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

The masses and radii of the earth and moon M_(1), R_(1) and M_(2), R_(

1.	The masses and radii of the earth and moon 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 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 kientic energy. `(1)/(2)MV^(2) = (2Gm(M_(1)+M_(2)))/(d)` or `v = 2sqrt((G(M_(1)+M_(2)))/(d))`