Robot Chasing

New puzzle time, because I actually have more than one in mind and I don't want to forget them.

I found this one somewhere random on the internets, I'll try to dig it up agian when I do the solution:
You are playing a game where you have to catch a robot. You and the robot are on an infinite 2-dimensional plane and you can see where the robot is. When the game starts the lights will be turned off and the robot will select a random direction and move in that direction at a constant speed. You may move around, and your top speed is some number which is greater than the robots constant speed (call them vr and vp for the robot speed and the player speed if you like). You naturally cannot see or hear the robot once the game begins. Find a strategy that guarantees you will find the robot.

So you must find a path (x(t), y(t)) which starts at a point (x0, y0) which never has a speed exceeding vp and that path much intersect the robots path such that there is a t where they meet.

As a bonus problem, assume the game is played on a disc of radius R, with the robot starting in the center of the disc and the player on some point a distance P away from the robot. If the robot reaches the edge of the disc it escapes and the player loses. What is the smallest R for which your strategy works?

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