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_{A}v_{A2}  m_{A}v_{A1}) + (m_{B}v_{B2}  m_{B}v_{B1}) = 0

m_{A}v_{A1} + m_{B}v_{B1} = m_{A}v_{A2} + m_{B}v_{B2}


Or simply,
m_{A}v_{A1} + m_{B}v_{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

Σm_{i}(v_{i})_{1} = Σm_{i}(v_{i})_{2}


