A particle is moving in a circular path of radius r with constant speed v . At a particular instant particle is at point A and then it reaches a point B which is diametrically opposite to point A . Average acceleration during this interval is .

Options

A. `(v^(2))/(pir)`
B. `(3v^(2))/(pi r)`
C. 0
D. `(2v^(2))/(pi r)`

Explanation

To find the average acceleration of a particle moving in a circular path from point A to point B (which is diametrically opposite to A), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**: The particle is moving in a circular path with a constant speed \( v \) and radius \( r \). The points A and B are on opposite ends of the diameter of the circle.

2. **Determine the Displacement**: The displacement of the particle when it moves from point A to point B is the straight line distance between these two points. Since A and B are diametrically opposite, the displacement \( \Delta \vec{s} \) is equal to the diameter of the circle:
\[ \Delta \vec{s} = 2r \]

3. **Calculate the Time Taken**: The time \( t \) taken to move from A to B can be calculated using the formula for the distance traveled in uniform circular motion. The distance traveled from A to B is half the circumference of the circle:
\[ \text{Distance from A to B} = \pi r \]
The time taken \( t \) is given by:
\[ t = \frac{\text{Distance}}{\text{Speed}} = \frac{\pi r}{v} \]

4. **Calculate the Average Velocity**: The average velocity \( \vec{v}_{avg} \) is given by the total displacement divided by the total time taken:
\[ \vec{v}_{avg} = \frac{\Delta \vec{s}}{t} = \frac{2r}{\frac{\pi r}{v}} = \frac{2v}{\pi} \]

5. **Determine the Initial and Final Velocity**:
- The initial velocity \( \vec{v}_i \) at point A is tangential to the circle.
- The final velocity \( \vec{v}_f \) at point B is also tangential but in the opposite direction. Since the speed is constant, the magnitudes are the same, but the directions are opposite:
\[ \vec{v}_i = v \hat{i}, \quad \vec{v}_f = -v \hat{i} \]

6. **Calculate the Change in Velocity**: The change in velocity \( \Delta \vec{v} \) is:
\[ \Delta \vec{v} = \vec{v}_f - \vec{v}_i = -v \hat{i} - v \hat{i} = -2v \hat{i} \]

7. **Calculate the Average Acceleration**: The average acceleration \( \vec{a}_{avg} \) is given by the change in velocity divided by the time taken:
\[ \vec{a}_{avg} = \frac{\Delta \vec{v}}{t} = \frac{-2v \hat{i}}{\frac{\pi r}{v}} = \frac{-2v^2}{\pi r} \hat{i} \]

### Final Answer:
The average acceleration during the interval from point A to point B is:
\[ \vec{a}_{avg} = \frac{-2v^2}{\pi r} \hat{i} \]