Coefficient of restitution: Derivation of Newton\'s Experimental Law from general energy considerations

TL;DR

This paper derives Newton's Experimental Law for the coefficient of restitution from fundamental energy considerations, providing a theoretical foundation for a widely used empirical collision law.

Contribution

It offers a novel derivation of Newton's Experimental Law based on energy conservation principles, linking it to fundamental physics.

Findings

Derivation of NEL from energy considerations

Clarification of the physical basis of the coefficient of restitution

Connection between energy loss and collision velocities

Abstract

In order to describe the velocity of two bodies after they collide, Newton developed a phenomenological equation known as "Newton\' s Experimental Law" (NEL). In this way, he was able to practically bypass the complication involving the details of the force that occurs during the collision of the two bodies. Today, we use NEL together with momentum conservation to predict each body\'s velocity after collision. This, indeed, avoids the complication of knowing the forces involved in the collision, making NEL very useful. Whereas in Newton\'s days the quantity of kinetic energy was not known, today it is a basic quantity that is in use. In this paper we will use the loss (or gain) of kinetic energy in a collision to show how NEL can be derived.