A satellite in low earth orbit travels to at 8 km /s. Taking the mass of Mars to be 0.1 earth's mass,and the radius of Mars to be 0.5 earth's radius, how fast would a satellite in a low Mars orbit travel?
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Answer:
Warm-up exercise: deriving acceleration in circular motion from Pythagoras’ theorem.
Imagine a cannon on a high mountain shoots a cannonball horizontally above the atmosphere at the right speed for it to go in a circular orbit. In one second, the ball will fall 5 meters below a horizontal line, at the same time traveling v meters horizontally, as in the diagram below (where the distances traveled are grossly exaggerated to make clear what’s going on).
Apply Pythagoras’ theorem to the right-angled triangle to establish that the appropriate speed for a circular orbit just above the earth’s atmosphere is given by
(Use the approximation that the distance traveled in one second is tiny compared to the radius of the earth.)
2. Properties of the Ellipse.
Take a point on the ellipse very close to P, and draw lines from the new point to the two foci.
Use the fact that the “rope” is the same length for P, to prove that a light ray from one focus to P will be reflected to the other focus.
3. Kepler’s Third Law states that has the same numerical value for all the sun’s planets.
For circular orbits, how are R, T related if the gravitational force is proportional to 1/R? to 1/R3? To R? What can we conclude from Kepler’s Third Law about the gravitational force?
4. 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.
5. An evil genius puts a spherical rock (made of ordinary stone) in the earth’s orbit, but moving around the sun the other way. It collides with the earth, landing in the desert. It is estimated that the crater is about the same as would have been caused by a one-megaton hydrogen bomb. How big was the rock?
6. Halley’s comet follows an elliptical orbit, its closest approach to the Sun is observed to be 0.587AU. Given that the orbital period is 76 years, what is its furthest distance from the Sun? What is the ellipticity of this orbit?
7. Halley’s Comet simplified.
(a) A comet having a period of 64 years has closest approach to the Sun 0.5 AU. Use Kepler’s Third Law, and comparison with the Earth, to figure out its farthest distance from the Sun.
(b) What is the ratio of its kinetic energy when nearest to the Sun to its kinetic energy at the farthest point?
(c) How does its kinetic energy at the closest approach to the Sun compare with that of an equal mass in a circular orbit around the Sun at that distance? (An approximate answer will do.)
Explanation:
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