Coulomb head-on closest approach with target recoil ($r_{\min }=k_e{q}_1{q}_2(m_1+m_2)/(m_2{K}_{\mathrm{lab}})$)

A projectile of mass $m_1$, electric charge $q_1$, and nonrelativistic laboratory kinetic energy $K_{\mathrm{lab}}$ approaches an initially stationary free target of mass $m_2$ and like-sign electric charge $q_2$. The relative-motion kinetic energy is $K_{\ast }=m_2{K}_{\mathrm{lab}}/(m_1+m_2)$. At a head-on turning point it is converted to the repulsive Coulomb potential $k_e{q}_1{q}_2/r_{\min }$, proving the displayed separation. The remaining laboratory kinetic energy is the translation of the centre of mass; it cannot be converted into the pair's separation energy. An externally fixed target is the infinite-target-mass limit.