elastic collision

In an elastic collision, both momentum and kinetic energy are conserved. The colliding bodies rebound without any net loss to heat, sound, or deformation. Truly elastic collisions exist most cleanly among atomic and subatomic particles: protons off protons, electrons off atoms. Macroscopically they are an idealisation well approximated by hard steel balls, rubber superballs, and the smooth spheres in idealised kinetic-theory gas models.

For a 1-D elastic collision between masses m_A (with incoming velocity v_A) and m_B (initially at rest), the outgoing velocities are determined uniquely by combining momentum and energy conservation: v_A' = ((m_A − m_B) / (m_A + m_B))·v_A and v_B' = (2m_A / (m_A + m_B))·v_A. Three cases illuminate the geometry: equal masses swap velocities (the Newton's cradle trick); a heavy projectile barely slows while a light target shoots off at twice the projectile's speed (the golf-club effect); a light projectile bounces straight back off an essentially stationary heavy target (the rubber ball on a wall).

Beyond its role as a pedagogical idealisation, the elastic collision is the conceptual backbone of scattering experiments in physics. When Rutherford in 1911 bounced alpha particles off gold-foil atoms and saw some deflect through large angles, he used the mathematics of elastic collisions to infer the existence of a small, heavy atomic nucleus. Every scattering experiment since — up to and including the LHC — is an elaborate extension of the same idea.