On the Stopping Time of a Bouncing Ball

TL;DR

This paper presents a detailed model of a bouncing ball considering elastic deformation and internal friction, showing it avoids inelastic collapse and providing asymptotic impact data, with comparisons to experiments.

Contribution

It introduces a physically realistic bouncing ball model that prevents inelastic collapse and derives asymptotic impact expressions, enhancing understanding of impact dynamics.

Findings

The model does not allow infinite impacts in finite time.

Asymptotic expressions for impact velocity and flight time are derived.

Contacts with zero impact velocity are possible but rare.

Abstract

We study a simple model of a bouncing ball that takes explicitely into account the elastic deformability of the body and the energy dissipation due to internal friction. We show that this model is not subject to the problem of inelastic collapse, that is, it does not allow an infinite number of impacts in a finite time. We compute asymptotic expressions for the time of flight and for the impact velocity. We also prove that contacts with zero velocity of the lower end of the ball are possible, but non-generic. Finally, we compare our findings with other models and laboratory experiments.