JEE Main · 2026 · 22 Jan 2026 · Shift 2 · Conservation of Momentum

Take the origin at the centre of the bar. From the figure the $2m$ mass strikes $2\mathrm{cm}$ from the centre on one side and the $m$ mass strikes $4\mathrm{cm}$ from the centre on the other side, moving in opposite directions. The centre of mass of the final system remains at the bar's centre, since the moments $2m(2)$ and $m(4)$ balance. Conserving angular momentum about this centre (both impacts drive the same sense of rotation), The moment of inertia of the combined body about the centre is With $L=I\omega$, $8mv=264m\omega$, so $\frac{v}{\omega }=33$. The ratio $v:\omega$ is $33$.