SIAM News Blog
SIAM News

# Run STEM Run!

By Janai Juarez

Think runners always sprint to the finish line? Think again! In a recently published paper in the SIAM Journal on Applied Mathematics, author Amandine Aftalion found that in 100 Meter races runners actually slow down at the last leg. In races that were longer, say 10,000 meters, the runners sprinted to the finish.

At first blush, it seems strange that short races are run with a decelerating final leg while long races are run with strong acceleration. But Aftalion’s mathematical model sheds light on why this is the case, illustrating why running sprints with an initial strong acceleration followed by deceleration to the finish line is the best use of one’s resources.

Aftalion’s model describes the evolution of the velocity, the anaerobic energy, and the propulsive force using a system of ordinary differential equations. The system of equations is coupled to the condition of optimizing the time to run a fixed distance, and factors in Newton’s second law and energy conservation. The model demonstrates that the monotony of the velocity curve versus time is in contrast to that of oxygen uptake over time. Oxygen uptake is seen to increase exponentially to its maximum, and then decrease for a short run. For long runs, oxygen uptake has an increasing start and a decreasing end, both mimicking the speed profiles for short and long races.

Aftalion uses data from real races to validate her model. Comparison of numerical simulations to time splits from world championships races for 100m, 400m, and 800m, show that the curves match quite well. Aftalion notes that “the method of running 100 meters, or any short race this way of running is not because they accelerated to strong at the beginning, or are exhausted, but because this is the best way to run 100 meters from the physiological point of view.”

The mathematical model used provides new insight into the complex world of running and gives new data to researchers in multiple fields of science. With this information, athletes can better their time, and researchers have a new perspective of variations of parameters and the effect on velocity.