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The masses and radii of the Earth and th...

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 ........

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Potential energy of m 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))=(-2Gm)/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=2Gm`(M_1+M_2)//d`
As this, work is provided by initial kinetic energy.
`1/2m upsilon^2 = (2Gm(M_1+M_2))/d` or `upsilon=2sqrt((G(M_1+M_2))/d)`
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