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Gravitaional potential energy

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 momentum
m1v1m2v2=0m1v1=m2v2   ...(i)
By conservation of energy 
change in P.E.=change in K.E.
Gm1m2r=12m1v12+12m2v22m12v12m1+m22v22m2=2Gm1m2r   ...(ii)
On solving equation (i) and (ii)
v1=2Gm22r(m1+m2)  and  v2=2Gm12r(m1+m2)
   vapp=|  v1|+  |  v2|=2Gr(m1+m2)

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