Question

Six particles situated at the corners of a regular hexagon of side a move at a constant speed $v$. Each particle maintains a direction towards the particle at the next corner. Calculate the time particles will take to meet each other.

Answer 1

The particle meet at the centroid $O$ of the triangle. At any instant the particles will form an equilateral $△ABC$ with the same centroid.Consider the motion of particle $A$. At any instant its velocity makes angle ${30}^o$, This component is the rate of decrease of the distance $AO$.Initially $AO=\frac{2}{3}\sqrt{a^2-{\left(\frac{a}{2}\right)}^2}-\frac{a}{\sqrt{3}}$Therefore, the time taken for $AO$ to becomes zero$=\frac{a/\sqrt{3}}{v\cos {30}^o}-\frac{2a}{\sqrt{3v}\times \sqrt{3}}-\frac{2a}{3v}$.