Two masses `m_(1)` and `m_(2)` at an infinite distance from each other are initially at rest, start interacting gravitationally. Find their velocity of approach when they are at a distance `r` apart.

Options

A. `[2G((m_(1)-m_(2)))/r]^(1//2)`
B. `[r/(2G(m_(1)-m_(2)))]^(1//2)`
C. `[(2G)/r(m_(1)+m_(2))]^(1//2)`
D. `[(2G)/rm_(1)m_(2)]^(1//2)`

Answer

C

Explanation

Let velocity of these masses at `r` distance form each other be `v_(1)` and `v_(2)` respectively.
By conservation of momentum
`m_(1)v_(1)-m_(2)v_(2)=0`
`implies m_(1)v_(1)=m_(2)v_(2).......(i)`
By conservation of energy
Change in `PE`=change `KE`
`(Gm_(1)m_(2))/r=1/2m_(1)v_(1)^(2)+1/2m_(2)v_(2)^(2)`
`implies (m_(1)^(2)v_(1)^(2))/(m_(1))+(m_(2)^(2)v_(2)^(2))/(m_(2))=(2Gm_(1)m_(2))/r........(ii)`
On solving Eqs. (i) and (ii)
`v_(1)=sqrt((2Gm_(2)^(2))/(r(m_(1)+m_(2)))` and
`v_(2)=sqrt((2Gm_(1)^(2))/(r(m_(1)+m_(2)))`
`:. v_(app)=|v_(1)|+|v_(2)|=sqrt((2G)/r(m_(1)+m_(2)))`