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jtan
Sweden5891 Posts
4 ants are sitting in the 4 corners of a table with side a. Suddenly all ants sees the ant to it's right, and starts moving towards it. How far will each ant walk before they all meet? Back up your amswer with calculations and don't post anything you didn't do yourself. Good luck!   | ||
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Cambium
United States16368 Posts
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KarlSberg~
731 Posts
+ Show Spoiler + From the point of view of an ant (any of them obviously), at any instant the speed of the ant you are following is orthogonal to your own speed. The relative speed simply is... your own speed (speed vector is not the same because of direction but its norm is the same). Which means you will close the gap in the exact same time as if the ant you are following was not moving. So the length of the spiral every ant walks is the side of the table. | ||
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15vs1
64 Posts
Using the symmetry of this problem (all ants are similar) one can infer that we can yield coordinates of the next anth by rotating corrdinates of the previous. Using this we can easy recieve following equations (in polar coordinates, origin in the centre of table) ![[image loading]](http://www.texify.com/img/%5CLARGE%5C%21dr%2Fdt%3D-%5Csqrt%7B2%7D%2F2V%5C%5Cdf%2Fdt%3D%5Cfrac%7B%5Csqrt%7B2%7D%2F2V%7D%7Br%7D.gif) Solution of this system is logarithmic spiral, it is not relevant for this problem because we can use first equation to obtain the time before they meet (when r=0, initial r is sqrt2/2*a) ![[image loading]](http://www.texify.com/img/%5CLARGE%5C%21t%3D%5Cfrac%7B%5Csqrt%7B2%7D%2F2a%7D%7B%5Csqrt%7B2%7D%2F2V%7D%3Da%2FV%5C%5Cs%3DtV%3Da.gif) where V-velocity of ants. | ||
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jtan
Sweden5891 Posts
 adding: give an equation for the curves of the ants paths. | ||
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jtan
Sweden5891 Posts
Where do you get those equation pics from? Looking sweet. | ||
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15vs1
64 Posts
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aseq
Netherlands4015 Posts
Length of circle = 2PI * R Length of quarter circle PI * R / 2 R is half a side, so: PI * side / 4 | ||
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jtan
Sweden5891 Posts
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aseq
Netherlands4015 Posts
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ToT)SiLeNcE(
Germany590 Posts
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jtan
Sweden5891 Posts
On February 25 2008 06:38 ToT)SiLeNcE( wrote: Hehe I was unable to solve this puzzle because I had to think of scvs circling around a vespene geyser all the time and therefore thought they wouldn't meet at all. lol. Haha! If you put 4 scvs in the corners of an empty map and order them to attack eachother, you will probably get the same effect though  | ||
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