A particle of mass m moving with a speed u strikes a smooth horizontal surface at an angle 45 ° . The…

A particle of mass $m$ moving with a speed $u$ strikes a smooth horizontal surface at an angle $45°$. The particle rebounds at an angle $\varphi$ with speed $v$, if coefficient of restitution is $\frac{1}{\sqrt{3}}$, then the angle $\mathrm{\varphi }$ is

1. ${30}^{\mathrm{o}}$
2. ${45}^{\mathrm{o}}$
3. ${60}^{\mathrm{o}}$
4. ${37}^{\mathrm{o}}$

Solution

In $y$ direction

$v\sin \mathrm{\varphi }=eu\sin {45}^{\mathrm{o}} \dots .\left(\mathrm{i}\right)$
In $x$ direction
$v\cos \mathrm{\varphi }=u\cos {45}^{\mathrm{o}} \dots .\left(\mathrm{i}\mathrm{i}\right)$
Equation $\left(\mathrm{i}\right)÷\left(\mathrm{i}\mathrm{i}\right)$
$\tan \mathrm{\varphi }=\frac{1}{\sqrt{3}}; \mathrm{\varphi }={30}^{\mathrm{o}}$

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