Spinning Robots

Ok, that was quite the break from blogging. Time to solve out the robot chasing puzzle I put up last time.

First of all, it is easiest to work in polar coordinates, r and θ, the robot starts at r=0 and moves along θ=θ₀ and you start at r=r₀ and θ=0 The robots path given

r(t) = v_rt
θ(t) = θ₀

We get to pick our r(t) and θ(t) and choose a path such that we will meet up with the robot path for any arbitrary value of θ₀.

First of all, we might as well move straight to the robot, assuming that θ₀ is 0. We move toward r=0 and would meet the robot at r=r₀*(v_r/(v_p+v_r)) This value, which I will call R, will be our effective starting point.

Next, we must move outward in a spiral, such that our r(t) continues to match the robot r(t), but our value of θ takes on all values between 0 and 2π. This is simple enough, find r(t) and θ(t) such that:

r'(t) = v_r
(r(t)θ'(t))²+r'(t)² = v_p²

Solving it out, you get

r(t) = v_rt+R
θ(t) = √(v_p²/v_r²-1)ln((v_rt+R)/R)

Note the problem we would have if the robot were faster than the player. Anyway, since the logarithm is unbounded from above, you know that this θ will eventually reach 2π, solving the problem. This also finds the smallest disc you can solve the problem on, since you just find that t where θ is 2π and put it back into r.

Not too tricky of a problem once you hit on the solution really, but at first it did look somewhat impossible.