Question

The engine of a truck moving along a straight road delivers constant power. The distance travelled by the truck in time $t$ is proportional to

• A. $t$
• B. $t^2$
• C. $\sqrt{t}$
• D. $t^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Answer 1

The correct option is D

$t^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Step 1: Given data

Power $\left(P\right)$ of the engine is constant.

The time taken by the truck to travel is, $t$

Step 2: Finding the proportional of $t$

The power $P$ can be written as $P=Fv=\mathrm{m}av$ [where $F$ is the force, $v$ is the velocity, $\mathrm{m}$ is the mass and $a$ is the acceleration]

On further expanding, $P=Fv=\mathrm{m}av=\mathrm{m}\frac{\mathrm{v}}{\mathrm{t}}v=\frac{\mathrm{m}{v}^2}{t}\left[\because a=\frac{\mathrm{v}}{\mathrm{t}}\right]$.

Now, using the relation $v=\frac{S}{t}$ where $S$ is the displacement it can be written $P=\frac{\mathrm{m}{v}^2}{t}=\frac{{\mathrm{mS}}^2}{t^3}$.

It is known that $P$ and $\mathrm{m}$ are constants.

So, it is established that $S^2$ is directly proportional to $t^3$ or $S$ is directly proportional to $t^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$.

Hence, option (d) is correct.