First slide

Collisions

Question

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

Moderate

A

$v = \sqrt{\frac{2}{3}} u$
B

$v = \frac{1}{\sqrt{6}} u$
C

$v = \frac{u}{\sqrt{3}}$
D

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

Solution

$\begin{array}{l} U \sin g m o m e n t u m c o n s e r v a t i o n, \\ {\vec{P}}_{i} = {\vec{P}}_{f} \\ \Rightarrow m (u \hat{i}) + 3 m (0) = m (v \hat{j}) + 3 m (\vec{v'}) \\ \Rightarrow \vec{v'} = (\frac{u}{3}) \hat{i} - (\frac{v}{3}) \hat{j} \\ U \sin g e n e r g y c o n s e r v a t i o n, \\ \frac{1}{2} m u^{2} = \frac{1}{2} m v^{2} + \frac{1}{2} 3 m {{(\frac{u}{3})}^{2} + {(\frac{v}{3})}^{2}} \\ \Rightarrow u^{2} = v^{2} + \frac{u^{2}}{3} + \frac{v^{2}}{3} \\ \Rightarrow \frac{2 u^{2}}{3} = \frac{4 v^{2}}{3} \\ \Rightarrow v = \frac{u}{\sqrt{2}} \end{array}$

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