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I came across this question which, asks,

3 particles, at the corner of an equilateral triangle of side s, start moving with constant speed v, such that each always heads directly towards its clockwise neighbor. What is the magnitude of the initial acceleration of each particle?

I know what time the particles would take to meet. However acceleration seems quite a bit tricky. Also I am not hundred percent sure if the question is correct (at least in its terminology). I am still working under the assumption that the constant speed v is actually the modulus of the velocity vector.

I would be grateful if anyone can give a proper analysis of this problem, or at least check, if the problem is at all correct, that is, if the acceleration can be evaluated in terms of the length of the side and the velocity.

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  • $\begingroup$ It is known that the trajectory is a very well-known spiral. Instead of providing a full solution, let me throw a link. $\endgroup$ – Sangchul Lee Oct 5 '18 at 21:18
  • $\begingroup$ Have you tried using polar coordinates? You can obtain expressions for the velocity and acceleration components along the radius and tangential vectors. $\endgroup$ – David Quinn Oct 5 '18 at 21:47
  • $\begingroup$ Thank you I figured it out! $\endgroup$ – Nothingham Oct 6 '18 at 6:15
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So I assume what you have done is use a relative velocity approach and use the fact that they meet at the centroid. That's good enough for finding time, but I am not certain that we can answer the acceleration question by doing the same. Without explicitly solving the problem, I'll try to provide a useful pointer.

The first thing I would do to be able to find this is find a trajectory of the snail as a function of time. Given this spiral, what can you say about the instantaneous velocity and/or acceleration?

I believe this might be of some help.

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  • $\begingroup$ Thanks, this one helped a lot. $\endgroup$ – Nothingham Oct 6 '18 at 6:20

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