Question

A racing car moves on a circular track of radius b. The car starts from rest and its speed increases at a constant rate $\alpha$. Find the angle between its velocity and acceleration vectors at time t.

Answer 1

$$|a_{\tau}|=\alpha, v=\alpha t$$

$$|a_r|=\dfrac{v^2}{b}=\dfrac{\alpha^2t^2}{b}$$

$$|a|=\sqrt{(\alpha)^2+(\dfrac{\alpha^2t^2}{b})^2}$$

$$\cos\theta=\dfrac{|a_{\tau}|}{|a|}=\dfrac{\alpha}{\sqrt{(\alpha)^2+(\dfrac{\alpha^2t^2}{b})^2}}$$

$$=\dfrac{b}{\sqrt{b^2+\alpha^2t^4}}$$

$$\theta=\cos^{-1}(\dfrac{b}{\sqrt{b^2+\alpha^2t^4}})$$