Table of Contents
Problem: Cyclist’s Round Trip
A cyclist starts from the centre \( O \) of a circular park of radius 1 km, reaches the edge \( P \) of the park, then cycles along the circumference, and returns to the centre along \( QO \). If the round trip takes 10 min, what is the
- net displacement,
- average velocity, and
- average speed of the cyclist?
Solution:
(a) Net Displacement
Displacement is the shortest distance between the initial and final positions of the cyclist. Since the cyclist returns to the center \( O \) after completing the round trip, the initial and final positions are the same.
Net Displacement: \( 0 \) km
(b) Average Velocity
Average velocity is defined as the total displacement divided by the total time taken. Since the net displacement is zero (the cyclist returns to the starting point), the average velocity will also be zero.
Average Velocity: \( 0 \) km/min
(c) Average Speed
Average speed is defined as the total distance traveled divided by the total time taken. Let’s calculate the total distance traveled by the cyclist.
- From \( O \) to \( P \): This is a straight line from the center to the edge of the circular park, which is the radius of the circle.
Distance: \( 1 \) km - Along the Circumference from \( P \) to \( Q \): This is a quarter of the circumference of the circle (since the cyclist travels from one point on the edge to another point that is a quarter of the circle away).
Circumference of the circle: \( 2 \pi \times 1 \) km
Quarter circumference: \( \frac{\pi}{2} \) km - From \( Q \) to \( O \): This is again a straight line from the edge to the center, which is the radius of the circle.
Distance: \( 1 \) km
Total Distance Traveled: \( 1 \) km (O to P) + \( \frac{\pi}{2} \) km (P to Q) + \( 1 \) km (Q to O) = \( 2 + \frac{\pi}{2} \) km
Total Time Taken: \( 10 \) min
Average Speed: Total distance traveled / Total time taken
\[
\text{Average Speed} = \frac{2 + \frac{\pi}{2}}{10} \text{ km/min}
\]
Simplifying further:
\[
\text{Average Speed} = \frac{4 + \pi}{20} \text{ km/min} = \frac{4 + \pi}{20} \text{ km/min}
\]