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Coefficient of restitution: Derivation of Newtonś Experimental Law from general energy considerations

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 bodyś velocity after collision. This, indeed, avoids the complication of knowing the forces involved in the collision, making NEL very useful. Whereas in Newtonś 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.