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Trouble-Shooting Free Throws with Calculus
By Rebecca Gehle
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Lang's article Modeling Basketball Free Throws was recently published in the prestigious mathematics journal Society for Industrial and Applied Mathematics. Inspired by some of his students, Lang began researching basketball free throws and applying mathematical formulae to them. He was able to determine the ideal release angle for players to maximize their chances of making a free throw, with the release angle changing according to each player's height.
In his article, Lang concentrates on Shaq's free throw issues, applying the math equations for the basketball star to discover his ideal release angle. Lang then proceeds to discover the optimal release angles for basketball players of any height. According to Lang's equations, v* = 6.7 m/s, the 7-foot-1-inch Shaq should release the ball at an angle of 52.37 degrees.
"One of the results documented in the paper is that the taller you are, the easier it should be to shoot free throws," said Lang. "When you work out the percentage error that's allowed when shooting, the taller you are, the bigger the error you're allowed to make." So, according to the mathematical principles, taller players should have higher free throw percentages; however, in looking at player statistics, the opposite is recorded.
Lang explained that "the contrary seems to be true where the tall players seem to have the most trouble, and the short guards are the ones with the high free throw percentages."
Although the players' heights are taken into consideration, the math equation does not incorporate other physical factors. Lang's article references other studies that show how larger basketball players, like Shaq, also have proportionally larger hands. "Very large, tall basketball players hold the ball in their hands, and it is like a grapefruit; it is very hard for them to shoot the ball in a consistent manner," explained Lang.
Lang enjoyed writing the article, and he created a very reader-friendly paper that draws practical solutions. "I always had in mind that this would be a paper of great interest to calculus students," said Lang. "The mathematics in the paper goes through stuff that calculus students can do up through senior level math."
A lunchtime basketball player himself, Lang enjoyed the research process for this article, but he is now ready to warm the bench and concentrate on teaching upper division math classes at ORU. Unfortunately for him, during the games Lang plays in with fellow ORU faculty and staff, the players do not shoot free throws for personal fouls. But if they did, you can bet that Lang would have a personalized equation for every member on the team.
Lang's article in SIAM may be read by visiting:
http://epubs.siam.org/sam-bin/getfile/SIREV/articles/33955.pdf
