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Suppose we have a regular polygon with $n$ sides. On each vertex, there is a particle. Every particle moves in such a way that its velocity vector $(\vec{v})$ always points towards particle next to it. The velocity vectors of all particles are equal in magnitude.
How do we show that the meet at centroid of the polygon.

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Actually, the problem was in physics, and it asked the time particle took to reach the centroid. The time can easily be calculated using resolution of components of vectors.

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    $\begingroup$ The short solution is because of symmetry. There is no other point where they could meet. $\endgroup$
    – Arthur
    Commented Feb 29, 2016 at 13:58
  • $\begingroup$ @Arthur Yes absolutely true! we used the argument that it was symmetrical, so the particles meet at the centroid. THen we could take the velocity component, the distance, and then then distance / speed gave us time. But this is my own question. $\endgroup$
    – jonsno
    Commented Feb 29, 2016 at 14:02
  • $\begingroup$ You could use simple physics to derive a solution - look at components of velocity tangent and radial. $\endgroup$
    – Moti
    Commented Mar 1, 2016 at 3:24
  • $\begingroup$ @Moti I wish to prove that they meet at centre, not find the time taken! $\endgroup$
    – jonsno
    Commented Mar 1, 2016 at 3:39
  • $\begingroup$ Actually, the statement need not be true unless you impose some condition on the the velocity (e.g. all velocity vectors has same magnitude). $\endgroup$
    – achille hui
    Commented Mar 1, 2016 at 3:55

1 Answer 1

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Based on simple physics the velocity of the particle may be broken into two components - one toward the center and one perpendicular to the line connecting a particle with the center. As all particle has the same velocity, all move in tandem along congruent curves. Since there is always a component to the center, equal for all, they will meet at the center after a finite time that could be calculated... do you know how to follow from here?

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