Explanation

(a) Given a=-$kx$
$a = \frac{dv}{dt} = \frac{dv}{dx}.\frac{dx}{dt} = - kx$
Or $\frac{vdv}{dx} = - kx$
Or $v\ dv = - kx\ dx$
Let for any displacement from 0 to $x,$the velocity changes from $v_{0}to\ v$.
$\Longrightarrow \int_{v0}^{v}{vdv = - \int_{0}^{x}{k\ \ x\ dx}}$
Or $\frac{v^{2} - v_{0}^{2}}{2} = - \frac{kx^{2}}{2}$
$or\ \ m\left( \frac{{v^{2} - v}_{0}^{2}}{2} \right) = - \frac{mkx^{2}}{2}$
Or $\mathrm{\Delta}K \propto x^{2}$ $(\mathrm{\Delta}K\ is\ loss\ in\ KE)$