It's a simple matter to derive some very useful relations between circular and linear motion. We begin with the relation between arc length \(s\) and angle \(\theta\) (in radians) for a circle of radi...It's a simple matter to derive some very useful relations between circular and linear motion. We begin with the relation between arc length \(s\) and angle \(\theta\) (in radians) for a circle of radius \(r\) : \[s=r \theta \text {. }\] Taking the derivative with respect to time of both sides gives a relation between linear veloctiy \(v=d s / d t\) and angular velocity \(\omega=d \theta / d t\) : \[v=r \omega\]