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A block of mass m moving with a velocity v₀ collides with a stationary block of mass M to which a spring of stiffness k is attached, as shown in figure. Choose the correct alternative(s).

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The velocity of the centre of mass is v₀.

When the spring is in the state of maximum compression, the kinetic energy in the centre of mass frame is zero.

The maximum compression in the spring is $v_{0} \sqrt{(\frac{mM}{m + M} \frac{1}{k})}$ .

The initial kinetic energy of the system in the centre of mass frame is $\frac{1}{4} (\frac{mM}{M + m}) v_{0}^{2}$ .

answer is C, D.

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Detailed Solution

Using conservation of linear momentum,

$\begin{array}{l} {mv}_{0} = (M + m) v \\ v = \frac{{mv}_{0}}{M + m} \end{array}$

KE in centre of mass frame:

${KE}_{c} = \frac{1}{2} m {(v_{0} - v)}^{2} + \frac{1}{2} {Mv}^{2} = \frac{1}{2} (\frac{Mm}{M + m}) v_{0}^{2}$

From energy conservation,

$\begin{array}{l} \frac{1}{2} {mv}_{0}^{2} - \frac{1}{2} (m + M) v^{2} = \frac{1}{2} {kx}^{2} \\ \Rightarrow x = v_{0} \sqrt{\frac{mM}{k (M + m)}} \end{array}$

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