Archive for Kepler’s Laws

The Problem of the Disintegrating Asteroid

Posted in Cute Problems, The Universe and Stuff with tags , , on September 30, 2019 by telescoper

I thought you might enjoy this entry in the Cute Problems folder.

An asteroid is moving on a circular orbit around the Sun with an orbital radius of 3AU when it spontaneously splits into two fragments, which initially move apart along the direction of the original orbit. One fragment has a speed which is a fraction 0.65 of the original speed, the other has a speed of 1.35 times the original speed. The original orbit (solid line) is shown above, along with the two new orbits (dashed and dotted).

  1. Which orbit does the fast fragment follow, and which the slow fragment?
  2.  Calculate the original orbital speed in AU/year.
  3. Calculate the angular momentum per unit mass, h, of the original asteroid and of each of the two fragments in units of AU2 per year. [HINT: Show that in these units, for a general orbit of eccentricity e and semi-major axis a, h2=4π2 a (1-e2).]
  4.  Calculate the eccentricities of the orbits of the two fragments.
  5.  Calculate the orbital periods of the two fragments in years.

Answers please through the Comments box. First complete set of answers wins a trip to the Moon on gossamer wings.

 

 

Transfer Orbit

Posted in Cute Problems, The Universe and Stuff with tags , on November 2, 2011 by telescoper

From time to time I like to post nice physics problems on here. Here is a quickie that I used to use in my first-year Astrophysical Concepts course which has now been discontinued, so I don’t need to keep it to myself it any longer.

A simple way to travel from one planet in the solar system to another is to inject a spacecraft into an elliptical transfer orbit, like the one shown by the dashed curve, which is described by Kepler’s Laws in the same way that the planetary orbits (solid curves) are.

Kepler’s Third Law states that the  period of an elliptical orbit is given by P^2 \propto a^3 where a is the semi-major axis of the ellipse. Assuming that the orbits of Earth and Mars are both approximately circular and the radius of Mars’ orbit is 50% larger than Earth’s, and without looking up any further data, calculate the time taken to travel in this way from Earth to Mars.