Planets A and B have masses MA and MB , respectively, where MA > MB , and radii RA and RB , respectively. The centers of the planets are separated by a distance D (as shown in image 1). A spaceship of mass m uses its engines to fly through the two-planet system a distance r from the center of Planet A along the dashed line shown. Point P is on the dotted line and on the line connecting the centers of the two planets. Express your answer in parts (a) – (c) in terms of MA , MB , 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) On the axis below (shown in image 2), sketch a graph of the gravitational potential energy Ug of the two planet–spaceship system as a function of position y along the dashed line shown in the original figure. On the horizontal axis, point y < P is above point P in the figure. (e) Derive an expression for the escape speed of the spaceship when it is at point P. Express your answer in terms of MA , MB , m, D, r, and physical constant, as appropriat
Planets A and B have masses MA and MB , respectively, where MA > MB , and radii RA and RB , respectively. The centers of the planets are separated by a distance D (as shown in image 1). A spaceship of mass m uses its engines to fly through the two-planet system a distance r from the center of Planet A along the dashed line shown. Point P is on the dotted line and on the line connecting the centers of the two planets. Express your answer in parts (a) – (c) in terms of MA , MB , 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) On the axis below (shown in image 2), sketch a graph of the gravitational potential energy Ug of the two planet–spaceship system as a function of position y along the dashed line shown in the original figure. On the horizontal axis, point y < P is above point P in the figure.
(e) Derive an expression for the escape speed of the spaceship when it is at point P. Express your answer in terms of MA , MB , m, D, r, and physical constant, as appropriate.
(f) If a similar spaceship that has twice the mass of the original spaceship passes along the same path between the planets, would the escape speed at point P for the new spaceship be greater than, less than, or equal to the escape speed of the original spaceship at point P ? Justify your answer
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