First slide

Power

Question

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

Moderate

A

$t^{1 / 2}$
B

$t^{3 / 4}$
C

$t^{3 / 2}$
D

$t^{2}$

Solution

$P = Fv = mav = m (\frac{dv}{dt}) v \Rightarrow \frac{P}{m} dt = v dv$
$\Rightarrow \frac{P}{m} \times t = \frac{v^{2}}{2} \Rightarrow v = {(\frac{2 P}{m})}^{1 / 2} {(t)}^{1 / 2}$
Now $s = \int_{}^{} v dt = \int_{}^{} {(\frac{2 P}{m})}^{1 / 2} t^{1 / 2} dt$
$∴ s = {(\frac{2 P}{m})}^{1 / 2} [\frac{2 t^{3 / 2}}{3}] \Rightarrow s \propto t^{3 / 2}$

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