Optimal trajectory for the basketball free throw

Human Performance Laboratory, The University of Calgary, Alberta, Canada.
Journal of Sports Sciences (Impact Factor: 2.25). 11/1997; 15(5):491-504. DOI: 10.1080/026404197367137
Source: PubMed


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[1]). 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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    • "Firstly, our article is the first to rigorously model what happens when a spinning elastic ball bounces on a series of surfaces that are arbitrarily-tilted at planes AE90 . Secondly, our study is the first to evaluate the effect of the variation in initial launch parameters in said situation, in the spirit of [12] [13]. "
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    Applied Mathematics and Computation 02/2015; 253:61-71. DOI:10.1016/j.amc.2014.12.061 · 1.55 Impact Factor
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    • "The ability to identify key elements that lead to successful shooting has been the focus of previous research. The kinematics of shooting has been investigated for jump shots (Huston and Grau, 2003; Miller and Bartlett, 1993; Miller and Bartlett, 1996), freethrow shooting (Hamilton and Reinschmidt, 1997; Satern and Keller-McNulty, 1992; Tan and Miller, 1981) and three-point shots (Erculj and Supej, 2009). Successful shooting is associated with a straight and high ball release, and forearm position and wrist flexion are critical for consistent straight projection of the ball with backspin (Penrose and Blanksby, 1976; Szymanski, 1967). "
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    • "In this particular case, as concluded by Hamilton and Reinschmidt (1987) and Miller (2002) in their studies in basketball shooting variability, and by Darling and Cooke (1987) and Messier and Kalaska (1999) in their analysis of grasping and reaching movements, we have found an inverse relationship between performance variability, the strength of performance and accuracy in serves. In addition, this variable could provide information about the characteristics present in the task of achieving the player’s desired performance, for strengthening, creating or changing the attractors of performance (Nashner and McCollum, 1985). "
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