Delayed Rebounds in the Two-Ball Bounce Problem

TL;DR

This paper investigates the delayed rebound phenomenon in a two-ball bounce system, revealing conditions where the second bounce exceeds initial expectations, using a simplified dynamical model accessible to undergraduates.

Contribution

It introduces a dimensionless dynamical model to explain delayed rebounds, extending beyond the independent contact model for inelastic collisions.

Findings

Delayed rebound occurs under specific parameter ranges.

First bounce is often lower than ICM prediction.

Second bounce can surpass initial expectations.

Abstract

In the classroom demonstration of a tennis ball dropped on top of a basketball, the surprisingly high bounce of the tennis ball is typically explained using momentum conservation for elastic collisions, with the basketball-floor collision treated as independent from the collision between the two balls. This textbook explanation is extended to inelastic collisions by including a coefficient of restitution. This independent contact model (ICM), as reviewed in this paper, is accurate for a wide variety of cases, even when the collisions are not truly independent. However, it is easy to explore situations that are not explained by the ICM, such as swapping the tennis ball for a ping-pong ball. In this paper, we study the conditions that lead to a "delayed rebound effect," a small first bounce followed by a higher second bounce, using techniques accessible to an undergraduate student. The dynamical model is based on the familiar solution of the damped harmonic oscillator. We focus on making the equations of motion dimensionless for numerical simulation, and reducing the number of parameters and initial conditions to emphasize universal behavior. The delayed rebound effect is found for a range of parameters, most commonly in cases where the first bounce is lower than the ICM prediction.