Two bodies of masses m1  and  m2 are initially at rest at infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance r between them is

Let velocities of these masses at r distance from each other be v1  and v2  respectively.By conservation of momentumm1v1−m2v2=0⇒m1v1=m2v2   ........(i)By conservation of energy change in P.E.=change in K.E.Gm1m2r=12m1v12+12m2v22⇒m12v12m1+m22v22m2=2Gm1m2r    .......(ii)On solving equation (i) and (ii)v1=2Gm22rm1+m2 and v2=2Gm12rm1+m2∴vapp =v1+v2=2Grm1+m2