This was a problem I gave on Exam 2 in 1994

The figure shows the Earth-Moon system (not to scale, of course). Consider the potential energy of a spaceship of mass m as it moves along a line joining the centers of the Earth and Moon. The distance between the centers of the Earth and Moon is D and we will use r to indicate the distance measured from the center of the Earth. The relevant parameters for this problem are given in the table below:

Earth - Moon distance	D = 3.21 x 108 m
Mass of the Earth	ME = 5.98 x 1024 kg
Mass of the Moon	MM = 7.34 x 1022 kg
Radius of the Earth	RE = 6.37 x 106 m
Radius of the Moon	RM = 1.74 x 106 m

(a) Write down the expression for the gravitational potential energy, U(r), of the spaceship.

(b) Sketch U(r) from the surface of the Earth to the surface of the Moon.

(c) For what value of r is U(r) a maximum ? Numerically express r as a fraction of D.

(d) Suppose you were to launch the spaceship from the surface of the Earth such that it would just surmount the maximum potential energy. What would be the minimum launch velocity v_o ?

(e) Compare your answer to part (d) with what you would compute for complete escape velocity, v_esc, from the surface of the Earth. Comment on any differences.

(f) Once the spaceship overcomes the potential energy maximum, it coasts to the surface of the Moon. What is the velocity (give a numerical answer) once it reaches the surface of the Moon ?