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Topic: Duck in Pond II (Read 3431 times) |
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James Fingas
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Duck in Pond II
« on: Feb 25th, 2003, 7:42am » |
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A hungry duck is swimming in a circular pond, with a hungry cat prowling the banks. Since the last time this happened, the cat has been working out, and can now move 4.5 times as fast as the duck.
The duck wishes to eat some of the grass growing on the bank of the pond, but fears the cat. Is it possible for the duck to swim over to the bank and grab a mouthful of grass before the cat gets there? If so, how? How fast does the cat need to move to catch the duck?
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SWF
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Re: Duck in Pond II
« Reply #1 on: Feb 27th, 2003, 5:15pm » |
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The maximum cat speed I can find for the duck to still reach the edge is 4.60333885 times the duck's speed.
If the pond has radius R, cat has velocity V, and duck has velocity 1, the duck can move a distance R/V from the pond's center while keeping the center in between itself and the cat. Starting from there, the duck should start swimming directly away from the cat until the cat commits to going either clockwise or counterclockwise around the pond. The cat should decide immediately or it will lose time. If the cat goes clockwise, the duck should also move clockwise and head toward the edge of the pond in a direction perpendicular to the line through the duck's starting position and the pond center. The duck should continue in a straight line to the pond's edge. If the cat reverses direction, the duck needs to compensate, but the cat will have lost some time.
Similar to the way geese naturally fly in a V pattern, I think this swimming path is an instinct for ducks. 
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James Fingas
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Re: Duck in Pond II
« Reply #2 on: Mar 7th, 2003, 1:42pm » |
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SWF,
Right. I never realized it was a straight line  so I guess I spent a little longer getting numerical solutions than I had to! My method was complicated, involving a mixed partial derivative (first time I ever used one in real life!)
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