$($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 mgRp than where g is the acceleration due to gravity at the planet$$s surface and Rp is the planet$$s radius. Ignore air resistance. $($b$)$ If the planet in question is the earth, at what temperature does the average translational kinetic energy of a nitrogen molecule $($molar mass $28\mathrm{.}0$ g$/$mol$.)$ equal that required to escape? What about a hydrogen molecule $($molar mass $2\mathrm{.}02$ g$/$mol$.)?($c$)$ Repeat part $($b$)$ for the moon, for which g $=1\mathrm{.}63$ m$/$s$2$ and Rp $=1740$ 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