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 particel of mass m should be projected from a point midway between the two centres so as to escape to infinity is ........

Explanation

Gravitational potential energy of the particle of mass `m` at a distance `r//2` from the centre of the
Earth `= - (GM_(1) m)/((r//2)) = - (2GM_(1) m)/(r )`
Gravitational potential energy of the particle of mass `m` at a distance `r//2` from the centre of the
Moon `= - (GM_(2) m)/((r//2)) = - (2 Gm_(2) m)/(r )`
Total potential energy of the particle,
`U = - (2 Gm_(1) m)/(r ) - (2 Gm_(2) m)/(r )`
`= - (2Gm)/(r )(M_(1) + M_(2))`
Since, the P.E. at infinity is zero, so work required so shift the mass from the given position
to infinity is, `W = 0 - U = (2Gm)/(r ) (M_(1) + M_(2))`
or `upsilon sqrt(4G (M_(1) + M_(2))//r)`