Question

A particle is moving unidirectionally on a horizontal plane under the action of a constant power supplying energy source. The $\text{displacement(S) - time (t)}$ graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale):

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

We know that
$\text{Power},P=Fv$

$\Rightarrow P=mav=m\frac{dv}{dt}v$

$\Rightarrow Pdt=mvdv$

Integrating both sides ${\int }_0^{t}Pdt=m{\int }_0^{v}vdv$

$P.t=\frac{1}{2}m{v}^2\Rightarrow v=\left(\sqrt{\frac{2P}{m}}\right)t^{1/2}$

Displacement,
$S={\int }_0^{1}vdt=\sqrt{\frac{2P}{m}}{\int }_0^t{t}^{1/2}dt$

$\Rightarrow S=\sqrt{\frac{2P}{m}}.\frac{t^{3/2}}{\left(3/2\right)}$

$\Rightarrow S=\sqrt{\frac{8P}{9m}}.t^{3/2}$

$\Rightarrow S\propto t^{3/2}$

Hence, graph $\left(B\right)$ is correct.