1. by D.N. Seppala-Holtzman St. Joseph’s College faculty.sjcny.edu/~holtzman

2. Analysis of the Basketball Free Throw To be published in the College Mathematics Journal of the Mathematical Association of America This presentation is for a general audience

3. The Free Throw Player stands behind free throw line and, un- encumbered by opponents and unassisted by team- mates, attempts to throw ball into basket There are two main components under the player’s control: The angle of elevation of the shot,  The initial velocity of the shot, v 0

4. The Problem Neither the angle of elevation of the shot nor the initial velocity of the ball can be completely and accurately controlled Errors will occur The problem we set ourselves was to find the “most forgiving shot” That is, we seek the shot that will not only succeed but will be maximally tolerant of error This notion of maximal tolerance is not well defined We will address this deficiency below

5. A Model Whenever one attempts to solve a real-world problem with mathematics, the first requirement is to make a model We seek to capture the essential features of the real situation while suppressing those whose absence aids the cause of simplicity without unduly sacrificing accuracy

6. Our Model No air, no friction, no spin The center of the ball will be confined to the plane that is perpendicular to the backboard and passes through the center of the basket Thus, we are disallowing shooting from the side and disregarding azimuthal error We will call this plane the trajectory plane We prohibit any contact with the rim We limit ourselves, initially, to the “swish shot” Nothing but net!

7. The Set-Up In addition to the constraints set out in our model, we will use parameters designated by letters to denote the dimensions of the court This will allow us to make general observations not limited to one set of values for these parameters

8. The Set-Up Here is our list of parameters: R = radius of rim H = height of rim from floor d = horizontal distance from point of launch to front end of rim r = radius of ball h = height of point of launch from floor All units are in inches

9. The Set-Up

10. We will endow the trajectory plane (the previous picture) with a coordinate system We will take the origin to be the point of launch Thus, the x-axis will be parallel to the floor but h units above it The y-axis will be the vertical line through the point of launch In this coordinate system, for any choice of angle and initial velocity, the trajectory of the center of the ball will have the following equation

11. Equation of the Trajectory

12. Important Observations Recall Note that if  is held fixed and v 0 is increased the graph of y as a function of x will be elevated in the plane Similarly, for fixed , decreasing v 0 will serve to lower the graph in the plane This agrees with one’s intuition

13. Higher and Lower Trajectories In the following slide, two trajectories are plotted They both have the same angle of elevation of about 54 degrees (0.95 radians) The blue trajectory (the one on top) was determined by an initial velocity value of 320 inches per second The lower red trajectory was determined by an initial velocity of 300 inches per second

14. Higher and Lower Trajectories

15. The Configuration Plane The equation for the trajectory of the center of the ball depends upon selecting values for  and for v 0 For each choice, we get a specific trajectory The plane determined by a horizontal axis indicating the value of  and vertical axis indicating the value of v 0 will be called the configuration plane. Each point in the configuration plane will be an ordered pair ( , v 0 ) which determines a unique trajectory

16. The Legal Region Each point in the configuration plane determines a trajectory Some of these trajectories will yield legal swish shots Others will not We will call the set of points in the configuration plane that correspond to proper swish shots, the legal region

17. Determining the Legal Region To determine this region, we shall compute a lower bound curve and an upper bound curve in the configuration plane

18. The Lower Bound For each choice of angle, , we will start with values of the initial velocity, v 0, that are too low and thus result in the ball hitting the front edge of the rim We continuously increase the value of initial velocity until we just succeed in going over the front edge of the rim Recording this boundary value of v 0 from illegal to legal for each choice of angle produces a formula for the lower bound curve

19. Lower Bound We compute this by observing that the trajectories that are just on the cusp between illegal and legal are those that are tangent to the upper right-hand quadrant of the circle of radius r (the radius of the ball) with center the leading edge of the rim

20. Trajectory Tangent to Circle about Front Edge of Rim

21. The Upper Bound Similar to the lower bound function, we compute the upper bound function that finds, for each choice of angle, , the precise value of v 0 that just succeeds in making it below the back edge of the rim This will coincide with the set of trajectories that are precisely tangent to the lower left-hand quadrant of the circle of radius r about the back edge of the rim

