Topics: Free throw, Basketball / Pages: 43 (10521 words) / Published: Dec 24th, 2011
SIAM REVIEW Vol. 47, No. 4, pp. 775–798

c 2005 Society for Industrial and Applied Mathematics

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