The masses and radii of the earth and moon are `M_(1),R_(1)" and"" " M_(2)R_(2)` respectively. Their centres are at
distance d apart. What is 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?

Explanation

Potential energy ofm when it is midway between `M_(1)` and `M_(2)`
U=m `(V_(1)+V_(2))=m` `[-("GM"_(1))/(d//2)+(-"GM"_(2))/(d//2)]=(-2"Gm")/(d)[M_(1)+M_(2)]`
And as potential energy at infinity is zero, so work required to shift m from the given position to infinity,
`W=0-U=2"Gm"(M_(1)+M_(2))//d`
As this work is provided by initial kinetic energy,
`(1)/(2)"mv"^(2)=(2"Gm"(M_(1)+M_(2)))/(d)` or `V=2 sqrt((G(M_(1)+M_(2)))/(d))`