Chapter 9, Problem 48E

A particle subject to a planet’s gravitational pull has a total mechanical energy given by $E_{\text{mechanical}}=\frac{1}{2}m{v}^2-GMm/r$ , where m is the particle’s mass, M the planet’s mass, and G the gravitational constant $6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^{\text{2}}{\text{/kg}}^{\text{2}}$ . It may escape if its energy is zero—that is if its positive KE is equal in magnitude to the negative PE holding it to the surface. Suppose the particle is a gas molecule in an atmosphere. (a) Temperatures in Earth’s atmosphere may reach 1000 K. Referring to the values obtained in Exercise 45 and given that $R_{\text{Earth}}=6.37\times {10}^6\text{m}$ and $M_{\text{Earth}}=5.98\times {10}^{24}\text{kg}$ , should Earth be able to “hold on” to hydrogen (1 g/mol)? to nitrogen (28 g/mol)? (Note: An upper limit on the number of molecules in Earth’s atmosphere is about ${10}^{48}$ .) (b) The moon’s mass is 0.0123 times Earth’s, its radius 0.26 times Earth’s, and its surface temperatures rise to 370 K. Should it be able to hold on to these gases?