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

Graviatational potential energy of the particle of mass `m` at a distance `r//2` from the centre of the
earth `= (-GM_(1) m)/((r//2)) = (-GM_(1)m)/(r )`
Gravitational potential energy of the potential mass `m` at a distance `r//2` from the centre of moon
`= (-GM_(2) m)/(r//2) = (-GM_(2)m)/(r )`
Total potential energy of the particle,
`U = (-2 GM_(1)m)/(r ) - (2GM_(2)m)/(r ) = (-2GM)/(r ) (M_(1) + M_(2))`
Since `PE` of particle at infinity is zero, so work required to shift the mass from given position to
infinity is, `W = 0 - U = (2GM)/(r )(M_(1) + M_(2))`
As this workdone is produced by initial `KE`, having escape velocity `upsilon`, so
`(1)/(2) m upsilon^(2) - (2Gm)/(r) (M_(1) + M_(2)) = 0`
or `upsilon = sqrt((4G (M_(1) +M_(2)))/(r))`