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Question

Two blocks $m_{1}$ and $m_{2}$ are placed on a smooth horizontal surface; and are joined together with a spring of stiffness $k$ as shown in the figure. Suddenly, the block $m_{2}$ receives a horizontal velocity $v_{0}$

A

v_{0} \sqrt{\frac{m_{1} m_{2}}{m_{1} + m_{2}}}

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B

v_{0} \sqrt{\frac{2 m_{1} m_{2}}{(m_{1} + m_{2}) k}}

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C

v_{0} \sqrt{\frac{m_{1} m_{2}}{2 (m_{1} + m_{2}) k}}

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D

v_{0} \sqrt{\frac{m_{1} m_{2}}{(m_{1} + m_{2}) k}}

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Solution

The correct option is D $v_{0} \sqrt{\frac{m_{1} m_{2}}{(m_{1} + m_{2}) k}}$
Let $x$ = maximum elongation of sprilng
Change of kinetic energy=Potential stored in spring
$\begin{matrix} (\frac{1}{2}) m_{2} v_{0}^{2} - (\frac{1}{2}) (m_{1} + m_{2}) {[\frac{m_{2} v_{0}}{(m_{1} + m_{2})}]}^{2} = (\frac{1}{2}) K x^{2} m_{2} v_{0}^{2} [1 - {\frac{m_{2}}{(m_{1} + m_{2})}}] = K x^{2} x = v_{0} \sqrt{\frac{m_{1} m_{2}}{(m_{1} + m_{2}) k}} \end{matrix}$
Hence,
Option $A$ is correct answer

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