Find the speed V reached by a car of mass 'm' that is driven with constant power P and X is the distance traveled from rest:

{(\frac{3 X P}{m})}^{1 / 3}

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{(\frac{2 X P}{m})}^{1 / 2}

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{(\frac{X P}{m})}^{1 / 3}

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{(\frac{2 X P}{3 m})}^{1 / 4}

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Solution

The correct option is B ${(\frac{3 X P}{m})}^{1 / 3}$
$P = F v = m a v$
$= m v \frac{d v}{d x} v$

$= m v^{2} \frac{d v}{d x} \Rightarrow P d x = m v^{2} d v$

$= \int_{0}^{x} P d x$

$= m \int_{0}^{v} v^{2} d v \Rightarrow P x$

$= \frac{m v^{3}}{3}$
$\Rightarrow v = {(\frac{3 P x}{m})}^{1 / 3}$

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