# A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude $P_{o}$. The instantaneous velocity of this car is proportional to Option 1) $t^{2} P_{o}$ Option 2) $t^{1/2}$ Option 3) $t ^{-1/2}$ Option 4) $\frac{1}{\sqrt{m}}$

V Vakul

As discussed in

Power if the force is variable -

$P_{av}= \frac{\Delta w}{\Delta t}= \frac{\int_{0}^{t}p\cdot dt}{\int_{0}^{t}dt}$

- wherein

$P\rightarrow power$

$dt\rightarrow short\: interval \: of\: time$

$P_{0} = FV = m \frac{dv}{dt} v$

$P_{0} dt = m v dv \Rightarrow P_{0}\int dt = m \int vdv$

$P_{0}t = \frac{mv^{2}}{2} = v = \sqrt{\frac{2P_{0}t}{m}}$

$v\propto \sqrt{t} \Rightarrow v\propto t^{\frac{1}{2}}$

Option 1)

$t^{2} P_{o}$

This solution is incorrect

Option 2)

$t^{1/2}$

This solution is correct

Option 3)

$t ^{-1/2}$

This solution is incorrect

Option 4)

$\frac{1}{\sqrt{m}}$

This solution is incorrect

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