Question

A box is moved along a straight line by a 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

The correct option is C

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

We know that, Power,
$P=Fv=m\frac{dv}{dt}.v\Rightarrow vdv=\frac{P}{m}dt$
Integrating both sides, we have
$\int vdv=\frac{P}{m}\int dt$ ($\because$ P = constant)
$\Rightarrow \frac{v^2}{2}=\frac{Pt}{m}$
$\Rightarrow v=\sqrt{\frac{2P}{m}}{t}^{\frac{1}{2}}$
$\Rightarrow \frac{dx}{dt}=\sqrt{\frac{2P}{m}}{t}^{\frac{1}{2}}$
$\Rightarrow dx=\sqrt{\frac{2P}{m}}{t}^{\frac{1}{2}}dt$
Integrating again, we have
$\int dx=\sqrt{\frac{2P}{m}}\int t^{\frac{1}{2}}dt$
$\Rightarrow x=\frac{2}{3}\sqrt{\frac{2P}{m}}{t}^{\frac{3}{2}}$
$\therefore x\propto t^{\frac{3}{2}}$.
Hence, the correct choice is (c).