Elastic One-Dimensional Collisions

in the Center of Mass Frame of Reference

For instance, suppose that mass m₁ with initial velocity v₁ collides elastically with mass m₂ with initial velocity v₂.

To switch to the center of mass frame of reference, subtract v_cm from each velocity. So, in the center of mass frame, the initial velocity of m₁ is v₁ - v_cm, and the initial velocity of m₂ is v₂ - v_cm.

Since the collision will be symmetric with respect to the center of mass, the velocity of m₁ after the collision will be -(v₁ - v_cm) = v_cm - v₁ in the center of mass frame, and the velocity of m₂ after the collision will be -(v₂ - v_cm) = v_cm - v₂.

To switch back to the Earth frame of reference, add v_cm to each velocity. The velocity of m₁ in the Earth frame will be 2v_cm - v₁ after the collision, and the velocity of m₂ will be 2v_cm - v₂.

In summary, if an object has a velocity v before a one-dimensional elastic collision, it will have a velocity 2v_cm - v after the collision. Simple, huh?