(a) Show that a projectile with mass m can escape from the

Chapter 18, Problem 18.73

(choose chapter or problem)

(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 \(mgR_p\), where g is the acceleration due to gravity at the planet’s surface and \(R_p\) is the planet’s radius. Ignore air resistance. (See Problem 18.72.) (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 (molar mass 2.02 g/mol?) (c) Repeat part (b) for the moon, for which \(g = 1.63 \ \mathrm{m/s}^2\) and \(R_p = 1740 \ \mathrm{km}\). (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.