Modeling the Dynamics of an American Football and the Stability Due to Spin

Abstract

This paper develops a mathematical model describing the motion through the air of an American football. The model is based on established equations used to describe spinning projectiles. While the equations are applicable to general motions, the emphasis of the paper is on the spiral pass and punt. Separate sections introduce formulas for the forces and moments understood to act on spun projectiles. The discussion of each force and moment includes an assessment of how well available experimental data characterizes the force or moment for an American football. For each force or moment, there is a description of how it affects the motion and trajectory. While the equations are valid for arbitrary motions, the available aerodynamic data is not. In parallel with the derivation of the nonlinear mathematical model, a linearized dynamics model is developed. The linearized model is used to help explain the behavior of the nonlinear model and to provide insight into the underlying physics. The linearized model is also used to derive a relationship between linear and angular velocity that ensures that the gyroscopic motion of a football is stable. The paper provides physical insights into what causes the apparent “wobble” of a spiral pass and what the character of the wobble says about the quality of the pass. Among the physical insights provided are the reason some passes have a rapid wobble and some slow, why a pass exhibits a lateral swerve, and why the Magnus effect may be neglected. The results are applicable to rugby footballs.