 Equal-speed elastic impact on a free rod

A point mass $m$ strikes a free rod of equal mass, initially at rest, perpendicularly at distance $h$ from its center of mass. Assume a smooth elastic collision and rod moment of inertia $I$ about its centre. The final translational speeds are equal precisely when $h^2=2I/m$. Indeed, momentum conservation then fixes both final velocities to $v_0/2$; angular impulse gives $I\omega =m{v}_{0}h/2$, and conservation of kinetic energy gives $I{\omega }^2=m{v}_0^2/2$. Eliminating $\omega$ proves the formula. A uniform rod has $I=m{l}^2/12$, hence $|h|=l/\sqrt{6}$, which lies within the rod.