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**Hint:**We are given that the path followed is circular. As radius is given we can find the circumference that is the distance travelled. Average speed is the ratio of total distance travelled to the total time taken to cover that distance. Average velocity is the total displacement divided by the total time taken.

**Complete step-by-step solution:**Let us first find the total distance covered in one revolution. As the radius $r$ is given as $r = 100\,m$ , therefore, the circumference $C$ will be:

$2\pi r = 2 \times \pi \times 100\,m$

$ \Rightarrow c = 200\pi \,m$

This is the distance travelled in one revolution.

( $i$ ) The average speed is the ratio of the total distance travelled to the total time taken to travel that distance. We are given that the cyclist completes one revolution in $4$ minutes. This implies that the cyclist covers the distance of $d = 200\pi \,m$ in $4$ minutes that is $4 \times 60 = 240\,s$ . The average speed ${s_{avg}}$ will be given as:

${s_{avg}} = \dfrac{{200\pi \,m}}{{240\,s}}$

$ \Rightarrow {s_{avg}} = \dfrac{{5\pi }}{6}m{s^{ - 1}}$

$ \Rightarrow {s_{avg}} = 2.618\,m{s^{ - 1}}$

This is the average speed of the cyclist.

( $ii$ ) average velocity in one full revolution is the ratio of the total displacement covered by the cyclist in one revolution to the total time taken to cover that displacement.

We know that displacement is the shortest distance between the starting point and the ending point. As the cyclist moves in a complete revolution, he reaches again to the starting point. As a result, the shortest distance between the starting point and the ending point will be zero.

**Therefore, the average velocity will also be zero as the displacement is zero.**

**Note:**Remember that average speed and average velocity are different. For calculating the speed, we consider the distance travelled and for calculating the velocity we consider the displacement. Remember that displacement is the shortest distance between the initial and final points.

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