Question

From the state of rest, a car accelerates while the engine supplied constant power p. The velocity of the car varies with time as

• A. $v=\frac{pt}{m}$
• B. $v=\frac{2pt}{m}$
• C. $v=\sqrt{\frac{2pt}{m}}$
• D. $v=\sqrt{\frac{pt}{m}}$

Answer 1

The correct option is C

$v=\sqrt{\frac{2pt}{m}}$

Step 1: Given

Initial velocity $u=0$

Power = $P$

Final velocity = $v$

Step 2: Find the power

Power is given by,

$$\begin{gathered} P=Fv \\ P=mav \end{gathered}$$

(Force has been substituted as $F=ma$ by the second equation of motion.)

Step 3: Find the acceleration

Acceleration can be rewritten as $a=\frac{dv}{dt}$

Step 4: Find the velocity

$$\begin{gathered} P=mav \\ P=mv\frac{dv}{dt} \\ \frac{P}{m}dt=vdv \end{gathered}$$

Integrating on both sides,

$$\begin{gathered} {\int }_0^t\frac{P}{m}dt={\int }_0^{v}vdv \\ \frac{Pt}{m}=\frac{v^2}{2} \\ \Rightarrow v=\sqrt{\frac{2Pt}{m}} \end{gathered}$$

Hence, option C is the correct option.