Elastic Collisions in 1D

Two objects with masses m1 and m2 and initial velocities v1 and v2 (because this example is a 1-dimensional problem, we have again omitted the vector arrows) collide elastically. What are their final velocities, v3 (for object 1) and v4 (for object 2)?

Answer:

The total momentum is always conserved in collisions. And because this collision is elastic, by definition total kinetic energy is also conserved. This means we have 2 equations for 2 unknowns, leading to:

v3 = v1 (m1-m2)/(m1+m2) + v2 (2m2)/(m1+m2)

v4 = v1 (2m1)/(m1+m2) + v2 (m2-m1)/(m1+m2)

These equations are the general solution for all one-dimensional elastic collision problems.