Explanation

(a)
Given, s=$\sqrt{at^{2} + 2bt + c}$
$\Rightarrow \ \ \frac{ds}{dt} = \frac{2at + 2b}{2\sqrt{at^{2} + 2bt + c}} = \frac{at + b}{\sqrt{at^{2} + 2bt + c}}$
$\frac{d^{2}s}{dt^{2}} = \frac{\left\lbrack \frac{\begin{array}{r}
\sqrt{at^{2} + 2bt + c + a} \\
- (at + b)\frac{(2at + 2b)}{2\sqrt{at^{2} + 2bt + c}}\ \
\end{array}}{\left( \sqrt{at^{2} + 2bt + c} \right)^{2}} \right\rbrack}{\left( \sqrt{at^{2} + 2bt + c} \right)^{2}}$
$\Rightarrow \ \ \frac{d^{2}}{dt^{2}} = \frac{ac - b^{2}}{s^{3}} \Rightarrow Acceleration \propto \frac{1}{s^{3}}$