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A body of mass $M$ is moving on a smooth surface with speed $V_{0}$ . Another body of same mass $M$ attached to a spring is kept on the same surface as shown. The moving mass makes an impact with the spring of spring constant $K$ and starts compressing it. The maximum compression in the spring is $V_{0} \sqrt{\frac{M}{P K}}$ , then $P$ value is----

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answer is 2.

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Detailed Solution

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At the moment of maximum compression both blocks will have same speed.
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By conservation linear momentum,
$2 M v = M V_{0} \Rightarrow V = \frac{v_{0}}{2}$

By conservation of mechanical energy
$\begin{array}{l} \frac{1}{2} M v_{0}^{2} + 0 = 2 \frac{1}{2} M (\frac{v_{0}^{2}}{4}) + \frac{1}{2} k x^{2} \\ \Rightarrow \frac{1}{4} M v_{0}^{2} = \frac{1}{2} k x^{2} \\ \Rightarrow x = v_{0} \sqrt{\frac{M}{2 k}} \end{array}$

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