The masses and radii of the earth and moon are M_(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 two centres so as to escape to infinity ?

The masses and radii of the earth and moon are M_(1), R_1 and M_(2)R_2

1.	The masses and radii of the earth and moon are M_(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 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 ot infinity , `W=0-U=2Gm(M_(1)+M_(2))//d` As this , work is provided by initial KINETIC energy. `1/2"m"upsilon^(2)=(2Gm(M_(1)+M_(2)))/(d)" or " upsilon=2sqrt((G(M_1+M_2))/d)`