Physics 152 Homework #1

Due Wednesday January 28

1. Television signals are relayed by synchronous satellites, placed in orbits such that they hover above the same spot on Earth. Use Kepler’s Laws and data about the Moon’s orbit to find how far above the Earth’s surface the synchronous satellites are. Could one be placed directly above Charlottesville? If you say no, explain your reasoning.

2. (a) Activate the Mars trip flashlet. The speed you enter is from a high parking orbit (say at ten Earth radii) so that the Earth’s own gravitational field has negligible effect. Find the minimum speed needed to reach Mars, sketch a picture of this most economical orbit, and estimate how long the trip would take.

(b) Give a qualitative explanation of how you would fire a rocket to get back to Earth from a parking orbit near Mars (so you neglect Mars’ own gravity).

3. (a) Find the orbital speed of a spaceship in low orbit around the Moon, just skimming the mountain tops.

(b) Suppose the pilot suddenly increases the speed by a factor of Ö2, but during the brief acceleration keeps the spaceship pointing the same way, that is, horizontally. Describe the path the spaceship will take after the engine cuts out.

4. In the year 3000, a group of bad guys fond of living in caves have excavated a huge spherical cave inside the Moon. Assuming the Moon is a sphere of rock of uniform density, prove that the gravitational field inside the cave is the same everywhere. (Hint: figure out the field for Moon with no cave, then think of the cave as a uniform sphere of negative mass density, and add the two contributions.)

5. The escape velocity from Earth is 11.2 km./sec. What is the escape velocity from the Solar System starting in a high parking orbit several Earth radii from Earth. (Hint: what is the Earth’s speed in orbit?) On the basis of this, estimate how much more fuel energy is needed to reach the outer planets compared with going to the Moon. Is there a way around this problem?

*6. Somewhere on the line from the Earth to the Sun there is a point, called a Lagrange point, such that a satellite placed there will orbit around the Sun in sync with the Earth. In fact, there’s already a satellite there, it monitors the Sun continuously. Come up with some estimate of how far from Earth this Lagrange point is (the Web might be helpful).