A Show A Projectile With Mass M Can Escape

(a) Show that a projectile with mass m can “escape” from the surface of a planet if it is launched vertically upward with a kinetic energy greater than mgRp, where g is the acceleration due to gravity at the planet’s surface and Rp is the planet’s radius. Ignore air resistance. (See Problem 18.76.)
(b) If the planet in question is the earth, at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass 28.0 g/mol) equal that required to escape? What about a hydrogen molecule (molarmass2.02gfmol)?
(c) Repeat part (b) for the moon, for which g = 1.63 m/s2 and R0 = 1740 km.
(d) While the earth and the moon have similar average surface temperatures, the moon has essentially no atmosphere. Use your results from parts (b) and (c) to explain why.

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