In Example 13.5 (Section 13.3) we ignored the gravitational effects of the moon on a spacecraft en route from the earth to the moon. In fact, we must include the gravitational potential energy due to the moon as well. For this problem, you can ignore the motion of the earth and moon.

(a) If the moon has radius R_M and the distance between the centers of the earth and the moon is R_EM, find the total gravitational potential energy of the particle–earth and particle–moon systems when a particle with mass m is between the earth and the moon, and a distance r from the center of the earth. Take the gravitational potential energy to be zero when the objects are far from each other. 

(b) There is a point along a line between the earth and the moon where the net gravitational force is zero. Use the expression derived in part (a) and numerical values from Appendix F to find the distance of this point from the center of the earth. With what speed must a spacecraft be launched from the surface of the earth just barely to reach this point?

(c) If a spacecraft were launched from the earth’s surface toward the moon with an initial speed 11.2 km/s, of with what speed would it impact the moon?

In Example 13.5:

In Jules Verne’s 1865 story with this title, three men went to the moon in a shell fired from a giant cannon sunk in the earth in Florida. (a) Find the minimum muzzle speed needed to shoot a shell straight up to a height above the earth equal to the earth’s radius R_E. (b) Find the minimum muzzle speed that would allow a shell to escape from the earth completely (the escape speed). Neglect air resistance, the earth’s rotation, and the gravitational pull of the moon. The earth’s radius and mass are R_E = 6.38 × 10⁶ m and m_E = 5.97 × 10²⁴ kg.