Question

A body is moved from rest along a straight line by a machine delivering constant power. Time taken by the body to travel a distance $S$ is proportional to:

• A. $S^{\frac{1}{3}}$
• B. $S^{\frac{2}{3}}$
• C. $S^{\frac{1}{2}}$
• D. $S^{\frac{1}{4}}$

Answer 1

The correct option is B

$S^{\frac{2}{3}}$

Step 1: Given data

Distance $=S$

Power is constant.

Step 2: Find time taken by the body

Energy supplied in time $"t"$ sec is,

$E=P\times t$ [Here, $P$ is the power delivered by the machine.]

As the body is moved from rest along a straight line by a machine delivering constant power, energy will be equal to the kinetic energy, $K.E.$

$E=K.E.$

$$\begin{gathered} P\times t=\frac{1}{2}m{v}^2 \\ v\propto \sqrt{t} \end{gathered}$$ [ $m$ is the mass, $v$ is the velocity]

Step 3: Writing the velocity in terms of the displacement of the body,

$\Rightarrow \frac{dS}{dt}=C\sqrt{t}$

Here $C$ is a constant.

$$\begin{gathered} {\int }_0^{S}ds=C{\int }_0^t{t}^{\frac{1}{2}}dt \\ \Rightarrow t^{\frac{3}{2}}=\frac{3S}{2C} \\ \Rightarrow t=S^{\frac{2}{3}}\left(\frac{2}{2C}\right) \\ \Rightarrow t\propto S^{\frac{2}{3}} \end{gathered}$$

Hence, option B is correct.