Expanding Our Limits with Calculus

June 21, 2019

The limit to human imagination does not exist. In the article “Usain Bolt’s Split Times and the Power of Calculus,” Steven Strogatz investigates the nature of motion and change and how mathematics can be used to model Bolt’s record-setting race. Through explorations of various methods of modeling the 100 metre dash, Strogatz contrasts the demise of historical paradoxes with the accomplishments of modern differential calculus. As past meets present, Strogatz attempts to determine not only when Bolt was sprinting at peak speed but also share the art of abstraction with readers.

On the evening of August 16, 2008, history was made by a twenty-one year old Jamaican runner named Usain Bolt. Relatively new to the 100 metre dash, Bolt started off the race in seventh place. Yet within the next few metres, Bolt pulled to the lead and maintained his pace with more than twenty metres to spare. At the time, the whole world held its breath in anticipation, wondering just how long it took the young runner to cross the finish line, only to cheer in unison as the 9.69 seconds raced across the screen. Upon discovering that a world record had been made, a new fury over Bolt’s race split began. Data analysts attempted to depict the race with distance-time graphs while Strogatz proposed a different question. As the world’s fastest man rose to fame, researchers including Strogatz, began to speculate over just how fast Bolt had run. In other words, what was Bolt’s instantaneous speed?

As with many concepts, the solution is much more complex than it seems. This is because of common knowledge that speed can only be measured over a time interval and not in a single instant, thus resulting in a paradoxical concept. Historically, the idea of instantaneous speed was first introduced in 450 B.C.E. when Zeno of Elea established his contradictory paradoxes (Internet Encyclopedia of Philosophy). Without diving too deep in Zeno’s arguments, including the famous arrow paradox and the paradox of Achilles and the tortoise, some believe that the root of many Greek philosophers at the time including Zeno, was their inability to accept the notion of a speed at an instant. To this day, ideas such as these are often left out of conversation because of the head scratching and uneasiness that ensues.

Using modern differential calculus, the answer was revealed. Founders of differential calculus developed a method of calculating instantaneous speed using limits which can be applied to Bolt’s race. After applying these limits, Strogatz concluded that Bolt’s top speed was a whopping 12.3 metres per second.

The next year, during the 2009 World Championships, Usain Bolt shattered his previous record with a race time of 9.58 seconds. This time, biomechanical researchers came prepared with hand held laser guns to create a more “accurate” graph of Bolt’s performance. Upon computing Bolt’s speed, a wiggly red mess resulted. Data analysts were frustrated by the red squiggles and decided to smooth over the wiggles in order to create a more “readable” graph. To author Steven Strogatz, the wiggles are a metaphor. As human nature has the tendency to extrapolate, Strogatz looked at the bigger picture and described the lesson. “[W]hen people try to push the limits of human capabilities to far, or analyze something with too much detail, we see chaos instead of harmony” (Strogatz 2019).

Through his analysis of a record breaking race, Strogatz reveals a valuable lesson. In his own words, “[i]n mathematical modeling, as in all of science, we always have to make choices about what to stress and what to ignore” (Strogatz 2019).  Furthermore, he encourages readers to realize that the world is faced with the challenge of deciding between what can and cannot be overlooked. Amidst this process, people often stress over intricate details and fail to see the beauty of their surroundings, just as data analysts pore of instantaneous speed only to settle for a more simplistic picture. Ultimately, as Strogatz was suggesting, creativity flourishes at the end of internalized boundaries, and the bigger picture is revealed when individuals come to terms with the unexpected, wiggly lines and all.

Works Cited
Strogatz, Steven. “Infinite Powers: Usain Bolt and the Art of Calculus.” Quanta Magazine, 3 Apr.
2019, www.quantamagazine.org/infinite-powers-usain-bolt-and-the-art-of-calculus-20190403/.