Impact dynamics for gravity-driven motion of a particle

Abstract

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.

Coefficient of restitution of a linear dashpot on a rigid surface

“…The coefficient of restitution ε is calculated in various models and compared to experimental data from [4]. The velocity dependence is seen in the experimental data and well matched by the models that include gravity.…”

“…For the sake of comparison, the characteristic velocities for a ping pong ball and a basketball are 0.01 m/s and 0.03 m/s, respectively. For the spring-loaded cart experiment [4], v * = 0.129 m/s.…”

“…The effects of the velocity-dependent coefficient of restitution are most clearly seen in experiments where an object is allowed to bounce multiple times until inelastic capture. In [4], the authors used a spring-loaded cart on an inclined track as a practical model of a spring with a linear dashpot. We use the data from that experiment to test the various models The mass and elastic coefficient are typically measured directly, but the damping coefficient β must be estimated from the bouncing motion.…”

“…We use each model recursively, solving for the successive impact velocities, and totaling the contact time and flight time between each bounce until inelastic capture. The damping coefficient β is determined by a best fit to the bounce time data [4], shown in Figure 6. The results from the various models are visually indistiguishable, so only the hybrid model is plotted.…”

“…Gravity can be neglected when two spheres collide in midair, as gravity affects each sphere equally. However, for a ball bouncing off of a rigid surface, gravity cannot be neglected, and its inclusion results in a coefficient of restitution that depends on impact velocity [4].…”