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Question

Two masses M1 and M2 at an infinite distance from each other and initially at rest, start interacting gravitationally. Find their velocity of approach when they are distances apart.

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Solution

Since they move under mutual attraction and no external force acts on them, their momentum and energy are conserved. Therefore,
0=12M1v21+12M2v22GM1M2s
It is zero because in the beginning, both kinetic energy and potential energy are zero.
0=M1v1+M2v2
Solving the equations,
v21=2GM22s(M1+M2)
and v22=2GM21s(M1+M2)
V(velocity of approach)=v1(v2)=v1+v2

=2G(M1+M2)s

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