Modeling Basketball Free Throws

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

References: [1] M. Bamberger, Everything you always wanted to know about free throws, Sports Illustrated, 88 (1998), pp. 15–21. [2] P. Brancazio, Physics of basketball, Amer. J. Phys., 49 (1981), pp. 356–365. [3] H. E. Donley, The drag force on a sphere, UMAP J., 12 (1991), pp. 47–80. [4] M. R. Eddings, Effect of Manipulating Angle of Projection on Height of Release and Accuracy in the Basketball Free Throw: A Biomechanical Study, Master’s thesis, California State University, 1996. [5] W. F. Gayton, K. L. Cielinski, W. J. Francis-Kensington, and J. F. Hearns, Effects of preshot routine on free-throw shooting, Perceptual and Motor Skills, 68 (1989), pp. 317– 318. [6] D. Halladay, R. Resnick, and J. Walker, Fundamentals of Physics, John Wiley & Sons, New York, 1997. [7] G. R. Hamilton and C. Reinschmidt, Optimal trajectory for the basketball free throw, J. Sports Sci., 15 (1997), pp. 491–504. [8] J. Hudson, A biomechanical analysis by skill level of free throw shooting in basketball, in Biomechanics in Sports, Academic Publishers, Del Mar, CA, 1982, pp. 95–102. [9] D. Indraneel and J. E. Dennis, Normal-Boundary Intersection (NBI), is a new technique for solving nonlinear multicriteria optimization problems., SIAM J. Optim., 8 (1998), pp. 631– 657. [10] C. T. Kelley, Iterative Methods for Optimization, Frontiers Appl. Math. 18, SIAM, Philadelphia, 1999. [11] B. J. Kolonay, Visual Motor Behavior Rehearsal, or VMBR, Master’s thesis, Hunter College, New York, 1977. [12] D. M. Onestak, The effects of progressive relaxation, mental practice, and hypnosis on athletic performance, J. Sports Behav., 14 (1997), pp. 247–282. [13] T. Pocock, Official Rules of Sports & Games, Kingswood Press, 1992. [14] P. C. Reddy, Physics of Sports (Basketball), Ashish Publishing House, New Delhi, India, 1992. [15] M. N. Satern, Comparison of adult male and female performance on the basketball free throw to that of adolescent boys, Biomech. Sports, 6 (1988), pp. 465–468. [16] K. Shibukawa, Velocity conditions of basketball shooting, Bull. Instit. Sport Sci., 13 (1975), pp. 59–64. [17] G. Thomas and R. Finney, Calculus and Analytic Geometry, Addison-Wesley, Reading, MA, 1996. [18] E. Tsarouchas, K. Kalamaras, A. Giavroglou, and S. Prassas, Biomechanical analysis of free throw shooting in basketball, Biomech. Sports, 6 (1988), pp. 551–560. [19] R. E. Vaughn and B. Kozar, Intra-individual variability for basketball free throws, Biomech. Sports, 11 (1993), pp. 305–308.