Q. 1.12 - "Problems in General Physics" by I.E. Irodov

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Three points are located at the vertices of an equilateral triangle whose side equals . They all start moving simultaneously with velocity constant in modulus, with the first point heading continually for the second, the second for the third and the third for the first. How soon will the points converge?

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By symmetry, the particles all converge when they reach the center of the triangle. Also, by symmetry, the particles maintain their relative configuration i.e. they always remain at the vertices of an equilateral triangle whose side keeps decreasing with time.

Consider an observer fixed to the first point. For this observer, the relative speed with which the second particle is coming towards the first particle is . This speed remains constant through-out the duration of traversal since the particles maintain their relative configuration as mentioned above.

Now, the initial distance between the first and second particle is . So the time taken for the first and second particle to collide is .