Question

Sohail cycles on a circular track in anticlockwise direction as shown in the figure. He travels with a speed ‘v’ to cover the path AB, next with speed ‘2v’ from B to C and with a speed of ‘3v’ from C to A. What is the average speed for the total journey?

Answer 1

Use the distance formula of circle i.e.
$d=2\pi r\dfrac{\theta }{360{}^\circ }$
Where d is the distance travelled in $\theta$ angle and r is the radius of the circle.
When the distance travelled in each segment of the circle is found use the given velocities to calculate the time taken to travel each segment.
After that use the average speed formula:
$\text{Average speed}=\dfrac{\text{Total distance}}{\text{Total time}}$
Or
${{v}_{avg}}=\dfrac{D}{T}$
Where D is the total distance travelled and T is the total time taken to travel the same distance.

Complete step by step solution

Distance travelled from A to B
$\begin{aligned} & {{d}_{1}}=2\pi r\dfrac{60{}^\circ }{360{}^\circ } \\\ & {{d}_{1}}=\dfrac{2\pi r}{6} \\\ \end{aligned}$
Time taken from A to B
$\begin{aligned} & {{t}_{1}}=\dfrac{{{d}_{1}}}{{{v}_{1}}} \\\ & {{t}_{1}}=\dfrac{2\pi r}{6v} \\\ & {{t}_{1}}=\dfrac{\pi r}{3v} \\\ \end{aligned}$
Distance travelled from B to C
$\begin{aligned} & {{d}_{2}}=2\pi r\dfrac{120{}^\circ }{360{}^\circ } \\\ & {{d}_{2}}=\dfrac{2\pi r}{3} \\\ \end{aligned}$
Time taken from B to C
$\begin{aligned} & {{t}_{2}}=\dfrac{{{d}_{2}}}{{{v}_{2}}} \\\ & {{t}_{2}}=\dfrac{2\pi r}{3(2v)} \\\ & {{t}_{2}}=\dfrac{\pi r}{3v} \\\ \end{aligned}$
Distance travelled from C to A
$\begin{aligned} & {{d}_{3}}=2\pi r\dfrac{180{}^\circ }{360{}^\circ } \\\ & {{d}_{3}}=\dfrac{2\pi r}{2} \\\ & {{d}_{3}}=\pi r \\\ \end{aligned}$
Time taken from C to A
$\begin{aligned} & {{t}_{3}}=\dfrac{{{d}_{3}}}{{{v}_{3}}} \\\ & {{t}_{3}}=\dfrac{\pi r}{3v} \\\ \end{aligned}$
Average speed of the total journey:
$\text{Average speed}=\dfrac{\text{Total distance}}{\text{Total time}}$
$\begin{aligned} & {{v}_{avg}}=\dfrac{{{d}_{1}}+{{d}_{2}}+{{d}_{3}}}{{{t}_{1}}+{{t}_{2}}+{{t}_{3}}} \\\ & =\dfrac{2\pi r}{\dfrac{\pi r}{3v}+\dfrac{\pi r}{3v}+\dfrac{\pi r}{3v}} \\\ & =\dfrac{2\pi r}{\dfrac{3\pi r}{3v}} \\\ & {{v}_{avg}}=2v \\\ \end{aligned}$
Therefore, the average speed of the total journey is 2v.

Note
Always remember to calculate the time of each segment separately to avoid the confusion during the calculation of average speed collectively at last. Distance travelled during motion and displacement can be different. So while calculating distance we have to consider the full path of journey.