# Coefficient of restitution

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## Definition

QUICK PHRASES: bounciness of a collision, final velocity difference divided by initial velocity difference

### For a collision in one dimension

Suppose two objects are moving along a one-dimensional axis, collide, and then again move along the same one-dimensional axis. If their initial velocities are (as signed real numbers) $u_1$ and $u_2$ respectively and their final velocities are $v_1$ and $v_2$ respectively, then the coefficient of restitution or bounciness of the collision is defined as:

$C_R := \frac{v_2 - v_1}{u_1 - u_2}$.

The coefficient of restitution is also sometimes denoted as $e$.

### For a collision in three dimensions

For a collision in three dimensions, the velocities are vectors rather than scalars, so the above formula does not make sense. In this case, we interpret $u_1,u_2,v_1,v_2$ as the components of velocities in the normal direction, i.e., the direction perpendicular to the plane of contact/tangent plane at the point of contact.

## Constitutive factors and measurement

The coefficient of restitution is dependent on the specific collision and cannot be inferred precisely from the geometry, velocity, and the kind of material in the colliding bodies. However, knowledge of the type of material and the geometry of the bodies usually allows for an approximate estimation of the coefficient of restitution.