Transcription
When we first got our kids a Nintendo Wii, I would beat my kids in Mario Kart maybe 80% of the time. Well, that's the way I remember it at least. It wasn't because I was great at video games in general. Honestly, I've never played them a lot. And it certainly wasn't because my mediocre Mario Kart skills from my teenage days had somehow reemerged better than ever after a two decade hiatus.
No, it was because I understood calculus. If you have played Mario Kart much, you know that you can get different combinations of riders and karts, and that different combinations perform differently. The combination impacts your acceleration, top speed, how well it turns, how fast it can go when you are off the track, etc. In the latest version, Mario Kart 8, you can even change wheels and gliders for even more customization.
Well, the best combination would be high on everything. But the combinations generally have a tradeoff between pairs of characteristics. And the ones that matter the most are acceleration and top speed. For example, the higher the top speed, the lower the acceleration, and vice-versa. This makes part of the strategy of Mario Kart picking a good combination to get effective characteristics.
When my kids first started to play Mario Kart, they tended to gravitate towards higher acceleration and lower top-speed combinations. This was their big mistake. Sometimes they would go for a mid-acceleration and mid-speed combination, but that is still not optimal. To understand why, let's do a little calculus.
Let's look at a possible speed graph of a high acceleration - low top-speed combination. The units on our axes would be time in seconds on the x-axis, and speed in units such as or feet per second or meters/second. The graph of a high acceleration, low top-speed vehicle would look something like this. The slope here would be relatively steep, because this vehicle gets up to a particular speed in a short amount of time, and it flattens out at a value that is not too high.
Calculus gives us tools to reason about how far this vehicle travels, based on this speed graph. For this part, we don't even need calculus. If the rate is 10 meters per second, then every second at that speed, the kart travels another 10 meters. In 2 seconds, it would go 20 meters, in 3.5 seconds, 35 meters. The hard part is this section here, where the speed is constantly changing.
We could think about this part differently; instead of constantly changing, we could think of constant average speeds over small intervals. We could approximate the distance by looking at small intervals during this period, and within each interval take an average speed over that interval. Perhaps the speed in the middle of the interval. Then we can use the same strategy we used when the speed was constant. If it took 5 seconds to get to a top speed of 10 meters per second, maybe we split this portion into 5 - 1 second intervals. Then the average speed for the first interval would be 1 meter per second, so in the first second of acceleration from zero, the distance the kart travels is about 1 m/s for 1 sec, so we end up with 1 meter. For the 2nd second, the average speed would be about 3 m/s, and over 1 second that would mean the kart would travel about 3 meters during the second interval. We can continue in the same way for all of these intervals and then add up all of those distances.
The reasoning we have done so far is at the heart of the meaning of integration, and integration is one of the main ideas in calculus. We referred to this way of thinking about integration in a previous video as the "adding up tiny-pieces" or the "adding up products" approach. If we represent our products of speed times time as rectangles, then it appears we are filling up the approximate area under the curve. If we take even smaller intervals, then the sum of the rectangles look like they are even closer to just representing the area. This is the basic argument that area under a speed or velocity curve represents the distance traveled. The Babylonians actually figured this out 1400 years before calculus was formalized! We'll try to talk about this more in a future video.
In our previous video where we talked about different ideas of integration, we talked about three different ways to think about integration. We downplayed the importance of integration as area under a curve, because it isn't helpful to reason through many real-world situations requiring integration. However, this is a situation where areas under curves will be helpful. So let's add in another kart onto the graph. A kart with high top speed but low acceleration. That curve might be something like this.
Now here is the question: at what point will the high-speed cart pass the high-acceleration kart? You might want to pause the video here while you think about this. A lot of students will say this point, where the graphs cross. Something is equal there, but it is not the position or distance the karts have traveled. That is the moment when the high-speed kart finally matches the speed of the high-acceleration kart. But the high-speed kart has been going slower than the other kart since the start of the race, so there is no way they could have both gone the same distance yet.
Now we don't really need values to reason through this situation. We can reason with areas. This area here is the difference in the distance between the two karts at the point when their speed is the same. But this area here represents the increased distance the top-speed kart will travel after the moment when the speeds are the same. So when will the high-speed car pass the high-acceleration car? When the area here, is equal to the area here. And every second after that, the high-speed kart is pulling further and further away in the race.
This is exactly the reasoning I used to pick combinations of players and karts in Mario Kart Wii. I picked high top-speed combinations and I would tend to win because you spend most of your time at top speed, not accelerating. I even tested it out with my preferred combination (in Mario Kart 7 it was Donkey Kong with the Flame Runner), and a combination one of my son's used. It only took seven seconds from us both starting at a standstill until I had caught up to him. Now if the graphs were more like this with a big difference in acceleration and little difference in top speed, that would be a different story. I might not have time to make up the distance before I have to restart again because of falling off or getting hit by a turtle shell.
Now, you might not consider the work that we've done so far as calculus, because it looks different than what you focused on in your calculus class. We never wrote down a function, never found an antiderivative, never applied the fundamental theorem of calculus, nor did we do any calculations. But the power of math is not just in calculating specific values; the ideas themselves are powerful. Outside of my work as a math teacher, I probably reason using ideas of math 50 times for every problem I actually sit down and calculate. Much like the Doctor that we mentioned in our Teaching Paradox video. He used the ideas of calculus every day in his work, but rarely needed to actually do calculus computations.
Well, it didn't take too long for my kids to catch on to this strategy. Now they are much better at Mario Kart than me, not only because of their driving skills, but they have really refined the combinations to get even better speed and acceleration metrics than I can. Even if I try to match their combination, it is hard for me to win, since they are better drivers than I am.
Do you have a favorite Mario Kart combination? Please share them in the comments below!
There are other considerations when picking a player and kart combination. As you get to the higher speed races, like the 150 or 200cc, then going fast is less of an issue, since all of the karts are faster. You still want a good top speed, but you often end up having to restart more because of the more competitive racers who are attacking you with items. And if you are not a great driver, but a mediocre one like me, an overly fast vehicle makes it harder to stay on the track, so I lose time falling off edges or trying to drive back to the track from the rough.
Now you know how I used calculus to beat my kids in Mario Kart. If you know of other games where doing the math helps get a winning strategy, then let everyone know what those are in the comments. Maybe we'll do a video about them later. Thank you for watching! Please subscribe and share our videos. Be sure to follow Math the World on Twitter, Instagram, and Facebook. Thank you so much for your support!