A person moves along a circular path by a distance equal to half the circumference in a given time. The ratio of his average speed to his average velocity is :
The distance covered by the person is half the circumference, $d = \frac{1}{2} C = \pi R$.
Let the time taken be $t$.
Average speed is defined as the total distance traveled divided by the total time taken.
Average speed = $\frac{d}{t} = \frac{\pi R}{t}$.
The person moves along a circular path by a distance equal to half the circumference. This means the person starts at one point on the circle and ends at the diametrically opposite point.
Let the starting point be A and the ending point be B, where AB is a diameter of the circle.
The displacement is the shortest straight-line distance from the initial position to the final position. In this case, the displacement is the length of the diameter.
Displacement = $2R$.
Average velocity is defined as the total displacement divided by the total time taken.
Average velocity = $\frac{\text{Displacement}}{t} = \frac{2R}{t}$.
The ratio of average speed to average velocity is:
Ratio = $\frac{\text{Average speed}}{\text{Average velocity}} = \frac{\pi R / t}{2R / t} = \frac{\pi R}{t} \times \frac{t}{2R} = \frac{\pi}{2}$.
The value $\frac{\pi}{2}$ is equivalent to $0.5\pi$.