Question

A particle moves along two paths $AOB$ and $AB$ of an arc $(r=\pi \text{m})$ as shown in the figure.
What will be the average speed of particle for the paths $AOB$ and $AB$ respectively, if it completes the journey from $A$ to $B$ in $\frac{\pi }{3}\text{sec}$ in each way?

• A. $6\text{m/s}$ and $12\text{m/s}$ respectively
• B. $2.5\text{m/s}$ and $\frac{\pi }{6}\text{m/s}$ respectively
• C. $6\text{m/s}$ and $2\pi \text{m/s}$ respectively.
• D. $2\text{m/s}$ and $4.5\text{m/s}$ respectively

Answer 1

The correct option is C $6\text{m/s}$ and $2\pi \text{m/s}$ respectively.

Given: Time interval, $\Delta t=\frac{\pi }{3}\text{sec}$
In case of path $AOB$,
Distance covered, $S=r+r=2r=2\pi \text{m}$
So, average speed is given by
$v_{\text{avg}}=\frac{2\pi }{\frac{\pi }{3}}=6\text{m/s}$

In case of path $AB$,
Distance covered, $S=\frac{2\pi r\theta }{{360}^{\circ }}\text{m}$
$\Rightarrow S=\frac{2\pi \times \pi \times {120}^{\circ }}{{360}^{\circ }}=\frac{2{\pi }^2}{3}\text{m}$
So, average speed is given by
$v_{\text{avg}}=\frac{2{\pi }^2}{3\times \frac{\pi }{3}}=2\pi \text{m/s}$
Hence the average speed for paths $AOB$ and $AB$ is $6\text{m/s}$ and $2\pi \text{m/s}$ respectively.