Explanation

(b) $P =$ constant
$\Rightarrow \ \ Fv = P\ \ $[$\because P =$ force $\times$ velocity]
$\Rightarrow \ Ma \times v = P\ \ \ \lbrack\because F = Ma\rbrack$
$\Rightarrow \ va = \frac{P}{M}$
$\Rightarrow \ v \times \frac{vdv}{ds} = \frac{P}{M}\ \ \ \left\lbrack \because a = \frac{vdv}{ds} \right\rbrack$
$\Rightarrow \ \ \int_{0}^{v}{v^{2}dv} = \int_{0}^{s}{\frac{P}{M}ds}$
[Assuming at $t = 0$ it starts from rest, $ie$, from $s = 0$]
$\Rightarrow \ \ \frac{v^{3}}{3} = \frac{P}{M}s$
$\Rightarrow \ v = \left( \frac{3P}{M} \right)^{1/3} \times s^{1/3}$
$\Rightarrow \ \ \frac{ds}{dt} = ks^{1/3}\left\lbrack k = \left( \frac{3P}{M} \right)^{1/3} \right\rbrack$
$\Rightarrow \ \ \int_{0}^{s}\frac{ds}{s^{1/3}} = \int_{0}^{t}{kdt}$
$\Rightarrow \ \ \frac{s^{2/3}}{2/3} = kt$
$\therefore\ \ s = \left( \frac{2}{3}k \right)^{3/2} \times t^{3/2}$
$\Rightarrow \ s \propto t^{3/2}$