Elastic Collisions in One Dimension

A collision between two bodies is called perfectly elastic if, in addition to the total momentum of the system being conserved (as it always is in any collision, so long as no external force acts during the brief interval of contact), the total kinetic energy of the system is also exactly conserved -- none of it is lost to heat, sound, or permanent deformation. Real macroscopic collisions only ever approximate this ideal (collisions between hard steel balls, or between billiard balls, come close), but it remains an extremely useful limiting case to work out in full.

Setting up the two conservation equations. Consider a head-on, one-dimensional collision between a body of mass $m_1$ moving with initial velocity $u_1$ and a body of mass $m_2$ moving with initial velocity $u_2$ (both measured along the same line, so either may be negative), after which their velocities become $v_1'$ and $v_2'$. Conservation of momentum gives

and conservation of kinetic energy (the defining property of an elastic collision) gives

Solving for the final velocities. Rearranging the momentum equation as $m_1(u_1 - v_1') = m_2(v_2' - u_2)$ and the energy equation (after cancelling the common factor of $\tfrac12$ and using the difference-of-squares identity) as $m_1(u_1-v_1')(u_1+v_1') = m_2(v_2'-u_2)(v_2'+u_2)$, dividing the second relation by the first (valid whenever $u_1 \neq v_1'$) gives the compact and very useful result $u_1 + v_1' = v_2' + u_2$, i.e. the relative velocity of approach equals the relative velocity of separation: $u_1 - u_2 = -(v_1' - v_2')$. Combining this with the momentum equation and solving the resulting pair of simultaneous linear equations gives the standard elastic-collision formulas:

Three instructive special cases, each obtained by substituting into these general formulas (taking $u_2 = 0$, a stationary target, for simplicity in each case):

• Equal masses ($m_1 = m_2$): the formulas reduce to $v_1' = 0$ and $v_2' = u_1$ -- the two bodies simply exchange velocities; the incoming body stops dead, and the target moves off with exactly the incoming body's original velocity (the everyday "Newton's cradle" effect, and the reason a cue ball that strikes a stationary, equal-mass ball squarely comes to a near-complete stop). …