Consider a system of two particles, and assume that no external forces are exerted on the system.

To describe the interaction between the two particles, begin with Newton's Third Law: "For every action, there is an equal and opposite reaction..."

The principle of impulse and momentum can be applied to each particle, giving

The subscript 1 and 2 represents the velocity before after the collision, respectively.

From Newton's Third Law, the reaction force magnitude on each object must be equal,

     F_AB = F_BA

In vector format, this means

     F_AB + F_BA = 0

Substituting, yields

     (m_Av_A2 - m_Av_A1) + (m_Bv_B2 - m_Bv_B1) = 0

Or simply,

     m_Av_A1 + m_Bv_B1 = constant

This equation states that if the effects of external forces are negligible, the total linear momentum of a system is conserved.

In general (multiple objects), conservation of momentum is

Center of Gravity for Point Masses

Conservation of Linear Momentum

Based on the center of mass concept derived in statics, the position of the center of gravity (CG) for the two particles is given by

A time derivative can be taken to obtain an expression for the velocity of the combined center of gravity, giving

If the total linear momentum of a system is conserved, the velocity of the combined center of gravity is also constant.