# How do you solve this Kinematic question? It's very non-standard [closed]

There are 4 beatles (John, Paul, George, and Ringo) on the vertices of a square with side length a, each beatle is moving in a constant speed v and in the direction of the beatle that started at the verice clockwise to the one the beatle started at. Their motion will follow a spiral. The question is how long will it take for them to meet?

• Beatles as in John, Paul, George, and Ringo? – noah Mar 14 at 11:22
• This is a great update for the Problem and Im adding it – sean python Mar 14 at 11:23
• Lmao that give me a good laugh @noah – Buraian Mar 14 at 12:03

This is quite a commonly posed problem. The easiest way to find a solution is to notice that for each individual Beatle at every moment in time, their own velocity vector is perpendicular to the velocity vector of the Beatle they are chasing. Therefore, in an infinitesimal timeslice $$\mathrm{d}t$$, the distance to the Beatle in front decreases by $$v\,\mathrm{d}t$$, if they are moving with velocity $$v$$ (the sideways motion by the Beatle in front is not changing the distance in an infinitesimal time step). So if the initial distance is $$a$$, and the Beatle separation decreases at a constant rate $$v$$, then the time for them to meet in the center and play their concert is $$t = a/v$$.