**The centre of mass**

The velocity with respect to the centre of mass is given by The coordinates of the centre of mass are given by:

In a 2-particle system, the coordinates of the centre of mass are given by:

With , the kinetic energy becomes: , with the reduced mass given by:

The motion within and outside the centre of mass can be separated:

**Collisions**

With collisions, where B are the coordinates of the collision and C an arbitrary other position, holds: is constant, and is constant. The changes in the relative velocities can be derived from: Further holds , = constant and with respect to B is constant.