A cyclist starts from the centre O of circular path of radius 10 m, covers the radius of circular path and reaches at point X on circumference, then cycles along the semi-circular path and reaches the point Y. If he takes 10 minutes to go from O to Y via X, then the net displacement and average speed of the cyclist would be

Options

A. `Zero, 10 ms^(-1)`
B. `10 m, ((1+ pi)/60)ms^(-1)`
C. `10 m, ((1+ pi)/6)ms^(-1)`
D. `10 m, ((1- pi)/60)ms^(-1)`

Answer

To solve the problem step-by-step, we will determine the net displacement and average speed of the cyclist as described in the question.

### Step 1: Understand the Path Taken by the Cyclist
The cyclist starts from the center O of a circular path with a radius of 10 m. He first travels to point X on the circumference of the circle, which is a distance equal to the radius (10 m). Then, he cycles along the semicircular path to reach point Y.

### Step 2: Calculate the Net Displacement
The net displacement is the straight-line distance from the starting point O to the endpoint Y. Since Y is directly opposite to O on the circumference of the circle, the displacement is equal to the radius of the circle.

- **Displacement (O to Y)** = Radius of the circle = 10 m

### Step 3: Calculate the Total Distance Traveled
The total distance traveled by the cyclist consists of two parts:
1. Distance from O to X (which is the radius of the circle).
2. Distance from X to Y (which is the length of the semicircular path).

- Distance OX = 10 m (radius)
- For the semicircular path from X to Y:
- The angle for a semicircle is π radians.
- The arc length (distance from X to Y) can be calculated using the formula:
\[ \text{Arc Length} = \theta \times r \]
- Here, \(\theta = \pi\) and \(r = 10\) m.
- Therefore,
\[ \text{Arc Length} = \pi \times 10 = 10\pi \text{ m} \]

- **Total Distance** = Distance OX + Arc Length XY
\[ \text{Total Distance} = 10 + 10\pi \text{ m} \]

### Step 4: Calculate the Average Speed
The average speed is calculated using the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
- The total time taken is given as 10 minutes, which we convert to seconds:
\[ \text{Total Time} = 10 \times 60 = 600 \text{ seconds} \]

- Now substituting the values into the average speed formula:
\[ \text{Average Speed} = \frac{10 + 10\pi}{600} \]
- Simplifying this gives:
\[ \text{Average Speed} = \frac{10(1 + \pi)}{600} = \frac{1 + \pi}{60} \text{ m/s} \]

### Final Results
- **Net Displacement** = 10 m
- **Average Speed** = \(\frac{1 + \pi}{60}\) m/s