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Perfectly Inelastic Collision

A particle of...

Question

A particle of mass $m$ moving in the $x -$ direction with speed $2 v$ is hit by another particle of mass $2 m$ moving in the $y -$ direction with speed $v$ . If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to

50 %

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56 %

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62 %

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42 %

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Solution

The correct option is B $56 %$

From conservation of momentum,
$P_{x i} = P_{x f}$
$2 m v = 3 m v_{x}$
$\Rightarrow v_{x} = \frac{2 v}{3}$
$P_{y i} = P_{y f}$
$2 m v = 3 m v_{y}$
$\Rightarrow v_{y} = \frac{2 v}{3}$

Final kinetic energy of system
$K E_{f} = \frac{1}{2} 3 m (v_{x}^{2} + v_{y}^{2}) = \frac{1}{2} 3 m [\frac{4 v^{2}}{9} + \frac{4 v^{2}}{9}]$
$K E_{f} = \frac{4 m v^{2}}{3}$

Initial kinetic energy of system
$K E_{i} = \frac{1}{2} m (2 v)^{2} + \frac{1}{2} 2 m (v^{2}) = 2 m v^{2} + m v^{2} = 3 m v^{2}$
$% Loss = \frac{K E_{i} - K E_{f}}{K E_{i}} = 100 = \frac{3 m v^{2} - \frac{4 m v^{2}}{3}}{3 m v^{2}} \times 100$
$= \frac{5}{9} \times 100 = 55.5 % \approx 56 %$
The correct option is (B)

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