Conservation of momentum. CM frame.

Conservation of momentum and center-of-mass (CM) frame explained in a class given at MIT by Prof. Walter Lewin (course 8.01, Physics I: Classical Mechanics, fall 1999).

Conservation of momentum is central to understanding collisions; it can be used as long as external forces (like friction) can be ignored. Consider a collision in one dimension of objects with masses and . The momentum before the collision is,

this is conserved. If the objects stick together after the collision, we therefore have

where is the speed they have after the collision. So, momentum does not disappear in a collision (generally); kinetic energy, on the other hand, can disappear completely. How?

Above we had two objects with masses and colliding. More generally, if we have two, three or more objects interacting, things could seem to be very complicated. However, there is a point defined by the system — the center-of-mass (CM) — which behaves very simple. If the objects have positions with respect to some inertial frame, and masses , then the location of the CM is,

The CM behaves as if the total mass of the system was located at this point:

where the sum on the lhs is over all external forces. This equation is identical to Newton’s second law for a single particle with mass . All this is explained at 35:30 in the video above.

(Young and Freedman, chap. 8.)