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There are four dogs/ants/people at four corners of a square of unit distance. At the same instant all of them start running with unit speed towards the person on their clockwise direction and will alw

Posted on 2011-02-21
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There are four dogs/ants/people at four corners of a square of unit distance. At the same instant all of them start running with unit speed towards the person on their clockwise direction and will always run towards that target. How long does it take for them to meet and where?
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Question by:dshrenik
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by:Aaron Tomosky
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They meet in the center in 1/2 unit time.
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by:dshrenik
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If possible, can you give an explanation.
I think they move in spirals.. but why does it take 1/2 unit time?
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by:Aaron Tomosky
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It's actually slightlymore that 1/2 unit time because it's an arc. So it's really 1/4 of the perimeter of a circle with a radius of 1/2 unit. pie*1/2 unit squared /4
Sorry that the best ican type on my phone.

So let's look at bottom left and bottom right. Bl starts moving up and br starts moving left, but see how br starts curving up since Bl is going up? Bl is also curving right since tl is moving right. So they all move in a nice arc toward the center
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by:TommySzalapski
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If they travel in perfectly circular paths (which I am not yet sure about, but it seems probable) then they would travel along a circle centered at the midway point on the edge of the side of the square.
Then they would travel a distance of pi/16 units which is what aarontomsky was trying to say.
pi/16 since the whole circle is pi*r^2 r is 1/2 unit and it's 1/4 of the circle so pi(1/2)^2*1/4 = pi/16
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by:TommySzalapski
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What? Oops. That's area. It's actually pi*d/4 so pi/4 since d is the unit side. My bad.
pi/4 is the distance travelled.
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by:TommySzalapski
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Scratch all that. They would not move in perfect circular curves. Try it out. Draw the square and the circles and you can see that at the halfway point the paths would not point at each other. I'll have to think about it for a while.
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TommySzalapski earned 500 total points
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The total distance travelled is 1. 1 unit is travelled.

Think about it like this. If you go 1/2 of the way before noticing that the other guy moved then you will travel 1/2 unit. Now look. They are all still in a square (but it's a rotated square) and the side of the new square is sqrt(2)/2 (pythagorean theorem) since the diagonal of the square is 1/2.
 step1So the solution for square of size 1 is S(1)
S(1) = 1/2 + S(sqrt(2)/2)
This is 1/2(1 + r^2 + r^3 + r^4 ...) where r is sqrt(2)/2.
Now if r > 0 and r < 1 then 1 + r^2 + r^3 + ... is a geometric series and equals 1/(1-r)
So in this case the answer is 1 + sqrt(2)/2
But what if we go only 1/3 of the way?
Using the pythagorean theorem again it is 1/3(1 + r^2 + r^3 ...) where r is now sqrt(5)/9
So the geometric series gives 1/3(1/(1-sqrt(5)/9)) = 3/(9 - sqrt(5))
So what if we go 1/n of the way and n tends toward infinity?
1/n(1 + r^2 + r^3 + r^4....) where r is sqrt(n^2 - 2n + 2)/n
Again geometric series makes it 1/n(1/(1-r)) or 1/(n - sqrt(n^2 - 2n + 2))
And (if you know limits from calculus) as n goes to infinity that goes to 1
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by:TommySzalapski
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If you don't believe it. Try plugging it into a calculator. Use bigger and bigger numbers for n.
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by:TommySzalapski
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If you didn't catch what happened as n tended to infinity this is what I was doing.
If you go halfway to the other guy each time you got approximately 1.7071 as the answer.
If you went 1/3 of the way before turning you got ~ 1.309 as the answer.
What if you keep turning constantly? Basically (applied math guys cover your ears) you go 1/infinity of the way before you turn. So you are always turning. So basic calculus limits get us the answer.
Here is an Excel spreadsheet for those who don't know limits. The number in column B is the portion of the distance you go before turning. Note how if you go the whole way you get an error. This is because you would end up at the other corner and would never get closer to the middle. So the math even works at the extreme ends.
SpiralIn.xls
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