Question

# A car drives along straight level frictionless road by an engine delivering constant power. Then velocity is directly proportional to?

A

$t$

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B

$\frac{1}{\sqrt{t}}$

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C

$\sqrt{t}$

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D

None of these

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Solution

## The correct option is C $\sqrt{t}$Step 1: GivenVelocity$=v$Time$=t$Power$=P$(Constant)Step 2: Formula usedPower is $P=Fv$Force is $F=ma$Acceleration is $a=\frac{dv}{dt}$Step 3: Determine the powerPower is given by, $P=Fv$Force can be written as $F=ma$Therefore, $P=mav$Acceleration can be rewritten as $a=\frac{dv}{dt}$. Hence,$P=mv\frac{dv}{dt}\phantom{\rule{0ex}{0ex}}Pdt=mvdv$Step 4: Integrate the equation$\int Pdt=\int mvdv\phantom{\rule{0ex}{0ex}}Pt=\frac{m{v}^{2}}{2}\phantom{\rule{0ex}{0ex}}⇒t\propto {v}^{2}\phantom{\rule{0ex}{0ex}}⇒v\propto \sqrt{t}$Therefore, option C is the correct option.

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