The total momentum of the system is

so the centre of mass moves as if it is a single particle of mass . Since internal forces cancel pairwise,

so the centre of mass accelerates like a point particle subject to the external force . In particular, implies : the total momentum of a system is conserved in the absence of external forces. This is why an extended object such as the Earth can be treated as a point particle: its internal forces cancel out.

Write each position as , where is the position of the -th particle relative to the centre of mass. Then , and since this holds for all time, . The kinetic energy therefore splits as

a centre-of-mass kinetic energy plus an internal kinetic energy.

For two particles with no external forces, let be the relative separation and . Then

and the kinetic energy becomes

where is the reduced mass. Moreover , so both the centre of mass motion and the relative separation behave like single particle problems. If , then : in this limit the heavy object is essentially still and the lighter object moves around it.