Kepler's Laws of Planetary Motion

Johannes Kepler used the data produced by Tycho Brahe's observations of the motion of the planets to come up with three laws of planetary motion. The first law states that all planets move in elliptical orbits around the Sun. The second law states that a line joining the Sun to a planet sweeps out equal areas in equal periods of time. Finally, his third law states that the period of revolution for a planet squared is proportional to the semi major axis of orbit cubed. Please watch the following presentation that will introduce Kepler's Laws.

Kepler's Laws of Planetary Motion Practice

Kepler's three laws can be used to predict the motion of planets in orbit around the Sun or stars. Please watch the following presentation of some example problems.

Kepler's Laws of Planetary Motion Self-Assessment

Now it's your turn. Complete the following Kepler's Laws problems to verify that you understand how to apply Kepler's Laws to the motion of bodies that orbit around planets or stars. You will notice that there is a lot of overlap with the Universal Gravitation problems.

A satellite placed in orbit had an apogee of 2.29x10⁶m above the Earth's surface and a perigee of 4.59x10⁵m above the Earth's surface. The period of orbit is 6760s. Find the ratio $LaTeX: \frac{v_p}{v_a}$ of the speed of the satellite.
In a series of science fiction books "Ringworld" is an artificial living space that encircles an entire star. The people of the world live on the inside surface where the sun is shining all day along. If Ringworld were revolving around our Sun fast enough to produce Earth normal gravity and the radius of orbit was the same as the Earth,
1. What centripetal acceleration would be produced by the surface of Ringworld pushing up on a person?
2. What would have to be the velocity of the construct as it orbited?
3. What would be the period of revolution?
4. If the radius of orbit were increased to 3r, what would be the period of revolution?
The orbital period of one of Saturn's moons is 81400s and the Orbital Radius is 1.85x10⁸m. The mass of Saturn is M_s.
1. Write an algebraic expression for the gravitational force between Saturn and this moon using standard and given constants.
2. Assuming that the orbit of the moon is circular, derive an equation for the orbital period T of the moon as a function of the Orbital Radius using only given or standard constants.
3. Assuming that you knew the period and orbital radius of other moons of Saturn, what quantities would be graphed to yield a straight line?

SOLUTIONS 1, 2, & 3

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