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Topic: Escape (Read 1164 times) |
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R
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A duck is in a circular pond with a menacing cat outside. The cat can run four times as fast as the duck can swim, but cannot enter the water. Can the duck get to the perimeter of the pond and escape?
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| « Last Edit: Oct 22nd, 2009, 11:15am by R » |
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gotit
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Re: Escape
« Reply #1 on: Oct 22nd, 2009, 11:49am » |
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The duck should move spirally
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R
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Re: Escape
« Reply #2 on: Oct 22nd, 2009, 11:52am » |
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on Oct 22nd, 2009, 11:49am, gotit wrote:| The duck should move spirally |
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That would need a proof, that the cat still cannot catch the duck, even if cat runs 4 time faster than duck.
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towr
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Re: Escape
« Reply #3 on: Oct 22nd, 2009, 12:03pm » |
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Take the pond to be a circle radius 4, the duck can keep up in a circle of radius 1 (with the same center) and keep a distance of 5 from the cat. It then only needs to swim 3 units to the opposite edge, whereas the cat needs to run more than 12 units to go around.
No real need to spiral.
[edit]First thread, duplicate, follow-up 4.5x, follow-up 5x[/edit]
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| « Last Edit: Oct 22nd, 2009, 12:08pm by towr » |
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R
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Re: Escape
« Reply #4 on: Oct 22nd, 2009, 12:09pm » |
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on Oct 22nd, 2009, 12:03pm, towr wrote:Take the pond to be a circle radius 4, the duck can keep up in a circle of radius 1 (with the same center) and keep a distance of 5 from the cat. It then only needs to swim 3 units to the opposite edge, whereas the cat needs to run more than 12 units to go around.
No real need to spiral. |
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You missed some details, about how the duck will create a distance of 5 from the cat.
BTW It will be 12.56 not 12. Right?
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R
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on Oct 22nd, 2009, 12:03pm, towr wrote:
Ok, let me come to the point then.
If the duck moves in a circle of radius R/4, and the cat and the duck starts from the aligning points on their circles, then duck and cat will complete their circles at the same time. For duck to move slightly faster it will have to move in a circle of radius slightly less than R/4. Right?
You kept it exactly at R/4.
[edit]By "aligning points of their circle" I meant:[/edit]
[edit2]I have read the first two links. I also reached to that solution but have an issue with it, so please pay attention to this.[/edit2]
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| « Last Edit: Oct 22nd, 2009, 12:27pm by R » |
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towr
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Re: Escape
« Reply #6 on: Oct 22nd, 2009, 12:36pm » |
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on Oct 22nd, 2009, 12:09pm, R wrote:| You missed some details, about how the duck will create a distance of 5 from the cat. |
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Well, within its circle it has the speed advantage, so it should be somewhat obvious.
Quote:| BTW It will be 12.56 not 12. Right? |
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4 pi = "more than 12"
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R
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Re: Escape
« Reply #7 on: Oct 22nd, 2009, 12:47pm » |
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on Oct 22nd, 2009, 12:36pm, towr wrote:
Well, within its circle it has the speed advantage, so it should be somewhat obvious.
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Okay, let me do the work.
If duck moves at the boundary circle of radius = R/4, with speed v. The cat is moving at the boundary of circle of radius R with speed 4v.
The time it takes for the duck to complete a circle is (2*pi*(R/4))/v = pi*R/2v.
The time it take for the cat to complete its circle is (2*pi*R)/4v = pi*R/2v.
Both take equal amount of time to complete the circle, virtually equal speed. How is the duck moving faster?
As I said earlier the duck has to move at the boundary of circle of radius slight less than R/4. Agree?
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| « Last Edit: Oct 22nd, 2009, 12:48pm by R » |
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towr
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Re: Escape
« Reply #8 on: Oct 22nd, 2009, 1:25pm » |
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on Oct 22nd, 2009, 12:47pm, R wrote:| Both take equal amount of time to complete the circle, virtually equal speed. How is the duck moving faster? |
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He's moving faster within the circle. Unlike the cat, the duck could go in a straight line.
Quote:| As I said earlier the duck has to move at the boundary of circle of radius slight less than R/4. Agree? |
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He'd get there quicker if he just moves straight to the point on his circle opposite the cat. There's no reason for the duck to constrain himself to going around on a circle, as long as he gets to the opposite side of his circle in the end, where he can then either stay or head for the edge.
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R
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Re: Escape
« Reply #9 on: Oct 22nd, 2009, 2:28pm » |
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on Oct 22nd, 2009, 1:25pm, towr wrote:
He's moving faster within the circle. |
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Okay. 
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ThudnBlunder
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Re: Escape
« Reply #10 on: Oct 23rd, 2009, 9:20am » |
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A thorough analysis can be found on page 128 here.
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R
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Re: Escape
« Reply #11 on: Oct 23rd, 2009, 9:39am » |
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on Oct 23rd, 2009, 9:20am, ThudanBlunder wrote:| A thorough analysis can be found on page 128 here. |
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 Wowww!!!
Am not going to read all that long mathematical explanation, but the cover page is nice. 
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ThudnBlunder
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Re: Escape
« Reply #12 on: Oct 23rd, 2009, 10:07am » |
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on Oct 23rd, 2009, 9:39am, R wrote:
Am not going to read all that long mathematical explanation, but the cover page is nice. |
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Hmm... just noticed that most of that problem is missing. But I own a hard copy.
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