Maximum Number of Collisions for Three Point Particles

A simple method of obtaining $n_{max}$, the maximum number of binary collisions for three point particles with masses $m_1$, $m_2$, and $m_3$ interacting through zero-range forces, is presented. The result is that $n_{max}$ is the smallest integer greater than or equal to $x$, where $cos\left(\frac{\pi }{x}\right)={\left[\frac{m_1{m}_2}{(m_1+m_3)(m_2+m_3)}\right]}^{\frac{1}{2}}$. (Here $m_3$ is the smallest mass, and $x\ge 3$.) Also, the magnitudes of the momenta after any number of classically allowed collisions are simply expressed in terms of the initial momenta.