Elastic collisions in one dimension

Consider two bodies of masses $m_1$ and $m_2$ moving along the same straight line (the positive $x$-direction) on a frictionless surface, with initial velocities $u_1 > u_2$ (so $m_1$ catches up to $m_2$) and final velocities $v_1, v_2$ after an elastic head-on collision.

Conservation of momentum gives:

Conservation of kinetic energy (the defining condition of an elastic collision) gives:

Using the difference-of-squares identity $a^2-b^2=(a-b)(a+b)$ on both sides and dividing by equation $(i)$ gives a remarkably clean result:

In any one-dimensional elastic collision, the relative velocity of approach before the collision equals the relative velocity of separation after it (in magnitude, with the sign reversed). Combining this with momentum conservation and solving simultaneously gives the final velocities explicitly:

Four special cases give physical insight:

• Equal masses ($m_1=m_2$): the formulas reduce to $v_1=u_2$ and $v_2=u_1$ -- the two bodies simply exchange velocities.
• Equal masses, target initially at rest ($m_1=m_2,\,u_2=0$): $v_1=0$ and $v_2=u_1$ -- the incoming body stops dead, and the target moves off with the incoming body's original speed (the classic "Newton's cradle" behaviour). …