a
(a)Initial momentum of the system (block \(C\)) \(= mv\)
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 maximum.
By the law of conservation of linear momentum
\(mv = (m + m) V\) ==> \(V = \frac{v}{2}\)
By the law of conservation of energy
\(K.E.\) of block \(C = K.E. \) of system \(+ P.E.\)  of system
\(\frac{1}{2}m{v^2} = \frac{1}{2}(2m)\,{V^2} + \frac{1}{2}k{x^2}\)
==> \(\frac{1}{2}m{v^2} = \frac{1}{2}(2m)\;{\left( {\frac{v}{2}} \right)^2} + \frac{1}{2}k{x^2}\)
==> \(k{x^2} = \frac{1}{2}m{v^2}\)
==> \(x = v\sqrt {\frac{m}{{2k}}} \)