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.