Recommended Reading
A question from a Cambridge First Year Applied maths paper from the 60s.
A malignant deity stops the moon dead in its tracks and the moon falls under the Earth's gravity. How long will the moon take to hit the earth? (You may assume that the orbit of the moon is circular with radius R=240,000 miles, and you may take the earth as a point.)
Ans. One can do a tedious integral.
Alternatively, one can use the theory of orbits. The moon takes 1 month to orbit the earth; 2 bodies orbiting the earth with equal major axes have the same orbital period (Kepler's 3rd Law: see http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion). The 'new' orbit of the moon is just a diameter of the original circular orbit (if we ignore collision with the earth), which can be considered a very flat ellipse with the same major axis 2R. It follows that the orbital period is unchanged so it will take ¼ of a month to hit the earth (around 7 days).
(The question and answer are from memory; the answer was provided by Prof R Lyttleton: http://en.wikipedia.org/wiki/Raymond_Lyttleton.)
