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 $\sin θ_{1} = \sqrt{n} \sin θ_{2}$ then value of n is

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Detailed Solution

$G i v e n \to \frac{1}{2} M V_{1}^{2} = \frac{1}{2} (\frac{1}{2} M u^{2}) \Rightarrow V_{1} = \frac{u}{\sqrt{2}}$

From conservation of momentum in y – direction
$V_{1} \sin θ_{1} = 10 V_{2} \sin θ_{2} \to (1)$
As collision is elastic KE remains conserved
$\frac{1}{2} M u^{2} = \frac{1}{2} M V_{1}^{2} + \frac{1}{2} (10 M) V_{2}^{2}$
$\Rightarrow \frac{1}{2} M u^{2} = \frac{1}{2} (\frac{1}{2} M u^{2}) + \frac{1}{2} (10 M) V_{2}^{2}$
$\Rightarrow \frac{1}{4} M u^{2} = \frac{10}{2} M V_{2}^{2}$
$\Rightarrow \frac{u^{2}}{20} = V_{2}^{2} \Rightarrow V_{2} = \frac{u}{\sqrt{20}}$
Put in equation (1)
$\Rightarrow \frac{u}{\sqrt{2}} \sin θ_{1} = 10 \frac{u}{\sqrt{20}} \sin θ_{2}$
$\Rightarrow \sin θ_{1} = \sqrt{10} \sin θ_{2}$