22. Trajectory Tangent to Circle about Back Edge of Rim

23. The Legal Region Armed with the upper and lower bound curves, we can now graph the legal region in the configuration plane This will necessitate our filling in specific values for the parameter letters we assigned at the beginning We will choose the standard NBA values: R = 9, r = 4.7, d = 156, H= 120 There is no standard value for h, the height of the point of launch, as this will vary from player to player We shall take h = 72

24. The Legal Region with Standard Values

25. Varying the Parameters The reason that we chose general parameters rather than specific values was so that we could examine the effects of changing them, one at a time In what follows, we plot the new legal regions that result from letting h, d and r get larger, one at a time In each case, we plot the standard legal region with dotted boundary for purposes of comparison

26. Legal Region with h = 84

27. Letting h Get Larger Increasing the value of h, the point of launch, from 72 to 84 inches moves the legal region lower in the configuration plane For each value of , far less force is needed to sink a swish shot As force and control have an mutually antagonistic relationship, we see that the shot is much easier for a taller player than a shorter one Once again, mathematics confirms the obvious!

28. Legal Region with d = 186

29. Letting d Get Larger Increasing d, the horizontal distance from the point of launch to the front edge of the rim, from 156 to 186 moves the legal region up in the configuration plane Clearly this means that much more force will be required for each value of , making the shot much harder to achieve Once again the obvious is confirmed

30. Legal Region with r = 7

31. Letting r Get Larger Letting r, the radius of the ball, increase from 4.7 to 7 inches, we see that the new legal region is strictly contained in the old one This, too, is obvious as any legal shot with a larger ball would clearly have been legal had the ball been smaller but not conversely

32. The Least Forgiving Shot Observe that the upper and lower bound curves meet at a point This point must represent a trajectory that is, simultaneously, tangent to the circles of radius r about both the front and the back edge of the rim This shot is the least forgiving shot as any increase or decrease in the value of v 0 will result in a failed shot No error, whatsoever, can be tolerated

33. Least Forgiving Shot

34. We Seek the Most Forgiving Shot In order to make clear what we mean by the most forgiving shot, we shall need some definitions For any point in the legal region of the configuration plane, consider the distances, left and right of that point to the two bounding curves We shall call the lesser of these two values (and their common value if they are equal) the θ-tolerance of the point

35. Θ-Tolerance

36. v 0 - Tolerance Similarly, for any point in the configuration plane, we find the vertical distances, up and down from the point, to the bounding curves We call the smaller of the two distances (and their common value if they are equal) the v 0 – tolerance of the point

37. v 0 - Tolerance

38. Inscribed Rectangle Inscribe a rectangle in the legal region and draw the vertical line that lies midway between the sides of the rectangle Now draw the horizontal line that lies midway between the top and bottom The point where these lines meet is the center of the rectangle

39. Inscribed Rectangle with Lines

40. Tolerances within the Rectangle Observe that every point on the vertical mid-line has θ-tolerance that is at least half the width of the rectangle and nearly all points on it have much greater θ-tolerance Similarly, every point on the horizontal mid-line has v 0 -tolerance that is at least half the height of the rectangle and nearly all of the points have much greater v 0 -tolerance

41. Inscribed Rectangle with Lines

42. Most Forgiving Shot We will define the most forgiving swish shot to be the shot determined by the center of the inscribed rectangle of maximal area Thus, we are seeking to maximize the product of the smallest θ and v 0 - tolerances

43. Most Forgiving Shot With the aid of a computer program written in Maple, I was able to determine that the values for the most forgiving swish shot to be θ = 0.943 radians (about 54 degrees) and v 0 = 290.85 inches per second

44. Most Forgiving Swish Shot

45. The Bounce Shot So far, we have concentrated on the swish shot. But what about the shot that bounces off the backboard and then goes cleanly into the basket without touching the rim? It turns out that this shot is equivalent to the swish shot that passes through the “phantom basket” that is the reflection of the real basket in the “mirror” of the backboard

46. The Bounce Shot

47. It turns out that the leading edge of the “phantom basket” is 186 inches horizontally from the point of launch We have already seen the legal region for this value of d in a previous slide Recall that the legal region for d = 186 was much higher than that for the NBA standard d = 156 and thus, the shot was much, much harder to achieve Once again, the obvious is confirmed

48. Legal Region with d = 186

49. Conclusion We conclude with the admission that this project was an exercise in mathematical analysis. Its purpose was not that of utility For those who find themselves aggrieved by this admission, I am prepared to accept the referee’s call of “foul” and offer, by way of recompense, one free throw Thank you