Bifurcation Adds Flavor to Basketball

TL;DR

This paper explores how bifurcation phenomena in basketball shooting dynamics influence shot difficulty, revealing that optimal launch angles change abruptly with speed and distance, emphasizing the importance of precise control.

Contribution

It introduces a phase-space volume approach to quantify shot difficulty and demonstrates bifurcation effects in basketball physics based on launch parameters.

Findings

Bifurcation occurs in optimal launch angles with increasing speed at certain distances.

Removing air resistance eliminates bifurcation, simplifying the phase-space volume distribution.

Precise control of launch speed is crucial near bifurcation points.

Abstract

We report an emergence of bifurcation in basketball, a single-particle system governed by Newtonian mechanics. When shooting the basketball, the obvious control parameters are the launch speed and the launch angle. We propose to use the three-dimensional velocity phase-space volume associated with the given launch parameters to quantify the difficulty of the shooting. The optimal launch angle that maximizes the associated phase-space volume undergoes a bifurcation as the launch speed is increased, if the shooter is farther than a critical distance away from the hoop. Thus, the bifurcation makes it very important to control the launch speed accurately. If the air resistance is removed, the bifurcation disappears and the phase-space volume distribution becomes dispersionless and shrinks in magnitude.