A point moves along an arc of a circle of radius $R$. Its velocity depends on the distance covered $

Question

A point moves along an arc of a circle of radius $R$. Its velocity depends on the distance covered $s$ as $v = a\sqrt{s}$, where $a$ is a positive constant. Let $\alpha$ be the angle between the total acceleration vector and the velocity vector. Which of the following statements is/are CORRECT?

Professor PC Thomas Classes · Answer

Tangential acceleration is $w_	au = v \frac{dv}{ds} = (a\sqrt{s}) \left(\frac{a}{2\sqrt{s}}ight) = \frac{a^2}{2}$ (A is incorrect). Normal acceleration is $w_n = \frac{v^2}{R} = \frac{a^2 s}{R}$, which is proportional to $s$ (B is correct). The velocity vector is always along the tangent. The angle $\alpha$ between total acceleration and velocity is given by $	an \alpha = \frac{w_n}{w_	au} = \frac{a^2 s / R}{a^2 / 2} = \frac{2s}{R}$ (C is correct).