A particle is moving with a constant speed v in a circle. What is the magnitude of average velocity after half rotation?

Options

A. 2v
B. `v(v/pi)`
C. `v/2`
D. `v/(2pi)`

Explanation

To find the magnitude of the average velocity of a particle moving with constant speed \( v \) in a circle after half a rotation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**:
The particle moves in a circular path. Let's denote the center of the circle as point O and the starting point as point A. After half a rotation, the particle will reach point B, which is directly opposite point A on the circle.

2. **Determine the Displacement**:
The displacement is the straight-line distance between the starting point A and the endpoint B. Since points A and B are diametrically opposite in the circle, the displacement can be calculated as the diameter of the circle.
\[ \text{Displacement} = AB = 2r \]
where \( r \) is the radius of the circle.

3. **Calculate the Time Taken**:
To find the average velocity, we need to determine the time taken to move from A to B. The distance traveled in half a rotation is half the circumference of the circle.
\[ \text{Distance traveled} = \frac{1}{2} \times 2\pi r = \pi r \]
The time taken to travel this distance at speed \( v \) is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{\pi r}{v} \]

4. **Calculate the Average Velocity**:
The average velocity is defined as the total displacement divided by the total time taken.
\[ \text{Average Velocity} = \frac{\text{Displacement}}{\text{Time}} = \frac{2r}{\frac{\pi r}{v}} = \frac{2r \cdot v}{\pi r} = \frac{2v}{\pi} \]

5. **Final Result**:
Therefore, the magnitude of the average velocity after half a rotation is:
\[ \text{Average Velocity} = \frac{2v}{\pi} \]