The Lever of Mahomet

Imagine a game: it uses a cart that can move forward and backward along a straight track. The cart can move with any finite speed and acceleration. A straight, uniform rod is attached to the bed of the cart with a hinge. We'll assume the hinge is frictionless, there is no air resistance, etc.

Player 1 assigns the cart a motion that gets it from Point A to Point B. It may start and stop several times, it may reverse direction, but it has to eventually get to Point B. Player 2 is given the motion Player 1 came up with and has to try to find an initial position of the rod such that the rod will never quite fall all the way down. He gets as many tries and as much time as he wants to try and accomplish it. 

The question: Is there any motion that Player 1 can choose that Player 2 can not eventually beat.

It turns out the answer is no. At least as a thought experiment, there is no dance Player 1 can come up with that can not be beat by Player 2 choosing just the right starting position. Think of the extremes. Given the motion, Player 2 knows that if he starts the rod far enough over to the back it will end up all the way down in back. He also knows that if he starts the rod far enough forward it end up all the way down in front. Well, everything we're talking about is continuous, so as he gradually changes the rod angle from back to front there must be some small range of angles where the result transitions from ending up down in back to ending up down in front. Any one of those transition points is a solution where it doesn't end up down at all. 

This problem is from an article by Richard Courant and Herbert Robbins called "The Lever of Mahomet" and can be found in The World of Mathematics.