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 degrees and 7.3 m s(-1) respectively over the domain of spins (-2 to +2 m s(-1) surface speed; -16 to +16 rad s). 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.
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ABSTRACT: This work is motivated by a staple carnival game. A player throws a ping-pong ball onto a grid of cups with the goal of having the ball land in a cup. Though there are many variations to this game, there is a common underlying characteristic. As the ball bounces on the cup grid, its sequence of bouncing trajectories becomes nonlinear. It is this nonlinearity which makes it impossible for an observer to predict the outcome, and makes the game difficult. The nonlinearity comes from the interaction of the ball’s linear motion, angular motion, and the how it bounces off the cup edges. The insight that led to the development of this model is that the ball bouncing on a cup edge is equivalent to it bouncing on a tilted surface. Thus, to develop a predictive model for this game, we modeled a spinning partially elastic ball as it bounces over a series of arbitrarily-tilted surfaces. We embedded this algorithm in a Monte Carlo simulation model which simulates a player throwing the ball while varying initial launch parameters. Using this model, we were able to track possible trajectories and make probabilistic statements about various outcomes of the game. Furthermore, we used our empirical results to suggest different scenarios for the game, then applied the model to assess and quantify their impacts on difficulty. Visual inspection and brief analysis of an actual game support our model’s credibility.
International Journal of Sports Science & Coaching 06/2012; 7(2):371-382. DOI:10.1260/1747-95184.108.40.2061 · 0.93 Impact Factor