A particle subject to a planet's gravitational pull has a total mechanical energy given by Emechanical = ½mv2-GMm/r, where m is the particle's mass, M the planet's mass, and G the gravitational constant 6.67 × 10-11 N • m2/kg2. 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 REarth = 6.37 X 106 m and MEarth = 5.98 × 1024 kg, should Earth be able to "hold on" to hydrogen (1g/mol)? to nitrogen (28 g/mol)? (Note: An upper limit on the number of molecules in Earth's atmosphere is about 1048,) (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?
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