The underlying physics of basketball shooting seems to be a straightforward example of the Newtonian mechanics that can easily be traced by numerical methods. However, a human basketball player does not make use of all the possible basketball trajectories. Instead, a basketball player will build up a database of successful shots and select the trajectory that has the greatest tolerance to small variations of the real world. We simulate the basketball player's shooting training as a Monte Carlo sequence to build optimal shooting strategies, such as the launch speed and angle of the basketball, and whether to take a direct shot or a bank shot, as a function of the player's court positions and height. The phase space volume that belongs to the successful launch velocities generated by Monte Carlo simulations are then used as the criterion to optimize a shooting strategy that incorporates not only mechanical, but human factors as well.
It may seem that the underlying physics of basketball is too well understood to merit another survey.
That may well be the reason why there is so little research into the physics of basketball, despite the huge popularity of the game. Furthermore, we possess enough computational power to realistically trace the motion of a basketball. These circumstances seem to suggest that we only need to calculate all the physical trajectories of a basketball and run the appropriate statistical analysis. This has been the approach of earlier researchers who studied the physics of basketball [1][2][3][4][5][6][7]. However, it is not possible for a human player to make use of all the possible basketball trajectories, nor is it solely on account of human shortcomings. High launch angle shots occupy less phase space volume because the velocity space volume element is proportional to the cosine of the launch angle measured from the horizontal plane. Smaller phase space volume means less tolerance for small inaccuracies in aim. Such an attempt would have a great entertainment value, but nobody would try it under a pressure to win.
We will introduce the phase space volume associated with the launch velocity as an important criterion in the optimization of the shooting strategy. We consider a basketball player shooting from a position in the court, given by We analyze the bank shots in Figure 2 by replacing each bouncing spots on the backboard with a Gaussian function. Then, the bouncing spots will become a continuous distribution on the backboard.
After normalization, it should correspond to the probability density of bank shots on the backboard.
We observe that the distribution is elongated upward (Figure 3). In Figure 3, only the right half of the backboard above the rim height is shown. The horizontal axis should correspond to the x-axis, and the vertical axis to the z-axis. This probability distribution is counter-intuitive in that it stretches too high on the backboard.
If we consider how a person learns to shoot, a basketball player will try a shot and then judge his own shooting by how far he has missed the hoop. Based on the deviations from the goal, he will try to improve his aim by making adjustments. Eventually, he will find optimal 0 and 0 v as a function of his court position and his height through a lengthy shooting practice. To do that, he will have to choose a part of the horseshoe shape, because the diagram extends to unlikely and v . But, a straightforward application of the deterministic Newtonian equations alone cannot offer a criterion for such a selection process.
Tran et al [4] introduce standard deviations and v about the optimal values of and v , and keep the successful shooting ratio at a value typical of professional players. The skill level of the player is reflected in the successful shooting ratio. This scheme would be equivalent to selecting a rectangular region in the v diagram centered at the 0 and 0 v . There exist arbitrariness in setting the optimal values ( 0 and 0 v ) and the standard deviations ( and v ).
We propose to view the basketball player’s shooting training process as a Monte Carlo sequence. A basketball player will judge his shootings by how far the basketball is located from the center of the rim when it passes the horizontal plane located at the height of the hoop. The skill of the basketball player will be described by the temperature parameter of the Monte Carlo simulation. There is no need for additional parameters in this approach.
As Silverberg et al [5] point out, the players tend to shoot more accurately when they are closer to the hoop. The ability of the shooter to control the launch parameters does not change as a function of his position. But, the nearer to the hoop the shooter is positioned, the more launch angle and speed (thus, the more phase space volume) is available to the shooter. Therefore, we choose to calculate the phase space volume occupied by successful launch velocities from the Monte Carlo sequence and take it as the criterion for the better strategy.
The dynamics of the basketball is governed by gravitational force and air resistance. The effect of the Magnus force for basketball is ignored because of its small magnitude. However, we assumed that the basketball is shot with a back spin and that the back spin results in a gain in the downward velocity as it bounces off the backboard, given by
, where n ˆ is the unit vector normal to the backboard and
. The basketball is assumed to be a rigid thin shell with a restitution coefficient of 0.75. The air resistance force is expressed by Okubo et al [6,7] as
where is the density of air,
the drag coefficient of the basketball in air, A the cross- section area of the basketball, and v is the velocity of the basketball. The equations were integrated using the fourth-order Runge-Kutta method with a time step of s
In the Monte Carlo simulation, the key variable is th
This content is AI-processed based on open access ArXiv data.
