Q: The velocity of a particle moving in the positive direction of the X – axis varies as v = K √s where K is a positive constant. v varies with time as

Sol: $\large v = K \sqrt{s}$

$\large \frac{ds}{dt} = K \sqrt{s}$

$\large \int_{0}^{s}\frac{ds}{\sqrt{s}} = K \int_{0}^{t} dt $

$\large 2 \sqrt{s} = K t $

$\large s = \frac{1}{4} K^2 t^2$

$\large v = \frac{ds}{dt} = \frac{1}{2} K^2 t $

$\large v \propto t $

The v – t graph is a straight line passing through the origin.