A particle is moving with a constant speed in a circular path. Find the ratio of average velocity to its instantaneous velocity when the particle rotates an angle `theta =((pi)/(2))` .

Explanation

To solve the problem, we need to find the ratio of average velocity to instantaneous velocity for a particle moving in a circular path when it rotates through an angle of \( \theta = \frac{\pi}{2} \) radians (90 degrees).

### Step-by-Step Solution:

1. **Understanding the Circular Motion**:
- The particle moves in a circular path with a constant speed.
- The radius of the circular path is denoted as \( R \).
- When the particle rotates through \( \theta = \frac{\pi}{2} \), it moves from point \( O \) to point \( M \) (90 degrees along the circular path).

2. **Finding the Displacement**:
- The displacement \( OM \) when the particle moves through \( \frac{\pi}{2} \) is the straight line connecting the starting point \( O \) and the endpoint \( M \).
- Using the Pythagorean theorem, the displacement can be calculated as:
\[ OM = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \]

3. **Calculating Time Taken**:
- The total time period \( T \) for one complete revolution is \( T \).
- Since the particle moves through \( \frac{\pi}{2} \) (or 90 degrees), the time taken to cover this angle is:
\[ t = \frac{T}{4} \]

4. **Finding Average Velocity**:
- Average velocity \( V_{avg} \) is defined as the total displacement divided by the total time taken.
- Therefore, the average velocity when the particle moves from \( O \) to \( M \) is:
\[ V_{avg} = \frac{OM}{t} = \frac{R\sqrt{2}}{\frac{T}{4}} = \frac{4R\sqrt{2}}{T} \]

5. **Finding Instantaneous Velocity**:
- The instantaneous velocity \( V_{inst} \) is defined as the speed of the particle at any point in time.
- The speed \( V \) for circular motion is given by:
\[ V = \frac{2\pi R}{T} \]
- Thus, the instantaneous velocity is:
\[ V_{inst} = \frac{2\pi R}{T} \]

6. **Calculating the Ratio**:
- Now, we can find the ratio of average velocity to instantaneous velocity:
\[ \text{Ratio} = \frac{V_{avg}}{V_{inst}} = \frac{\frac{4R\sqrt{2}}{T}}{\frac{2\pi R}{T}} = \frac{4\sqrt{2}}{2\pi} = \frac{2\sqrt{2}}{\pi} \]

### Final Answer:
The ratio of average velocity to instantaneous velocity when the particle rotates through an angle of \( \theta = \frac{\pi}{2} \) is:
\[ \frac{2\sqrt{2}}{\pi} \]