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Bugs Problem

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I'm thinking about the following problem at the moment:


Four bugs sitting at the corners of the unit square begin to chase one another with constant speed, each maintaining the course in the direction of the one pursued. Describe the trajectories of their motions. What is the law of motion (in cartesian/polar coordinates)?


I heard the problem is fairly known but I think I need some guidance now.


Now I started with polar coordinates (with the centre of the square being in the origin) and got stuck with what to do with r(t) in: [math]\frac{d}{dt}\vec{r(t)}=\frac{d}{dt}{r(t)}\vec{e_r}+r(t)\frac{d}{dt}\vec{e_r}[/math]

I mean the objective is a diff. equation isn't it?... but what should I do with the variable radius? Is the length of a side of the square of some importance? What is the implication of the fact that the course is in the direction of the other bug?


I went on like this: [math]\frac{d}{dt}\vec{r(t)}=\frac{d}{dt}{r(t)}\vec{e_r} +r(t)\frac{d}{dt}\vec{e_r}=\frac{d}{dt}{r(t)}\vec{e_r}+r(t)\frac{d}{dt}\phi\vec{e_{\phi}}[/math]

Here we see, that [math]v_r=\frac{d}{dt}r(t),v_{\phi}=r(t)\frac{d}{dt}\phi[/math]

Since the angle between the radius-vector and the velocity vector does not change (only the lenght of the radius-vector changes with time), we get:

[math]v_r=-v\frac{\sqrt{2}}{2},v_{\phi}=v\frac{\sqrt{2}}{2}[/math] ... here I stopped.


I looked up a definition of a spiral, to be exact - the Archimedean spiral: [math]r=a\phi+c;a,c=const[/math]. Now, looking at this equation, I can't get which parameter the radius-vector should depend on in the end...is it the angle phi or time?


Am I on the right way with all this?

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