1. Volcanos on Io are observed to reach as high as 400 km from the surface of Io. Let's see how fast the material must be shooting out of the surface in order to attain that height. The straightforward way is to use mgh = 1/2 mv², but this is not quite correct. (a) What is the acceleration of gravity, g, at the surface of Io? (b) Use it with the above relation to calculate the speed v. Express your speed in both m/s and mph. (c) The reason this is not correct is that the acceleration of gravity decreases as the material goes up, and since 400 km is an appreciable height compared to the radius of Io, this effect can be important. Using instead the relation

		R+h
m		g dr		= 1/2 mv2
		R

show that the correct expression for the velocity is v = sqrt{2GM[1/R - 1/(R+h)]}. (d) Use this expression to calculate the speed v, and compare with your answer to (b). (e) Expand the second term using the binomial expansion (Lecture 12) and show that it to first order it is the same as in part (b). Keep the second order term and recalculate to compare with your answer to (d).

2. The rings of Saturn are made of particles orbiting the planet according to Kepler's third law (eq. 2.37 of the text). Calculate the distance for which the particles orbit synchronously with Saturn's rotation (period 10h 45m). Comparing to the distances of the rings, describe where in the rings this distance would occur. If you could stand on Saturn and view the rings from its cloudtops, the rings closer than this distance would move across the sky from west to east, while the rings farther away would move across the sky from east to west.