\( = \frac{1}{2}\left[ {\frac{1}{2}\left( {\frac{M}{2}} \right)\,{v^2}} \right]\)

\(\left[ {{\rm{As}}\,m = \frac{M}{2}{\rm{ given}}} \right]\)==>\({V^2} = \frac{{{v^2}}}{4}\)

\( \Rightarrow \,V = \frac{v}{2}\).....(i) \(\frac{1}{2}M{(V + 1)^2} = \frac{1}{2}m\,{v^2} = \frac{1}{2}\left( {\frac{M}{2}} \right)\,{v^2}\)

==> \({(V + 1)^2} = \frac{{{v^2}}}{2}\)

==> \(V + 1 = \frac{v}{{\sqrt 2 }}\).....(ii)

\(V = \frac{1}{{\sqrt 2 - 1}}m/s\).