The time derivative of u ^ ^ θ θ ( t ) is found the same way as for u ^ ^ R ( t ) . Again, u ^ ^ θ θ ( t ) is a unit vector and its tip traces a unit circle with an angle that is π /2 + θ. Hence, an increase in angle dθ by r ( t ) implies u ^ ^ θ θ ( t ) traces an arc of magnitude dθ, and as u ^ ^ θ θ ( t ) is orthogonal to u ^ ^ R ( t ) , we have: d u ^ ^ θ θ d t = − − d θ θ d t u ^ ^ R ( t ) = − − ω ω u ^ ^ R ( t ) , where a negative sign is necessary to keep u ^ ^ θ θ ( t ) orthogonal to u ^ ^ R ( t ) . (Otherwise, the angle between u ^ ^ θ θ ( t ) and u ^ ^ R ( t ) would decrease with an increase in dθ.) See the unit circle at the left of Figure 4. Consequently, the acceleration is: a ( t ) = R ( d ω ω d t u ^ ^ θ θ ( t ) + ω ω d u ^ ^ θ θ d t ) = R d ω ω d t u ^ ^ θ θ ( t ) − − ω ω 2 R u ^ ^ R ( t ) .