Question

A box is moved along a straight line by machine delivering constant power. The distance moved by the body in time $t$ is proportional to:

• A. $t^{\frac{1}{2}}$
• B. $t^{\frac{3}{4}}$
• C. $t^{\frac{3}{2}}$
• D. $t^2$

Answer 1

Answer: (C)

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

Power delivered, $P=Fv$

$\Rightarrow$ $P=mav$ where, $a=\frac{dv}{dt}$

$\Rightarrow P=mv\frac{dv}{dt}$

Integrating w.r.t time, $m\int vdv=P\int dt$ $(\because P=constant)$

$\Rightarrow m\frac{v^2}{2}=Pt$

$\Rightarrow v=\sqrt{\frac{2Pt}{m}}$

$\Rightarrow \frac{dx}{dt}=\sqrt{\frac{2Pt}{m}}$

Integrating w.r.t time, $\int dx=\sqrt{\frac{2P}{m}}$$\int \sqrt{t}dt$

$\Rightarrow$ $x=\sqrt{\frac{2P}{m}}$ $\frac{t^{\frac{3}{2}}}{\frac{3}{2}}=\frac{2}{3}\sqrt{\frac{2P}{m}}{t}^{\frac{3}{2}}$

$\Rightarrow$ $x\propto (t{)}^{\frac{3}{2}}$