Jupiter Propp, the fictitious composite yachtsman, uses eddies and the Gulf Stream to great advantage when racing from Connecticut to Bermuda. To show his technique, he proposes a simple game that can be played by remote-controlled cars and a ground-level rotating disk.
The goal is to go from the northern horizontal bar to the southern one—to be precise, from the point where the northern horizontal bar meets the dotted-line segment ("Connecticut") to the point where the southern horizontal bar meets the dotted-line segment ("Bermuda”") in the least time. The disk in between spins clockwise, so trying to follow the dotted line directly is a losing proposition. (As my nonfictitious climbing friend Kentucky Pete puts it, "The shortest path between two points is a straight line, but the easiest path wanders." This one wanders in a big way.)
In fact, it may be good to start off going slightly toward the southeast, then navigate somehow across the disk, which is spinning slightly east of the dotted line, and then go southwest toward the finish, as shown in the illustration.
http://cs.nyu.edu/cs/faculty/shasha/papers/eddyfig1.doc
b = car speed (100 meters per minute)
d = distance from the start point to the northernmost edge of the disk (25 meters)
e = distance from southern end of the disk to the end point (25 meters)
r = radius of the disk (25 meters)
s = spin speed of disk (30 degrees per minute)
A = the angle away from the straight line through the center of the disk leaving Connecticut
B = the angle away from the straight line through the center of the disk approaching Bermuda.
Notice that if the disk didn't rotate at all, then A and B should be 0. Just take the straight path indicated by the dotted line.
Warm-up 1:
If the disk rotates superfast (say 10,000 revolutions per minute), then what values should A and B take?
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Remarkably, for intermediate rotational speeds of the disk, we want A and B to be positive. But now the question becomes: how should the car travel across the disk to go from the point where the northern segment arrives at the disk to the point where the southern segment leaves the disk?
Warm-up 2:
What do you think?
Problems:
1. Okay, and unusual for this column, I would now like to invite you to calculate. I myself used some trigonometry and a computer language, but the solution will explain how to set up the problem as well as give the answer.
Hint: The main challenge is to figure out how to set it up as though A and B were known. (The H angle in this illustration can then be derived.)
http://cs.nyu.edu/cs/faculty/shasha/papers/eddyfig5car.doc
Then the computer can search different A and B values. If you have an elegant closed solution to the problem, I'm very interested.
2. Now for an advanced problem. For which rotational speeds of the disk will the average of A and B be close to its maximum? What is that maximum?
