Question

in a straight line motion the distance travelled is proportional to the square root of the time taken. the acceleration of particle is proportional to:
a) velocity (b)v2 (c)v3 (d)?v

Answer 1

Dear Student,

Given
$$\begin{gathered} s\propto t^{\frac{1}{2}} \\ \Rightarrow s=k{t}^{\frac{1}{2}} \end{gathered}$$
differentiating s w.r.t time, we get
$v=\frac{ds}{dt}=\frac{1}{2}k{t}^{-\frac{1}{2}}$
differentiating v w.r.t time we get,

$$\begin{gathered} a=\frac{dv}{dt}=\frac{d}{dt}\frac{1}{2}k{t}^{-\frac{1}{2}}=\frac{1}{4}k{t}^{-\frac{3}{2}} \\ \Rightarrow a=\frac{1}{4}k{v}^3 \end{gathered}$$

Regards