Explanation

(c) $P = Fv = m.\frac{dv}{dt} \bullet v$
$\int_{}^{}{v\ dv} = \int_{}^{}{\frac{p}{mdt};}\ \frac{{\ \ v}^{2}}{2} = \frac{pt}{m}\ $
$v = \sqrt{\frac{2p}{m}}t^{1/2\ ;}\frac{dx}{dt} = \sqrt{\frac{2p}{m}}t^{1/2}$
$\int_{}^{}{dx = \sqrt{\frac{2p}{m}}\ \int_{}^{}{t^{1/2}dt;}\ }$
$x = \sqrt{\frac{2p}{3}}\ \frac{t^{3/2}}{3/2} = \frac{2}{3}\sqrt{\frac{2p}{3}}t^{3/2}$
$x \propto t^{3/2}$