$ a)\bigg(\sqrt {\large\frac{3k}{2m}}\bigg)\\ b)\bigg(\sqrt {\large\frac{2k}{3m}}\bigg)\\ c)\sqrt {\large\frac{2km}{m}}\\ d)\sqrt {\large\frac{3km}{2x}} $

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Let $v_r$ be the relative velocity pf the two.

Let the reduced mass of the 2 block system be $\mu$

$\mu=\large\frac{(m)(2m)}{m+2m}=\frac{2}{3}$$m$

From conservation of mechnical energy we have

$\large\frac{1}{2}$$kx^2=\large\frac{1}{2}$$\mu v_r^2$

$kx^2=\mu v_r^2$

$\qquad= \large\frac{2}{3} $$mv_r^2$

$v_r= \bigg(\sqrt {\large\frac{3k}{2m}}\bigg)$$x$

Hence a is the correct answer.

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