A body of mass (4m) is lying in the plane at rest. It suddenly explodes into three pieces. Two pieces, each of mass (m), move perpendicular to each other with equal speeds (v). The total kinetic energy generated due to explosion is [2014]

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Solution

Q.
A body of mass (4m) is lying in the $x - y$ plane at rest. It suddenly explodes into three pieces. Two pieces, each of mass (m), move perpendicular to each other with equal speeds (v). The total kinetic energy generated due to explosion is [2014]

1 $m v^{2}$

2 $\frac{3}{2} m v^{2}$

3 $2 m v^{2}$

4 $4 m v^{2}$

Ans.
(2)

Let $\vec{v'}$ be the velocity of third piece of mass $2 m .$

Initial momentum, $\vec{p_{i}} = 0$ (As the body is at rest)

Final momentum, $\vec{p_{f}} = m v \hat{i} + m v \hat{j} + 2 m \vec{v'}$

According to law of conservation of momentum

$\vec{p_{i}} = \vec{p_{f}} or, 0 = m v \hat{i} + m v \hat{j} + 2 m \vec{v'}$ $or, \vec{v'} = - \frac{v}{2} \hat{i} - \frac{v}{2} \hat{j}$

The magnitude of $v'$ is

$v' = \sqrt{{(- \frac{v}{2})}^{2} + {(- \frac{v}{2})}^{2}} = \frac{v}{\sqrt{2}}$

Total kinetic energy generated due to explosion

$= \frac{1}{2} m v^{2} + \frac{1}{2} m v^{2} + \frac{1}{2} (2 m) v'^{2}$

$= \frac{1}{2} m v^{2} + \frac{1}{2} m v^{2} + \frac{1}{2} (2 m) {(\frac{v}{\sqrt{2}})}^{2}$ $= m v^{2} + \frac{m v^{2}}{2} = \frac{3}{2} m v^{2}$

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