A body is moving unidirectionally under the influence of a source

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

(i) $t^{\frac{1}{2}}$

(ii) $t$

(iii) $t^{\frac{3}{2}}$

(iv) $t^{2}$

solution:

(iii) $t^{\frac{3}{2}}$

Power is given by the relation:

P = Fv

$=m a v=m v \frac{d v}{d t}=$ Constant $($ say,$k)$

$\therefore v d v=\frac{k}{m} d t$

$m$ Integrating both sides:

$\frac{v^{2}}{2}=\frac{k}{m} t$

For displacement $x$ of the body, we have:

$v=\frac{d x}{d t}=\sqrt{\frac{2 k}{m} t^{\frac{1}{2}}}$

$d x=k^{\prime} t^{\frac{1}{2}} d t$

Where $k^{\prime}=\sqrt{\frac{2 k}{3}}=$ New constant

On integrating both sides, we get:

$x=\frac{2}{3} k^{\prime} t^{\frac{3}{2}}$

$\therefore x \propto t^{\frac{3}{2}}$