Impact dynamics for gravity-driven motion of a particle

Introductory mechanics courses use the bouncing ball model to familiarize students with the principles of binary inelastic collisions. Nonetheless, in undergraduate courses, the modeling of binary systems typically disregards the time of contact and the effects of gravity, which yields to a constant coefficient of restitution (COR) and, as a consequence, prevents students from elucidating the real dependence of COR with impact speed. In this work, we proposed a simple experimental setup to investigate the impact dynamics of a spring-mass body that bounces on a rigid plate as well as a theoretical framework that captures the velocity dependence of the COR for low-impact speeds. Our analytical expression for the COR highlights the role-played by gravity, impact speed and collision time on the collision dynamics. Our results suggest that the inclusion of gravity force allows an adequate estimation of the critical impact speed and maximum deformation distance, crucial parameters to differentiate between repulsive and attractive interaction regimes. Our experimental setup enables the clarification of several key concepts of mechanics while it is easy to be performed in most undergraduate laboratories.