Relation of Usain Bolt's Running Speed to Calculus

Usain Bolt is factually the fastest sprinter in the world, of the human race that is. Many people have analyzed his running speeds in relation to physiology and training factors but in the articles I focused on, they performed the math behind it. These articles studied things such as how Usain can improve his speed based on specific factors, his peak and lowest speed, and research on Usains speeds using differential calculus and equations used in the beginning of math. All of these things combined explain his top speeds and also how he can improve it without any extra effort.

A mathematician named John D. Barrow discovered how to do just that, improve Usains speed by things with a simple fix. He discovered these things by using a significance study. The first factor is Usains reaction time. Barrow discovered that Bolt had the slowest reaction time out of all his competitors. By improving his reaction time by. 10 seconds he can shorten his record to 9.53 seconds (his original record is 9.58 seconds). The second factor he found was weather conditions, more specifically wind speed. If Usain could somehow have the tail wind at 2 miles per hour it would shave another. 05 seconds off his time to get it down to 9.48 seconds. The final factor he found is altitude. Running at an altitude reduces air density, reducing air density increases speed. While races aren’t permitted to be help at altitudes over 1000 meters above sea level. If Usain ran at an altitude of 1000 m he would shave off an additional. 03 seconds, totalling at a new record of 9.45 seconds.

Another article focused more on how fast Usain can run at his peak speed. The three mathematicians who discovered this were Sebastian Schreiber, Wayne Getz, and Karl Smith. They took speeds from 10 meters throughout the whole race and created a graph illustrating the differences (Figure 1). They found that his fastest speeds were between 50 and 80 meters. The peak velocity they found from the graph was 28.45 miles per hour. From this data they also found he was the slowest at the last two seconds, approximately the last 20 meters. This can be accounted for by him relaxing at the end of the race due to his distance ahead of his competitors.

The last and original article I found was much more interesting. This article also broke down Usains speeds from his original record in increments of 10 meters. Again they found the fastest speeds were between 50-80 meters per increment. Each 10 meter was ran in. 82 seconds each. The last ten meters he ran at 11.1 meters per second which equals approximately 4 miles per hour slower. They graphed each 10 meters on a line graph which formed a polygonal curve (Figure 2). From this graph they found the slope was shallowest at the beginning, which correlated with slower speeds associated with his acceleration. After this the slope becomes more upward representing when his speeds increased to his max. From this graph they were trying to figure out the exact time he was at his peak speed. This showed to be significantly different from the data they had, they couldn’t pinpoint it because “speed” requires two points, not one. They also didn’t have the data for the specific speeds in between each 10 meters, just how long it took to get from one point to another.

The earliest history of math dating back to 450 B.C. didn’t focus or have any data really on the laws and math of movement. 2000 years after this, the founders of differential calculus solved the calculations for instantaneous speed, “the solution was to define instantaneous speed as a limit- specifically, the limit of average speeds taken over shorter and shorter time intervals.” Basically this was the beginning of learning how to calculate speeds based off of the smallest intervals they could get. For example the speed between 20 seconds would average out a number but it wouldn’t be specific to each second within that time frame, by making the time fragment smaller they would then be able to find specific speeds for each of the 20 seconds. This was an issue with trying to figure out Usains man speed, this issue also related to distances. The only information they had from his first record was a time every 10 meters, not by every second or every meter so it was harder to define. To figure out the speeds in between they had to make educated guesses. To do this they used interpolation. Interpolation is the process of drawing smooth curves around the hard data they already had. This gave them access to some of the unknowns between the data(figure 3). From the curves they discovered the peak speed he was at in the race. The fastest Usain Bolt ran in his race was 27.5 miles per hour. After that peak is when he started to decline significantly to 22.6 miles per hour. If he didn’t relax towards the end, his record could’ve been higher. By the next time Usain ran they were better equipped with laser guns to capture his speed 100 times per second. They tracked his instantaneous speed as well as his average speed. In figure 4 you can see how there's two sets of lines, waves as well as a more solid line. The solid line is his average speed while the waves are his instantaneous speed. The instantaneous speed varies so much because of how his body naturally slows down when his feet are planted and then speeds up when he pushes off his feet.

These articles and studies have shown us that the calculus of speed is not as straightforward as we think. Looking from the outside in, it seems like figuring out max speed and slowest speed patterns would be fairly easy. At Least I would’ve never guessed so much math would need to go into figuring out his speeds per millisecond. The closer we start to analyze things the more complicated it gets, displayed in something as simple as running speeds.