A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to

The correct option is C $t^{3/2}$

$P=C$, where C is constant.

$P=F\times v=m\times a\times v$

$a=\frac{dv}{dt}$

$\therefore C=m\times \frac{dv}{dt}\times v$

$\therefore C\times dt=m\times v\times dv$

Integrating both sides,

$\int C\times dt=\int m\times v\times dv$

$\therefore C\times t=m\times \frac{v^2}{2}$

$\therefore v=\frac{ds}{dt}={\left(\frac{2\times C\times t}{m}\right)}^{\frac{1}{2}}$

Let, $C^{\prime }={\left(\frac{2\times C}{m}\right)}^{\frac{1}{2}}$, where C' is constant.

$\therefore ds=C^{\prime }\times t^{\frac{1}{2}}\times dt$

Integrating both sides,

$\int ds=\int C^{\prime }\times t^{\frac{1}{2}}\times dt$

$s=C^{\prime }\times \frac{2}{3}\times t^{\frac{3}{2}}$

$s\propto t^{3/2}$