\[\begin {array} {1 1} (a)\;v_o \sqrt {\frac{m_1m_2}{v(m_1+m_2)}} & \quad (b)\;v_o \sqrt {\frac{2m_1m_2}{(m_1+m_2)R}} \\ (c)\;v_o \sqrt {\frac{m_1m_2}{2(m_1+m_2)k}} & \quad (d)\;v_o \sqrt {\frac{m_1m_2}{(m_1+m_2)k}} \end {array}\]

$V_{c}=\large\frac{m_2V_0}{(m_1+m_2)}$

At maximum extension, velocity of both $m_1$ and $m_2$ are equal and equal to velocity of Centre of mass .

By energy conservation

$\large\frac{1}{2}$$m_2 V_0^2=\large\frac{1}{2} $$(m_1+m_2)V_{c}^2+\large\frac{1}{2}$$ kx^2$

$\large\frac{1}{2}$$m_2 V_0^2=\large\frac{1}{2} \frac{(m_1+m_2)m_2^2V_0^2}{(m_1+m_2)^2}$

$\qquad+ \large\frac{1}{2}$$kx^2$

$\large\frac{m_1m_2V_0^2}{(m_1+m_2)}$$=kx^2$

$x =v_{0}\sqrt {\large\frac{m_1m_2}{k(m_1+m_2)}}$

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