The anatomy of the bank shot in men’s basketball

Appendix

As stated in the paper, the shooter was assumed to successfully make the free throw 70% of the time and to shoot other shots with the same statistics. The optimal shots were selected from a set of free throws having a range of launch velocities v and launch angles α for a ground height of 7 ft. The launch speed v varied between 22.70 ft/s (6.92 m/s) and 28.45 ft/s (8.67 m/s) in 144 steps and α varied from 22° to 82° in 300 steps (43,200 cases). Each of the trajectories is a highly accurate trajectory that accounts for ball contact with the hoop and the backboard (Silverberg et al. 2003). The free throws were launched with 3 Hz of back spin which was determined to be optimal (Tran and Silverberg 2008). As an illustration, one set of data is shown in Figure A1.

Figure A1

V-α curve for a 7.0 ft (2.13 m) release height and a ω=3 Hz back spin.

Notice that the successful shots appear in bands. The large left band is associated with direct shots, the two thin middle bands are associated with the ball striking the back of the hoop (closest to the backboard), and the large right band is associated with bank shots. Our interest lies in the large left band. The probability of success depends on the means and standard deviations of v and α, in which the mean values of v and α are the values that the shooter seeks to prescribe and where the standard deviations correspond to shooter consistency. Improvements in shooting depend on the values of v and α that the shooter prescribes (understanding the nature of the highest probability shot) and consistency. The standard deviations used in this paper were associated with a 70% shooter (Tran and Silverberg 2008). The mean values were determined using the segment bisection shown in Figure A1. In previous work it was verified that the best mean v and α can be accurately determined by bisecting the line segment between the point of the smallest v on the left boundary of the band and the point of the smallest v on the right boundary of the band. The v and α of the bisecting point are the best mean values.

Referring to Figure A2, the center of the basketball is initially located at (x₀ y₀ z₀), upon contact with the backboard is located at (x_b y_b z_b), its contact point is projected onto the backboard at the aim point (x_a y_a z_a), the center of the ball enters the plane of the hoop at (x_F y_F 0), and is imagined to pass through the backboard and reach the plane of the hoop at (x_R y_R 0).

Figure A2

Trajectory and terms.

The components of the initial position and the launch velocity are expressed in terms of polar coordinates and in terms of the pitch angle and the side angle by (See again Figure 1)

in which the ball is launched 7 ft from the floor (3 ft below the hoop). For the centered shot (x_F=y_F=0), the side angle is related to the physical parameters by (Silverberg et al. 2003)

where γ is the coefficient of restitution between the basketball and the backboard. The velocity components of the center of the ball, the time of flight T₁ to the backboard, and the coordinates y_b and z_b of the center of the ball when it makes contact with the backboard are

The aim points projected onto the backboard are related to the contact points by

The components of the velocity of the center of the ball immediately after contact are (Silverberg et al. 2003)

Next, the time of flight T₂ from the backboard to the plane of the hoop and the coordinates x_F and y_F of the center of the ball in the plane of the hoop are determined.

Finally, the time of flight T₃ from launch to the instant the ball reaches the plane of the reflection hoop and the coordinates x_R and y_R of the center of the ball in the plane of the reflection hoop are determined.

The range of successful shots is determined by incrementing with respect to pitch angle and side angle and testing whether (x_Fy_F) falls inside the circle

Figure A3

Colors used for quadrants.