As a consequence of the principle of action and reaction, all collisions conserve the total linear momentum. The following derivation is a limited proof of the previous statement.
Consider the kinetic conditions previous to the collision illustrated below where the two masses are sliding over a frictionless surface at constant velocity,
The impulse on mass one is while the impulse on the mass two is . Adding these two relation member by member leads to the relations,
but accordingly with Newton's Third Law, action and reaction, . Thus, where represents the total linear momentum of the system (vector addition of the linear momentum of the individual parts of the system).
Notice that the previous derivation is independent of the length of the time because accordingly to Newton's Third Law of motion. In this case, represents the change in the total linear momentum of the system. The total momentum of the system is the vector addition of the linear momentum of each part, . Thus, the linear momentum of the individual masses is not conserved. The total linear momentum of the system is conserved.
Since the total linear momentum of the system is conserved independently of the duration of the collision, independent of , all collisions conserve the total linear momentum of the system. A more careful proof of this statement can be given using elementary calculus.
The mechanical energy of the system is not always conserved as a consequence of mechanical energy loss to other forms of energy during a collision. Thus, collisions are classified accordingly to mechanical energy and linear momentum conservation.