Orange County, California

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  • Greg Hodel

How does a sprinter’s movement compare to projectile motion?

I can't resist beginning with a joke from https://www.math.utah.edu/~cherk/mathjokes.html :

A mathematician, a physicist, an engineer went to the races and laid their money down. Commiserating in the bar after the race, the engineer says, "I don't understand why I lost all my money. I measured all the horses and calculated their strength and mechanical advantage and figured out how fast they could run..." The physicist interrupted him: "...but you didn't take individual variations into account. I did a statistical analysis of their previous performances and bet on the horses with the highest probability of winning..." "...so if you're so hot why are you broke?" asked the engineer. But before the argument can grow, the mathematician takes out his pipe and they get a glimpse of his well-fattened wallet. Obviously here was a man who knows something about horses. They both demanded to know his secret. "Well," he says, "first I assumed all the horses were identical and spherical..."

It is very rare that people laugh at this joke, but it is one of my favorites and it makes me laugh every time (you can ask in the comments why I think it is funny).

An object’s (sphere’s) horizontal component of velocity determines its horizontal speed and its vertical component determines its time in air. Both contribute to how far the object travels in the air and the optimal release angle is about 45 degrees. If the vertical component is not large enough, the object will not be in the air long enough for it to move its maximum horizontal distance. If the vertical component is too large relative to the horizontal component, then the object will be in the air longer, but will not travel its maximum horizontal distance.

These two components apply to running, but in the best sprinters, their hips appear to stay near a constant height. The vertical component of the hip’s displacement appears to be very small relative to what is seen in a projectile with a 45 degree launch angle. The parabolic path of the projectile is not apparent in the runner, but the runner’s center of mass must be following a path close to the parabolic path of a projectile.

What might be occurring to make the runner appear to have such a shallow trajectory? At the time when both feet are in the air, both arms are at the highest points of their paths. These positions of the limbs would raise the runner’s center of mass even if the hips did not rise. After the limbs have rotated downward, they collectively lower the center of mass. When the knees cross, the foot contacts the ground, and the arms are at the lowest point in their path, the runner’s center of mass would be lower even if the hips did not move lower. This shows that the center of mass is oscillating vertically more than what is seen in the vertical movement of the hips. Due to the vertical displacement of the limbs, the center of mass can change while the hips stay near the same height. With this understanding, the simple physics of a projectile can be considered in a runner’s forward motion. The many variables associated with the movement of the limbs can be reduced so that a simple representation of running mechanics can be analyzed.