b
Mass of pieces by \(\frac{ M }{4}, \frac{ M }{4}, \frac{ M }{2}\)

conserving momentum

\(\overrightarrow{ P }_{1}+\overrightarrow{ P }_{2}+\overrightarrow{ P }_{3}=0\)

\(\overline{ P }_{3}=-\left(\overline{ P }_{1}+\overline{ P }_{2}\right)\)

As \(\overline{ P }_{1} \,and \,\overline{ P }_{2}\) are perpendicular

so \(P _{3}=\sqrt{ P _{1}^{2}+ P _{2}^{2}}\)

\(P _{3}=(50) \frac{ M }{4}\)

\( and \,P _{3}=\frac{ M }{2} v\)

so \(v =25\)