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What sort of spiral is traced out by each of the ants?

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Four ants are situated at each vertex of a unit square. Suddenly, each ant begins to chase its counterclockwise neighbor. All the ants travel at the same speed and eventually,they all meet at the centre of the square. What sort of spiral is traced out by each of the ants? :confused:
 
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  • #2
Tide
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From the symmetry of the problem, the direction of travel always makes a -45 deg angle with the radius vector so

[tex]\frac {r d\theta}{dr} = -1[/tex]

from which

[tex]r = r_0 e^{-(\theta - \theta_0)}[/tex]

Curiously, if the ants travel at a constant speed they meet at a finite time but only after making an infinite number of loops about the center!
 
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I failed to mention that all four ants are travelling at the same speed. I've edited the first post.
 
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  • #4
HallsofIvy
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Here's how I did that problem (many years ago).
Set up a coordinate system so that corners of the square are at (0,0), (1,0), (1,1) and (0,1). The ant at (0,0) starts toward its neighbor at (1,0), which starts toward its neighbor at (1,1), etc. At first I thought I would have to set up a system of 4 related differential equations but then I came to my senses and used symmetry as Tide said!

Let (x,y) be the position at time t of the ant that starts at (0,0). Then, by symmetry (Draw a picture!), the position of its neighbor at the same time is (1-y, x).
The line from (x,y) to (1-y,x) has slope [itex]\frac{x-y}{1-x-y}[/itex] and that is the slope of the tangent line to the ant's path:
[tex]\frac{dy}{dx}= \frac{x-y}{1-x-y}[/tex]
Solve that differential equation with the initial condition y(0)= 0.

If you let u= x-1/2, v= y- 1/2 (i.e. shift so the center of the square is at (0,0) which is what I should have done to start with) then change to polar coordinates, the differential equation becomes dr/dθ= -r so that is an exponential spiral.

(Edited: I had the fraction for the slope upside down!)
 
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