On the flight of an American Football

TL;DR

This paper develops a detailed theoretical model of an American football's flight, incorporating spin and air resistance, and compares it to gyroscope and rocket trajectories, with corrections for previous errors.

Contribution

It introduces a realistic air resistance model for football speeds and corrects prior errors in moment of inertia calculations, enhancing flight prediction accuracy.

Findings

Football shape resembles a parabola of revolution

Air resistance modeled as proportional to v or v^2 depending on speed

Corrected moments of inertia for more accurate trajectory predictions

Abstract

In this paper we examine the detailed theory of the American football in flight, with spin and air resistance included. We find the theory has much in common with the theory of a gyroscope and also rocket trajectory with a misaligned thruster. Unfortunately most of the air resistance data, for rocketry and ballistics, is for speeds of Mach 1 or higher, where the air resistance increases dramatically. We shall approximate a realistic air resistance, at the slower speeds of football flight, with a drag force proportional to cross sectional area and either $v$ or $v^2$, depending on speed, where $v$ is velocity of the football. We begin with a discussion of the motion, giving as much detail as possible without the use of complex analytic calculations. We point out the previous errors made with moments of inertia and make the necessary corrections for more accurate results. We show that the shape of an American football closely resembles a parabola of revolution.