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Two particles of masses `m` and `M` are initially at rest at an infinite distance part. They move towards each other and gain speeds due to gravitational attracton. Find their speeds when the separation between the masses becomes equal to `d`.

Answer» Let `v_(1)` and `v_(2)` be the speeds of two masses `m` and `M`, respectively, when the are at a separation d.
As they approach each other, the kinetic energy increases and `GPE` decreases. Hence for the system
Loss is `GPE=` Gain in `KE`
`implies(GPE)_(i)=(GPE)_(f)=KE_(f)-KE_(i)`
`implies0-(-(GMm)/d)=(1/2mv_(1)^(2)+1/2Mv_(2)^(2))-0`
`(GMm)/d=1/0mv_(1)^(2)+1/2MV_(2)^(2)`
As there is no external force on this system its total momentum remains conserved.
`P_(i)=P_(f)`
`0=mv_(1)-MV_(2)`
Combining the two equations, we have
`v_(1)=sqrt((2GM^(2))/(d(m+M)))`


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