The science behind making the perfect pitch

Mathematical model describes the optimal way to throw

New research uses mathematical models to figure out the best strategies to throw something at a target.

Baseball legend Satchel Paige, one of the greatest pitcher in the history of the sport, had a simple philosophy when it came to pitching: Keep the ball away from the bat.

But as anyone who has thrown anything knows, it’s not that easy. Throwing is one of the most complex actions humans perform. Even tossing a crumpled piece of paper into a wastebasket two feet away requires a series of complex neurological and mechanical calculations.  Should you toss overhand or underhand? How fast should you throw? At what angle should you hold your arm?

Applied mathematicians at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) decided to use mathematical models to figure out the best strategies to throw something at a target.

“There are many different ways to get an object to a target,” said L. Mahadevan, the Lola England de Valpine Professor of Applied Mathematics, Physics, and Organismic and Evolutionary Biology at SEAS and senior author of the study. “How do you choose? Our hypothesis was that you choose based on a strategy that minimizes the error at the target while giving yourself the greatest room for error at the release.”

The team found that while underhand throws are best for reaching a target close by and above the shoulder, overhand throws are more accurate for targets below the shoulder — like a wastepaper basket — and are more forgiving to errors over long distances.

The research is published in Royal Society Open Science.

The study shows that the so-called "granny throw" in basketball may be more accurate than the overhand free throw (Image courtesy of Giphy)

As all pitchers, quarterbacks and bowlers know, once an object is released, the thrower loses control over where it goes. Mahadevan and M. Venkadesan, of Yale University, analyzed the parabolic trajectories of thrown objects to understand how release errors affect the accuracy of the throw.

“We asked, how do errors introduced in the release of the thrown object propagate at the location of the target, as a function of the distance, orientation and height of the target,” said Mahadevan, who is also a core faculty member of the Wyss Institute of Biologically Inspired Engineering at Harvard University.

The researchers also modeled the tradeoff between speed and accuracy when throwing an object. 

The team found that regardless of the target location, the most accurate throw is slightly faster than the minimum speed needed to reach the target. The faster the throw, the less likely it is to be accurate, which explains why even the best pitchers still throw a lot of balls. The researchers found that at both high speeds and longer distances, the overarm throw beats the underhand throw in accuracy.

Errors introduced in the release of the pitch, combined with the speed of the object, can lead to an inaccurate throw. (Image via Giphy)

The findings shed light on how humans evolved to throw, said Mahadevan. After all, the ability to hit a target with a thrown object was key to human evolution. Without claws or sharp teeth, humans’ ability to throw a stone or spear was a primary method of hunting for food. 

“This research demonstrates the theoretically best way to throw. But most of us are not born throwers of anything. We learn how to throw through trial and error,” said Mahadevan. “Now, we have a mathematical framework to think about how learning about the physical world requires interacting with the world. We can’t think about tasks unless we think about the way in which we interact with the physicality of the environment.”

Press Contact

Leah Burrows | 617-496-1351 |