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

The Correct Option is C

(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)  \(\text t^{\frac{3}{2}}\)