Introduction
I was interested in the running race before even the calculus in out curriculum. I was watching the 2016 Olympic with my family, I was concentrating on the running race. In that Usain BOLT got the first place in the 100m. I knew that he is going to be a first place in the running race. But I asked one question that really Usain BOLT is the fastest runner or not. And my father said he is the fastest runner in the world. But actually we don’t know whether he is the fastest or in 100m alone, he is the fastest person. So I got interested in that topic. And I realize that to calculate the momentary acceleration, we need the calculus to find it. Therefore, I chose this topic as it has a link with calculus and while learning the concept I found it very interesting. Right from the beginning I wanted to do my IA related to sports. So I was trying to co-relate my IA with sports and calculus and I found that we can link calculus and running. Therefore, I decided to investigate the speed of different runners and prove that the runners are not running at a constant speed throughout the race. For this IA, I went through the calculus chapter in my textbook and additional book for higher level students from where I found there many formulas. Using this, I will be: “Determine the fastest runner in the world using the instantaneous velocity”
Background information
Calculus is one of the topic which branch of mathematics that involves the study of rates of change. Using this we can find the properties of derivatives and integrals of functions. There are 2 types of calculus, which is integral calculus and differential calculus. But in this IA, I am going to use only differential calculus. The basic formula for the calculus itself is: h0f(x+h) - f(x)h
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This formula is to get the derivative of f(x). There are several number of calculus functions are there. To find out what if the functions are combine, multiply, divided or extras. In calculus, we also can see different types of formulas related to trigonometric. The equations for there are:
- ddxxn=nxn-1
- ddxg(f(x)) = g'(f(x))f'(x)
- ddx(ab)=a'b+ab'
- ddx(ab)=a'b-ab'b2
Experiments
To do this experiment, I would like to use the Olympic which happen in all around the world. I got the data from the videos and the website which written it down all about the timing and the distance that how much they run. So, to invest that the speed is not constant. I used the data for Olympic which conducted on 2016.
First I am going to measure the speed of the runner negligible the air resistance, using the distance which is 100m and time taken. And same for all the runners. They all did the running race for the same amount of distance so the numerator will be same for all the runners. But air resistance, the increasing in speed and the acceleration is present during the running race which is not include when I was calculating the average for each runners.
After I got that, I found that I have to think about those things when I want to find out the speed for one specific point. And I assume that the first 10m, the speed of Usain BOLT will be:
- time = 9.8110100=0.981s
- speed=100.98110.91m/s
But actually, in this the speed of each runners will be same for all the point. But there is no air resistance, acceleration and friction at all. So if I think about all of those. The speed will not constant for all point. So I found the instantaneous velocity for all the runners. So that my statement which is the speed is not constant during the running race.
Through this we can notice that all of the runners are not running in constant speed throughout the race. Instantaneous velocity is which the velocity of an object in motion at a particular point of time, so it can define as average velocity. To find the instantaneous velocity we have to get the slope of the distance and time graph. Through the table 1.2, we can clearly see the speed difference and the time difference. But still this is just rough sketch about it. So to get the exact speed.
This is the graph for the Usain BOLT. The gradient of the Usain BOLT’s graph will be the instantaneous velocity for the Usain BOLT. Using this graph, we can find the gradient and y-intercept to find the function for Usain BOLT: y=12.138x-23.291
I am going to do two more runners to prove my experiment. So it will be more provable. For that I am going to take a Andre DE GRASSE and Trayvon BROMELL.
In the 100m race, the Usain BOLT was faster than Andre DE GRASSE. But according to the function, Andre DE GRASSE’s initial speed was faster than Usain BOLT. At the 20m Andre DE GRASSE was faster than Usain BOLT, but there were not much difference in the timing for each 20m. The difference was reducing while they were doing the running race. At the beginning like, when they were in the 20m, the difference was 0.2s, till 60m, the difference was 0.2s. But at the 80m, the difference became 0.1 and finally at the 100m, Usain BOLT was 0.28s faster than Andre DE GRASSE. And through the function also the gradient is Andre DE GRASSE is larger than Usain BOLT. And when I plot the speed-time graph, I found that Andre DE GRASSE’s gradient was larger than Usain BOLT’s gradient. And when I extended the graph, the speed of Andre DE GRASSE was faster than Usain BOLT.
Now we can clearly see that the Andre DE GRASSE was faster than Usain BOLT. And through the data which I found it, I tried to get the average speed, time and acceleration. First I would like to calculate the average speed for both of them. For Usain BOLT and :
- Average speed(Usain BOLT)=5.88+7.55+8.57+9.09+10.195=8.256
- Average speed(Andre DE GRASSE)=5.56+7.41+8.33+8.99+10.095=8.074
- And for average time for both runners are:
- Average time(Usain BOLT)=3.4+5.3+7.0+8.8+9.815=6.862
- Average time(Andre DE GRASSE)=3.6+5.4+7.2+8.9+9.915=7.002
- Finally for average acceleration for both the runners are:
- Average acceleration(Usain BOLT)=8.2566.862=1.203
- Average acceleration(Andre DE GRASSE)=8.0747.002=1.153
There is not much difference between them. And if the race is goes on, Andre DE GRASSE will catch up the Usain BOLT. Through this we can notice that the Usain BOLT’s instantaneous velocity and acceleration is fastest but, if we see the average acceleration and velocity is not the fastest in the world.
Conclusion
Through this experiment, I wanted to check whether he is the fastest runner in the world or not. In 100m, through the result, we can clearly know that the Usain BOLT is the fastest in that. But, we are not sure about whether Usain BOLT is the fastest runner even if the distance of the running race is increasing. It could be the instantaneous acceleration of Usain BOLT might high, not the average acceleration. That is mean, the instantaneous velocity might be low. The difference between the Usain BOLT’s and Andre DE GRASSE’s speed is not much. That is mean if the distance is increasing, there is a possibility for Andre DE GRASSE to become a first place.
Bibliography
- https://www.olympic.org/rio-2016/athletics/100m-men
- https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-rates/v/usain-bolt-s-average-speed
- https://www.khanacademy.org/science/ap-physics-1/ap-one-dimensional-motion/instantaneous-velocity-and-speed/v/instantaneous-speed-and-velocity
- https://byjus.com/calculus-formulas/
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