Note: All bold and underlined variables in this post are base vectors I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The wheel rotates with uniform angular velocity dθ/dt = ω radians per second about an axis fixed in space. At t = 0 the spoke is along the x axis, and the bead is at the origin. The book then says that the velocity of the bead at time t in polar coordinates is ur + uωtθ. Elaborating, the text says "at time t, the bead is at radius ut on the spoke." What I don't understand is why u can be used in this calculation without any modification. If the bead is at radius ut at time t then the velocity would increase indefinitely and the spoke would have a position vector longer than the wheel it was attached to, which obviously doesn't make sense. Am I misunderstanding something about polar coordinates/vectors here or am I misunderstanding the example?