Case of the 'mice problem' for $n=3$

Question

Three particles $A,$ $B$ and $C$ are situated at the vertices of an equilateral $ΔABC$ with sides each of length $l.$ Particle $A$ moves towards particle $B$ with a speed of $s.$ Simultaneously, particle $B$ moves with the same speed towards particle $C$ and $C$ moves towards particle $A.$ Tracing out a pursuit curve, they meet each other.

What is the distance covered by particle $A$ when it revolves around by $2π$ radians?

I could find out the total time taken by the particle $A$ to collide with $B$ (which is $\frac{2l}{3s}$) and total distance $l$ covered by $A$ but I could not figure out the distance traced to revolove 360°.

What is their angular velocity? This is constant, so once you figure that out you can find out how long it takes, and hence how far it has gone. — Jaap Scherphuis, Jun 01 '18 at 14:12
@JaapScherphuis Logic dictates that the mice are moving with a constant speed, only their direction changes. — peterh, Jun 01 '18 at 14:22
@peterh and their direction changes at a constant rate, which is what you can use to solve this particular question.. — Jaap Scherphuis, Jun 01 '18 at 14:32
@JaapScherphuis I think it is not, their direction change is accelerating. They make infinite many loops in finite time. — peterh, Jun 01 '18 at 14:34
Oh wait, I think you are right. At any moment the relative locations and angles all stay the same, everything is just scaled down and rotated, but the linear speed is not scaled accordingly so everything is relatively speeded up, including the angular velocity. So ignore my previous comments, sorry. Thanks @peterh. — Jaap Scherphuis, Jun 01 '18 at 14:40
Does this answer your question? Particle on vertex of a polygon moving towards adjacent particle. — Clemens Bartholdy, Jul 05 '21 at 15:05

score 1 · Answer 1 · answered Jun 01 '18 at 14:31

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The point is to recognize that the spiral is equiangular. The angle between the spiral and the radius is constant at $\frac {\pi} 3$. This means it is a logarithmic spiral of the form $r=ae^{b\theta}$ with $\arctan \frac 1b=\frac \pi 3, b=\frac 1{\sqrt 3}$. If we choose the origin of $\theta$ to go through one of the corners of the triangle we have $a=\frac {\sqrt 3}3l$

answered Jun 01 '18 at 14:31

Ross Millikan

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So, is the answer $l(1-e^{\frac{-2π}{\sqrt3}})$ ? – Ryan Scott Jun 01 '18 at 14:38
No, you need to integrate the arc length along the curve from $0$ to $-2\pi$. I couldn't convince Alpha to do it for me. – Ross Millikan Jun 01 '18 at 14:39

Case of the 'mice problem' for $n=3$

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