b
(b) Let velocities of these masses at \(r\) distance from each other be \({v_1}\) and \({v_2}\) respectively. By conservation of momentum \({m_1}{v_1} - {m_2}{v_2} = 0\)

\( \Rightarrow \,{m_1}{v_1} = {m_2}{v_2}\)… (i)

By conservation of energy change in \(P.E.\) = change in \(K.E.\)

\(\frac{{G{m_1}{m_2}}}{r} = \frac{1}{2}{m_1}v_1^2 + \frac{1}{2}{m_2}v_2^2\)

\( \Rightarrow \,\frac{{m_1^2v_1^2}}{{{m_1}}} + \frac{{m_2^2v_2^2}}{{{m_2}}} = \frac{{2\,G{m_1}{m_2}}}{r}\) …(ii)

On solving equation (i) and (ii) \({v_1} = \sqrt {\frac{{2\,Gm_2^2}}{{r({m_1} + {m_2})}}} \) and \({v_2} = \sqrt {\frac{{2\,Gm_1^2}}{{r({m_1} + {m_2})}}} \)

\(\therefore \,{v_{{\rm{app}}}} = \,|\,{v_1}\,|\, + \,|\,\,{v_2}\,|\, = \,\sqrt {\frac{{2G}}{r}({m_1} + {m_2})} \)