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 of finding the ratio of average velocity to instantaneous velocity for a particle moving in a circular path, we will follow these steps:

### Step 1: Understand the Motion
The particle is moving in a circular path with a constant speed and rotates through an angle of \(\theta = \frac{\pi}{2}\) radians (90 degrees). This means the particle moves from point A to point B along a quarter of the circular path.

### Step 2: Identify the Points
Let:
- O be the center of the circular path.
- A be the initial position of the particle.
- B be the position of the particle after rotating through \(\frac{\pi}{2}\).

### Step 3: Determine the Displacement
The displacement \(AB\) can be calculated using the geometry of the situation. Since the particle moves from A to B, which are at right angles to each other, we can form a right triangle:
- The radius \(OA = R\) (where R is the radius of the circular path).
- The length \(OB = R\).

Using the Pythagorean theorem:
\[ AB = \sqrt{OA^2 + OB^2} = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \]

### Step 4: Calculate the Average Velocity
The average velocity \(V_{avg}\) is defined as the total displacement divided by the time taken. The time taken to move through a quarter of the circular path is \(\frac{T}{4}\), where \(T\) is the time for a complete revolution.

The average velocity can be calculated as:
\[ V_{avg} = \frac{AB}{\text{time}} = \frac{R\sqrt{2}}{\frac{T}{4}} = \frac{4R\sqrt{2}}{T} \]

### Step 5: Calculate the Instantaneous Velocity
The instantaneous velocity \(V_{inst}\) is defined as the speed of the particle at any point in time. For a complete revolution, the instantaneous velocity can be calculated as:
\[ V_{inst} = \frac{\text{Circumference}}{\text{Time}} = \frac{2\pi R}{T} \]

### Step 6: Find the Ratio of Average Velocity to Instantaneous Velocity
Now, we need to find the ratio of average velocity to instantaneous velocity:
\[ \text{Ratio} = \frac{V_{avg}}{V_{inst}} = \frac{\frac{4R\sqrt{2}}{T}}{\frac{2\pi R}{T}} = \frac{4R\sqrt{2}}{2\pi R} \]

After simplifying, we get:
\[ \text{Ratio} = \frac{4\sqrt{2}}{2\pi} = \frac{2\sqrt{2}}{\pi} \]

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