The masses and radii of the earth and moon are M_1, R_1 and M2, 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?

The masses and radii of the earth and moon are M_1, R_1 and M2, R_2 re

1.	The masses and radii of the earth and moon are M_1, R_1 and M2, 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?
Answer» Solution :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)`