The masses and radii of the earth and moon are `1M_(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 of 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 G m (M_(1) + M_(2)) // d`
`(1)/(2) mv^(2) = (2 G m(M_(1) + M_(2)))/(d) or v = 2 sqrt((G (M_(1)+ M_(2))/(d))`