Question

In the example 12.5 (Section 12.3) in the textbook we ignored thegravitational effects of the moon on a spacecraft en route from theearth to the moon. In fact, we must include the gravitationalpotential energy due to the moon as well. For this problem, you canignore the motion of the earth and moon.

Part A
If the moon of mass m_{\rm M} has radius R_{\rm M} and the distancebetween the centers of the earth and the moon is R_{\rm EM}, findthe total gravitational potential energy of the particle-earth andparticle-moon systems when a particle with mass m is between theearth and the moon, and a distance r from the center of the earth.Take the gravitational potential energy to be zero when the objectsare far from each other. Take the mass of earth as m_{\rm E}.

Part B
There is a point along a line between the earth and the moon wherethe net gravitational force is zero. Use the expression derived inpart (a) to find the distance of this point from the center of theearth.

Part C
With what speed must a spacecraft be launched from the surface ofthe earth just barely to reach this point?

Part D
If a spacecraft were launched from the earth's surface toward themoon with an initial speed of 11.2 {\rm km/s}, with what speedwould it impact the moon?