Question

A particle of mass $m$ with an initial velocity $u\hat{i}$ collides elastically with a mass $3m$ at rest. It moves with a velocity $v\hat{j}$ after collision, then, $v$ is given by :

• A. $v=\sqrt{\frac{2}{3}u}$
• B. $v=\frac{u}{\sqrt{3}}$
• C. $v=\frac{u}{\sqrt{2}}$
• D. $v=\frac{1}{\sqrt{6}}u$

Answer 1

The correct option is C $v=\frac{u}{\sqrt{2}}$


From the law of conservation of linear momentum,

$mu\hat{i}+0=mv\hat{j}+3m\overrightarrow{v^{\prime }}$

$\Rightarrow \overrightarrow{v^{\prime }}=\frac{u}{3}\hat{i}-\frac{v}{3}\hat{j}$

$\Rightarrow |v^{\prime }|=\sqrt{{\left(\frac{u}{3}\right)}^2+{\left(\frac{v}{3}\right)}^2}$

As the collision is elastic, from the conservation of kinetic energy,

$\frac{1}{2}m{u}^2=\frac{1}{2}m{v}^2+\frac{1}{2}\left(3m\right)({\left(\frac{u}{3}\right)}^2+{\left(\frac{v}{3}\right)}^2)$

$\Rightarrow u^2=v^2+\frac{u^2}{3}+\frac{v^2}{3}$

$\therefore v=\frac{u}{\sqrt{2}}$

Hence, option $\left(C\right)$ is correct.