Final answer:
The relationship between velocity (v) and radius (r) in rotational motion is quantified by the equation v = rw, meaning that tangential speed is proportional to distance from the center of rotation. Angular velocity remains the same for all points on a rotating body, but linear speed increases with radius.
Step-by-step explanation:
The relationship between velocity (v) and radius (r) often involves the concept of angular velocity (w) and can be expressed by the equation v = rw or w = v/r. In the context of rotational motion, a point farther from the center of rotation (larger r) will have a higher tangential speed if it has the same angular speed as a point closer to the center, because it has to cover a longer distance in the same amount of time. This is easily seen with examples such as a CD player or the tire of a car; points on the rim move faster than points closer to the center. On the other hand, in the context of a charged particle in a magnetic field, r is the radius of curvature of the particle's path and velocity v is the component perpendicular to the magnetic field, which determines the shape of the path (circular or spiral).