a
Linear momentum is conserved

\(2 m v-2 m v=(2 m+m+M) V_{c m}\)

\(\Rightarrow V_{c m}=0\)

Angular momentum is conserve

\(2 m v \frac{L}{6}+2 m v \times \frac{L}{3}+0=I w\)         \(...(i)\)

\(I=\mathrm{M.I.}\) of rod \(+\mathrm{M.I.}\) of mass \(2 \mathrm{m}+\mathrm{M.I}\) of mass \(\mathrm{m}\)

\(=\frac{8 m L^{2}}{12}+2 m\left(\frac{L}{6}\right)^{2}+m\left(\frac{L}{3}\right)^{2}\)

\(=m L^{2}\left(\frac{2}{3}+\frac{1}{18}+\frac{1}{9}\right)=\frac{5}{6} m L^{2}\)

eqn \((1)\) becomes

\(\left(\frac{m v L}{3}+\frac{2}{3} m v L\right)=\frac{5}{6} m L^{2} w\)

\(\frac{m L}{3}(V+2 v)=\frac{5}{6} m L^{2} w\)

\(\frac{6 V}{5 L}=w\)