# 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 ......... Option: 1 1 Option: 2 10 Option: 3 20 Option: 4 30

$\begin{array}{l} Given \ \ \\ \frac{1}{2} m v_{1}^{2}=\frac{1}{2}\left(\frac{1}{2} \mathrm{mu}^{2}\right) \\ v_{1}^{2}=\frac{u^{2}}{2} \\ v_{1}=\frac{u}{\sqrt{2}} \quad \ldots .(i) \end{array}$

Also

Apply energy conservation

$\begin{array}{l} \frac{1}{2} \mathrm{mu}^{2}=\frac{1}{2} m v_{1}^{2}+\frac{1}{2}(10 \mathrm{~m}) \times v_{2}^{2} \\ \frac{1}{2} \times 10 \mathrm{~m} \times v_{2}^{2}=\frac{1}{2} \times \frac{1}{2} \mathrm{mu}^{2} \\ v_{2}^{2}=\frac{u^{2}}{20} \text { or }, v_{2}=\frac{u}{\sqrt{20}} \end{array}$

Now

From momentum conservation in the perpendicular direction of the initial motion

$\begin{array}{l} m v_{1} \sin \theta_{1}=10 m v_{2} \sin \theta_{2} \\ \frac{u}{\sqrt{2}} \sin \theta_{1}=10 \times \frac{u}{\sqrt{20}} \sin \theta_{2} \\ \sin \theta_{1}=\frac{10}{\sqrt{10} \sin \theta_{2}} \\ \sin \theta_{1}=\sqrt{10} \sin \theta_{2} \\ \therefore n=10 \end{array}$

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