Coefficient of restitution

The coefficient of restitution compares normal separation and approach speeds at the actual contact points of a collision. Choose a unit normal vector $\mathbf{n}$ from the first body toward the second, and let ${\mathbf{u}}_{-}$ and ${\mathbf{u}}_{+}$ be the second contact point's velocity relative to the first before and after impact. For approach ${\mathbf{u}}_{-}\cdot \mathbf{n}<0$,For a rigid body, the contact velocity includes the rotational contribution $\mathbf{\omega }\times \mathbf{r}$ about its centre of mass. Substituting only centre velocities can therefore give the wrong coefficient.

For two smooth point masses with unchanged tangential relative motion, the loss of kinetic energy is $\frac{1}{2}\mu (1-e^2)({\mathbf{u}}_{-}\cdot \mathbf{n}{)}^2$, where $\mu$ is the reduced mass. In that model $e=1$ gives an elastic collision. The value $e=0$ only removes immediate normal separation; a tangential relative velocity can remain, so it does not imply a perfectly inelastic collision with complete sticking. A vertical bounce on a fixed plate gives the measurable relation $e=\sqrt{h_{\mathrm{rebound}}/h_{\mathrm{drop}}}$ when constant gravity and negligible air resistance connect each height to its impact speed.

An experiment by S. F. Foerster, M. Y. Louge, H. Chang (collision mechanics) and K. Allia released small spheres without initial rotation and inferred incoming and outgoing velocities from photographs made with timed flashes. They measured normal restitution together with friction and tangential rebound properties, illustrating why one normal coefficient cannot describe all contact dynamics.^[ref]

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