Modeling Basketball Free Throws

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SIAM REVIEW Vol. 47, No. 4, pp. 775–798

c 2005 Society for Industrial and Applied Mathematics

Modeling Basketball Free Throws∗
Joerg M. Gablonsky† Andrew S. I. D. Lang‡
Abstract. This paper presents a mathematical model for basketball free throws. It is intended to be a supplement to an existing calculus course and could easily be used as a basis for a calculus project. Students will learn how to apply calculus to model an interesting real-world problem, from problem identification all the way through to interpretation and verification. Along the way we will introduce topics such as optimization (univariate and multiobjective), numerical methods, and differential equations. Key words. basketball, mathematical modeling, calculus projects AMS subject classifications. 00-01, 00A71, 26A06 DOI. 10.1137/S0036144598339555

1. Introduction. In these days of superstar basketball players, you would think that shooting free throws should be as much a formality, and just as exciting, as the extra point in professional football. Not so. Take for example Shaquille O’Neal, the subject of our first model, who as of the end of the 2004–2005 regular season had a career free throw percentage of 53.1%. His troubles seemed to increase during the playoffs, where he shot around 45% from the line. Shaquille is not alone in his free throw shooting troubles. In fact nearly one-third of all NBA players shoot less than 70% from the foul line. When a basketball player steps up to shoot a free throw he does not usually think (unless he also happens to be a mathematician), “I wonder if my free throw shooting percentage would improve if I changed my initial shooting angle,” or “I wonder how air resistance affects the trajectory of my shot,” or even “Should I be aiming for the back rim, front rim, or the middle of the basket?” We present here a calculus-based model for basketball free throws to show that they should address some of these musings. We begin by conjecturing that some players shoot poorly from the line because they are shooting the ball at the wrong angle. Therefore, the focus of our model will be the release angle, a simple place to start, and we will extend it later. Some of the more interesting facts that we’ll discover by refining and interpreting our model are: 1. The best way to shoot free throws depends upon the person shooting. The two most important factors are their height and

∗ Received by the editors May 22, 1998; accepted for publication (in revised form) July 20, 2005; published electronically October 31, 2005. † The Boeing Company, Mathematics and Engineering Analysis, P.O. Box 3707, MC 7L-21, Seattle, WA 98124-2207 ( ‡ Department of Computer Science and Mathematics, Oral Roberts University, 7777 South Lewis Ave., Tulsa, OK 74171 (




how consistent they are in controlling both the release angle and the release velocity. 2. In general, the taller you are, the lower your release angle should be. We’ll actually see that taller players are allowed more error in both their release angles and release velocities and thus they should have an easier time shooting free throws than shorter players. 3. It is much more important to consistently use the right release velocity than the right release angle. 4. The best shot does not pass through the center of the hoop. The best trajectories pass through the hoop somewhere between the center and the back rim. Taller players should shoot closer to the center while shorter players should aim more towards the back rim. 2. Mathematical Modeling. Before we jump into modeling a basketball free throw, it would help for us to tell you exactly what we mean by mathematical modeling: Mathematical modeling is the process of formulating real world situations in mathematical terms. Less formally, mathematical modeling takes observed real-world...
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