A Car Talk Puzzler that Thinking Physics people can solve

Car Talk Puzzler for the week of 11/23/13: Two trains are on a collision course. The trains are heading toward each other with identical speeds. When they are a certain distance apart from each other, a bee flies from the front of one train toward the other at constant speed. When it reaches the front of the other train, the bee turns around and flies back to the first train. The process repeats until the trains collide and the bee is squished.

The question: How far does the bee fly in the process?

To answer, you'll need the particulars: the distance between the trains when the bee begins its journey, the speed of the trains, and the speed of the bee. For these particulars (and the means by which you can enter your correct answer for the chance to win a prize), proceed to

The Car Talk Puzzler.

(Lewis Carroll Epstein's Thinking Physics)