Optimal trajectory for the basketball free throw

Abstract

Using a theoretical approach, we studied the basketball free throw as a function of angle, speed and spin at release. The ball was constrained to the sagittal plane bisecting the hoop and normal to the backboard, and was permitted to bounce and change spin on both backboard and hoop. Combinations of angle, speed and spin resulting in a successful shot were calculated analytically. Standard deviations for a shooter's angle and speed were used to predict the optimal trajectory for a specific position of release. An optimal trajectory was predicted which had an initial angle and speed of approximately 60°and 7.3 m s-1 respectively over the domain of spins (-2 to +2 m s-1 surface speed; -16 to +16 rad s1). The effect of air resistance and the sagittal plane constraint on the predicted optimal trajectory were discussed and quantified. The optimal trajectory depended on both the anthropometric characteristics and accuracy of the shooter, but generally a high backspin with an angle and speed combination which sent the ball closer to the far rim of the basket than the near rim was advantageous. We provide recommendations for shooters as a function of the height of ball release.