Explanation

(a) Velocity $v = \alpha\ \sqrt{x}$
$\frac{dx}{dt} = \alpha\sqrt{x}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( \because v = \frac{dx}{dt} \right)$
Or $\frac{dx}{\sqrt{x}} = \alpha\ dt$
Integrating
$\ \ \ \int_{0}^{x}\frac{dx}{\sqrt{x}} = \int_{0}^{t}\alpha dt$
$\ \ \ \lbrack\because at\ t = 0,\ x = 0\ ad\ let\ at\ any\ time\ t,\ paricle\ is\ at\ x\rbrack$
Or $\ \ \ \frac{x^{1/2}}{1/2} = \alpha t$
Or $x^{1/2} = \frac{\alpha}{2}t$
Or $\ \ \ \ \ x = \frac{\alpha^{2}}{4}\ \times t^{2}\ \Rightarrow x \propto t^{2}$