A particle of mass $m$ is moving along the $x$-axis with initial velocity $u\hat i$. It collides elastically with a particle of mass $10m$ at rest and then moves with half its initial kinetic energy (see figure). If $\sin\theta_1 = \sqrt{n}\,\sin\theta_2$ then value of $n$ is ______.
Numerical value type. Enter your answer.
Answer: 10
After the collision, $m$ has speed $v_1 = \dfrac{u}{\sqrt2}$ (half the KE). The other half goes to $10m$: $\tfrac12(10m)v_2^{2} = \tfrac14mu^{2} \Rightarrow v_2 = \dfrac{u}{\sqrt{20}}$.
Momentum perpendicular to the $x$-axis is zero:
$$mv_1\sin\theta_1 = 10m\,v_2\sin\theta_2 \Rightarrow \sin\theta_1 = \frac{10\,u/\sqrt{20}}{u/\sqrt2}\sin\theta_2 = \sqrt{10}\,\sin\theta_2$$
So $n = 10$.
Solution by Sreeraj P, M.Sc Physics