Planets A and B have masses M sub A and M sub B , respectively, where M sub A is greater than M sub B , and radii uppercase R sub A and uppercase R sub B , respectively. The centers of the planets are separated by a distance D. A spaceship of mass m uses its engines to fly through the two-planet system a distance r from the center of Planet A. Point P is between the two planets, also being located a distance r from planet A. Express your answer in parts a – d in terms of M sub A, M sub B, m, D, r, and physical constants, as appropriate. A. Derive an expression for the sum of the gravitational potential energy of the spaceship-planet A and spaceship-planet B systems when the spaceship reaches point P.
B. Derive an expression for the magnitude of the net gravitational force on the spaceship when the spaceship reaches point P.
C. Determine an expression for the work done by gravity as the spaceship moves from a very large distance away to point P.
D. Derive an expression for the escape speed of the spaceship when it is at point P. Express your answer in terms of M sub A, M sub B, m, D, r, and physical constant, as appropriate.