# The theory of colliding bodies

The principle used to solve these collision problems was first enunciated by Huygens and involves the existence of something called the Coefficient of Restitution $e$.

Consider the problem of two balls of masses $m_{1}$ and $m_{2}$ sliding on a line (rolling involves a more complicated theory) with velocities $u_{1}$ and $u_{2}$.

If they collide, then linear momentum is conserved.
i.e. the velocities $v_{1}$ and $v_{2}$ after collision satisfy
$m_{1}u_{1} + m_{2}u_{2} = m_{1}v_{1} + m_{2}v_{2}$
In addition the relative velocities before and after collision satisfy
$u_{1} - u_{2} = -e(v_{1} - v_{2})$
The fact that $e$ is independent of the size of the relative velocties and only depends on the material of the balls is the underlying assumption of this model. It is at least a reasonable first approximation of what happens in practice.

For balls moving in two or three dimensions, similar methods apply, except in those cases when a collision occurs between balls it is only the component of the relative velocity along the line of centres that is affected. The component perpendicular to the line of centres remains unchanged. Similar considerations apply if the collision is with a "wall" or a "corner".