<Dd> d u ^ R d t = d θ d t u ^ θ (t), (\ displaystyle (\ frac (d (\ hat (u)) _ (R)) (dt)) = (\ frac (d \ theta) (dt)) (\ hat (u)) _ (\ theta) (t) \,) </Dd> <P> where the direction of the change must be perpendicular to u ^ R (t) (\ displaystyle (\ hat (u)) _ (R) (t)) (or, in other words, along u ^ θ (t) (\ displaystyle (\ hat (u)) _ (\ theta) (t))) because any change d u ^ R (t) (\ displaystyle d (\ hat (u)) _ (R) (t)) in the direction of u ^ R (t) (\ displaystyle (\ hat (u)) _ (R) (t)) would change the size of u ^ R (t) (\ displaystyle (\ hat (u)) _ (R) (t)). The sign is positive, because an increase in dθ implies the object and u ^ R (t) (\ displaystyle (\ hat (u)) _ (R) (t)) have moved in the direction of u ^ θ (t) (\ displaystyle (\ hat (u)) _ (\ theta) (t)). Hence the velocity becomes: </P> <Dl> <Dd> v → (t) = d d t r → (t) = R d u ^ R d t = R d θ d t u ^ θ (t) = R ω u ^ θ (t). (\ displaystyle (\ vec (v)) (t) = (\ frac (d) (dt)) (\ vec (r)) (t) = R (\ frac (d (\ hat (u)) _ (R)) (dt)) = R (\ frac (d \ theta) (dt)) (\ hat (u)) _ (\ theta) (t) = R \ omega (\ hat (u)) _ (\ theta) (t) \ .) </Dd> </Dl> <Dd> v → (t) = d d t r → (t) = R d u ^ R d t = R d θ d t u ^ θ (t) = R ω u ^ θ (t). (\ displaystyle (\ vec (v)) (t) = (\ frac (d) (dt)) (\ vec (r)) (t) = R (\ frac (d (\ hat (u)) _ (R)) (dt)) = R (\ frac (d \ theta) (dt)) (\ hat (u)) _ (\ theta) (t) = R \ omega (\ hat (u)) _ (\ theta) (t) \ .) </Dd>

In circular motion inertia takes the form of