Q: The acceleration (a) of a particle moving in a straight line varies with its displacement (S) as a = 2 s . The velocity of the particle is zero at zero displacement. Find the corresponding velocity – displacement equation.

Sol: a = 2 s

$\large v\frac{dv}{ds} = 2 s $

$\large \int_{0}^{v}v dv = \int_{0}^{s} 2 s ds $

$\large [\frac{v^2}{2}]_{0}^{v} = 2 [\frac{s^2}{2}]_{0}^{s}$

$\large \frac{v^2}{2} = s^2 $