The particle m is at a distance d/2 from the centre of the earth (mass M1) as well as from the centre of the moon (mass M2). Therefore, the total gravitational potential energy of the particle due to the earth and the moon is
−GM1md/2−GM2md/2=−2G(M1+M2)md
Clearly, the work required to move the particle from the point midway between the earth and the moon to infinity would be 2G(M1+M2)md.
Thus, the minimum velocity of projection for the particle m to escape is given by
12mve2=2G(M1+M2)md
or ve=2√G(M1+M2)md
ve=√4G(M1+M2)md