CP (a) Show that a projectile with mass m can “escape”

QUESTION:

CP (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 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 hydro-gen molecule (molar mass 2.02 g/mol?) (c) Repeat part (b) for the moon, for which g = 1.63 m/s2 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. 18.72 . ?Hydrogen on the Sun. The surface of the sun has a temperature of about 5800 K and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is 1.67 X 10-27 kg.) (b) The escape speed for a particle to leave the gravitational influence of the sun is given by (2GM/R)1/2, where M is the sun’s mass, R its radius, and G the gravitational constant (see Example 13.5 of Section 13.3). Use Appendix F to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain.