Two light, rigid rods of length $2 \ell$ have a mass $m$ attached to each end. Both are free to move in two dimensions. The two rods are placed so that their two ends are located at $(-d,+2 \ell),(-d, 0)$, and $(+d, 0),(+d,-2 \ell)$ respectively, where $d$ is positive. They are set in motion with no rotation, with centre-of-mass velocities $(+V, 0)$ and $(-V, 0)$, so that the lower mass on the first rod collides head on with the upper mass on the second rod at the origin $(0,0)$. [You may assume that the impulse is directed along the $x$-axis.]

Assuming the collision is elastic, calculate the centre of-mass velocity $\boldsymbol{v}$ and the angular velocity $\boldsymbol{\omega}$ of each rod immediately after the collision.

Assuming a coefficient of restitution $e$, compute $\boldsymbol{v}$ and $\boldsymbol{\omega}$ for each rod after the collision.