A cyclist moves along a circular path of radius 70 m. If he completes one round in 11 s, calculate (i) total length of path, (ii) magnitude of the displacement, (iii) average speed, and (iv) magnitude of average velocity.

Answer

To solve the problem step by step, we will address each part of the question systematically.

### Given Data:
- Radius of the circular path, \( r = 70 \, \text{m} \)
- Time for one complete round, \( t = 11 \, \text{s} \)

### (i) Total Length of Path (Circumference of the Circle)
The total length of the path is the circumference of the circular path, which can be calculated using the formula:
\[ C = 2 \pi r \]
Substituting the given radius:
\[ C = 2 \pi \times 70 \, \text{m} = 140 \pi \, \text{m} \]

### (ii) Magnitude of the Displacement
Displacement is defined as the shortest distance from the initial position to the final position. Since the cyclist completes one full round and returns to the starting point, the initial and final positions are the same. Therefore, the magnitude of the displacement is:
\[ \text{Displacement} = 0 \, \text{m} \]

### (iii) Average Speed
Average speed is defined as the total distance traveled divided by the total time taken. The total distance traveled in one round is the circumference, which we calculated as \( 140 \pi \, \text{m} \). Thus, the average speed can be calculated as:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Time}} = \frac{140 \pi}{11} \, \text{m/s} \]

### (iv) Magnitude of Average Velocity
Average velocity is defined as the net displacement divided by the total time taken. Since the net displacement is \( 0 \, \text{m} \) (as established in part (ii)), the average velocity is:
\[ \text{Average Velocity} = \frac{\text{Net Displacement}}{\text{Time}} = \frac{0}{11} = 0 \, \text{m/s} \]

### Summary of Results:
1. Total Length of Path: \( 140 \pi \, \text{m} \)
2. Magnitude of Displacement: \( 0 \, \text{m} \)
3. Average Speed: \( \frac{140 \pi}{11} \, \text{m/s} \)
4. Magnitude of Average Velocity: \( 0 \, \text{m/s} \)