## Collisions

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,

Before Collision

 The velocity v1 is greater than the velocity v2 permitting the mass m1 to catch up with the mass m2  and producing a collision. Before the collision, the individual linear momentum of the masses are and  .

During Collision

 The force F12 is the force on the mass one applied by the mass two. The forces F12 and F21 are an action and reaction pair; thus,  F12 + F21 = 0. Therefore, the resulting impulse acting in each of the masses obey

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.

 Collisions Inelastic Collisions Conserve Linear Momentum Do NOT conserve Mechanical Energy Elastic Collisions Conserve Linear Momentum Conserve Mechanical Energy

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com