The masses and radii of the Earth and the Moon are `M_1, R_1 and M_2,R_2` respectively. Their centres are at a distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway between the two centres so as to escape to infinity is ........

Options

A. `sqrt((2G(M_(1)+M_(2)))/d)`
B. `sqrt((4G(M_(1)+M_(2)))/d)`
C. `sqrt((4GM_(1)M_(2))/d)`
D. `sqrt((G(M_(1)+M_(2)))/d)`

Answer

B

Explanation

Potential energy of mass `m` when it is midway between masses `M_(1)` and `M_(2) ` is
`U=-(GM_(1)m)/(d//2)-(GM_(2)m)/(d//2)=-(2Gm)/d(M_(1)+M_(2))`
According to law of conservation of energy
`1/2mv_(e)^(2)=(2Gm)/d(M_(1)+M_(2))`
Therefore, escape velocity
`v_(e)=sqrt((4G(M_(1)+M_(2)))/d)`