Question

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

• A. $t$
• B. $\frac{1}{\sqrt{t}}$
• C. $\sqrt{t}$
• D. None of these

Answer 1

The correct option is C

$\sqrt{t}$

Step 1: Given

Velocity$=v$

Time$=t$

Power$=P$(Constant)

Step 2: Formula used

1. Power is $P=Fv$
2. Force is $F=ma$
3. Acceleration is $a=\frac{dv}{dt}$

Step 3: Determine the power

Power 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,

$$\begin{gathered} P=mv\frac{dv}{dt} \\ Pdt=mvdv \end{gathered}$$

Step 4: Integrate the equation

$$\begin{gathered} \int Pdt=\int mvdv \\ Pt=\frac{m{v}^2}{2} \\ \Rightarrow t\propto v^2 \\ \Rightarrow v\propto \sqrt{t} \end{gathered}$$

Therefore, option C is the correct option.