Three points are located at the vertices of an equilateral triangle each of whose sides measure a. They all start simultaneously with speed v, each aiming at the next in order. How soon will the points converge ?

Explanation

Strategy : From the symmetry of the problem all the three points are always located at the vertices of equilateral triangles of varying side length and finally meet at the centroid of the initila equilateral triangle whose side length is a in the sought time interval (say t)
Let us consider an arbitrary equilateral triangle of edge length / (say) . Then the rate by which 1 approaches, 2,2 approches 3 and 3 approches, 1 ,becomes
`(-dt)/(dt) = v- v cos ( (2pi)/(3))`
On intergrating `-overset(0)underset(a)intdl= (3v)/2overset(t)underset(0)intdt`
` a = 3/2 vt , so, t = (2a)/(3v)`