Worked example 7.2: Circular race track

Question: A car of mass $m=2000 {\rm kg}$ travels around a flat circular race track of radius $r = 85 {\rm m}$. The car starts at rest, and its speed increases at the constant rate $a_\theta = 0.6 {\rm m/s}$. What is the speed of the car at the point when its centripetal and tangential accelerations are equal?

Answer: The tangential acceleration of the car is $a_\theta = 0.6 {\rm m/s}$. When the car travels with tangential velocity $v$ its centripetal acceleration is $a_r=v^2/r$. Hence, $a_r = a_\theta$ when

$$\frac{v^2}{r} = a_\theta,$$

or

$$v = \sqrt{r a_\theta} = \sqrt{85\times 0.6} = 7.14 {\rm m/s}.$$