Center of Mass Velocity:
* The center of mass velocity of a system of particles is the average velocity of all the particles weighted relative to their mass.

or

Total Momentum and Center of Mass Velocity:
*
The total momentum of all the particles in a system is equal to the momentum of a single particle with a mass m = m1 + m2 + ...mN moving at the velocity of the center of mass vcm.

* Since momentum is always conserved, the velocity of the center of mass before any collisions will always be equal to the velocity of the center of mass of the system after the collisions.

Energy Loss and the Kinetic Energy of the Center of Mass:
*
If all the particles collided and stuck together in a totally inelastic collision, the final velocity of the clump will be equal to the velocity of the center of mass.

* The maximum energy loss during any collision occurs during a total inelastic collision. After such a collision, the final kinetic energy will be equal to the kinetic energy of the center of mass.

* For a partially elastic collision, the energy loss is given by

Frame of Reference in a Collision
Three different shaped objects collide with each other. One can view the collision from either the normal lab frame, a frame moving with each of the objects, or the center of mass frame.

Frames of Reference for a Moving Spring
Two blocks are connected by an initially unstretched spring and one block is given an initial velocity. The subsequent motion can be viewed in either normal lab frame, a frame moving with each of the objects, or the center of mass frame. Gravity can also be turned on or off to observe the motion of the blocks.