Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector u ^ ^ R ( t ) has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of r ( t ) . If the particle displacement rotates through an angle dθ in time dt, so does u ^ ^ R ( t ) , describing an arc on the unit circle of magnitude dθ. See the unit circle at the left of Figure 4. Hence: d u ^ ^ R d t = d θ θ d t u ^ ^ θ θ ( t ) , where the direction of the change must be perpendicular to u ^ ^ R ( t ) (or, in other words, along u ^ ^ θ θ ( t ) ) because any change d u ^ ^ R ( t ) in the direction of u ^ ^ R ( t ) would change the size of u ^ ^ R ( t ) . The sign is positive because an increase in dθ implies the object and u ^ ^ R ( t ) have moved in the direction of u ^ ^ θ θ ( t ) . Hence the velocity becomes: v ( t ) = d d t r ( t ) = R d u ^ ^ R d t = R d θ θ d t u ^ ^ θ θ ( t ) = R ω ω u ^ ^ θ θ ( t ) .