Question

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

• A. $\text t^{\frac{1}{2}}$
• B. $\text t$
• C. $\text t^{\frac{3}{2}}$
• D. $\text t^2$

Answer 1

Answer: (C)

$\text t^{\frac{3}{2}}$

Answer 2

(iii) $\text t^{\frac{3}{2}}$
Power is given by the relation :
P = Fv
= $\text {mav}$ = $\text {mv}\frac{\text {dv}}{\text{dt}}$ = Constant (say, k)

∴ $\text {vdv}$ = $\frac{k}{m}dt$
Integrating both sides:
$\frac{\text v^2}{2}$ = $\frac{\text k}{\text m}\text t$

$\text v$ = $\sqrt {\frac{2kt}{m}}$
For displacement x of the body , we have :
$\text v$ = $\frac{dx}{dt}$= $\sqrt{\frac{\text {2k}}{\text m}}\text t^{\frac{1}{2}}$

$\text {dx}$ = $\text {k' }t^{\frac{1}{2}}\text d\text t$
Where $\text k'$ = $\sqrt {\frac{\text {2k}}{3}}$ = New Constant
On integrating both sides, we get: $\text x$ = $\frac{2}{3}$ $\text k'\text t^{\frac{3}{2}}$
∴ $\text x$ $\propto \text t^{\frac{3}{2}}$

Therefore, the correct option is (C) $\text t^{\frac{3}{2}}$