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

Question

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

Infinity Learn · Accepted Answer

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 $$=v1+v2=2Grm1+m2$$

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

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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

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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