Question

A constant power $P$ is applied to a car starting from rest. If $v$ is the velocity of the car at time $t$, then

• A. $v\propto t$
• B. $v\propto 1/t$
• C. $v\propto \surd t$
• D. $v\propto 1/\surd t$

Answer 1

The correct option is C

$v\propto \surd t$

Explanation for the correct answer:

Option (C): $v\propto \surd t$

As we know that,

Power, $P$ is rate of change of work with respect to time.

$P=\frac{Work}{Time}$

In differential form, above equation is written as

$P=\frac{dW}{dt}$

$\Rightarrow P=\frac{d(Fxs)}{dt}$ $\left[Work\left(W\right)=Force\left(F\right)\times Displacement\left(s\right)\right]$

$\Rightarrow P=\frac{F\times ds}{dt}$

$\Rightarrow P=mav$ $\left[F=maandv=\frac{ds}{dt}\right]$

$\Rightarrow P=m\times \frac{v}{t}\times v$ $\left[a=\frac{v}{t}\right]$

$$\begin{gathered} \Rightarrow P=m\times \frac{v^2}{t} \\ \Rightarrow v^2=\frac{2Pt}{m} \\ \therefore v\propto \surd t \end{gathered}$$

Hence, correct option is (C).