The conservation of linear momentum was one of the great discoveries of physics. It explains the behavior of billiard balls where a motionless b all upon impact by a moving ball flies away while the impacting ball rests at the point of impact.

A Rebound

Consider a spherical body traveling at a horizontal velocity of v to the right. When it hits a vertical wall it rebounds with a horizontal velocity v to the left.

Since momentum is mass times velocity there would be a tendency to say momentum has been conserved. But momentum has changed from +mv to −mv. One then notes that the absolute value of momentum has been conserved and looks for a more general statement of the conservation of momentum.

Consider a body traveling with a velocity vector v=(v_x, v_y) where v_x is the horizontal component and v_y the vertical component. Upon impact the body the is reflected off the vertical wall. This means its velocity vector becomes v'=(−v_x, v_y).

The magnitude of the body's velocity vector is the same before and after the relection off the vertical wall; i.e.,

||v|| = ||v'|| = (v_x² + v_y²)½

This strongly suggests that there is some generalization of the Principle of the Conservation of Momentum in terms of the magnitude of the linear momentum vector being conserved in collisions. That suggestion however is wrong.

Explanation of What Happens
to Linear Momentua in Rebounds
and Reflections

Consider the wall being covered with springs. As the body compresses one of those springs its kinetic energy is converted into potential energy. The body slows down until all of its kinetic energy has been converted into potential energy; i.e., its velocity becomes zero. Then the spring pushes the body in the opposite direction. If total energy is conserved the compressed spring pushes the body until all of its potential energy has been converted into kinetic energy for the body.Thus the body's velocity has become −v.

In the case of a reflection it is only the component of velocity perpendicular to the wall that gets converted into potential energy.

Linear momentum is obviously not converved in the collision because at some point the velocity and hence the momentum are zero.