a way to find the instantaneous speed of an object. She realises that understanding Calculus is easy: it is just our knowledge of algebra stretched to a limit.
Now let me tell you that this is the basic idea on which the concept of differentiation is based. What do we mean by it? The speed of an object measures how its position changes with respect to time. It is nothing but the RATE of change of the position of the object. What we did here near point B, is that we looked at how the position of the ball changes in very very small time intervals. And then we found the instantaneous speed of the ball at ‘B’. The process by which we found this rate of change at an instant is called differentiation. We will see that differentiation helps us in analysing how things change. Using the concept, we can find the instantaneous rate of change of any quantity. We will learn more about it in our upcoming videos. In the next video, we will further explore this idea and see how it is related to another problem we discussed earlier, which was about finding the areas of shapes.