d
Initial momentum of the system block \((C)\)

\( = m{v_0}\) .After striking with \(A\), the block \(C\) comes to rest and now both block \(A\) and \(B\) moves with velocity \(v\) when compression in spring is \({x_0}.\)

By the law of conseravtion of linear momentum  

\(m{v_0} = \left( {m + 2m} \right)v \Rightarrow \frac{{{v_0}}}{3}\)

By the law of conservation of energy 

\(K.E. \,of\, block \,C = K.E.\ of\, system + P.E. of system\)

\(\begin{array}{l}
\frac{1}{2}mv_0^2 = \frac{1}{2}\left( {3m} \right){\left( {\frac{{{v_0}}}{3}} \right)^2} + \frac{1}{2}kx_0^2\\
 \Rightarrow \,\frac{1}{2}mv_0^2 = \frac{1}{6}mv_0^2 + \frac{1}{2}kx_0^2\\
 \Rightarrow \,\frac{1}{2}kx_0^2 = \frac{1}{2}mv_0^2 - \frac{1}{6}mv_0^2 = \frac{{mv_0^2}}{3}\\
\therefore \,\,\,\,\,\,\,k = \frac{2}{3}m{\left( {\frac{{{v_0}}}{{{x_0}}}} \right)^2}
\end{array}\)