A point traversed 3/4 th of the circle of radius R in time t. The magnitude of the average velocity of the particle in this time interval is

Options

A. `(pi R)/(t)`
B. `(3pi R)/(2t)`
C. `(R sqrt(2))/(t)`
D. `(R )/(sqrt(2)t)`

Explanation

To find the magnitude of the average velocity of a particle that traverses 3/4 of a circle of radius R in time t, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**: The particle travels along the circumference of a circle. Given that it covers 3/4 of the circle, we need to identify the initial and final positions of the particle.

2. **Identify Initial and Final Points**:
- Let's denote the center of the circle as point O.
- Assume the particle starts at point A (0 degrees) and moves to point B (270 degrees) after covering 3/4 of the circle.

3. **Determine Displacement**:
- Displacement is defined as the shortest distance between the initial and final points.
- The initial point A is at coordinates (R, 0) and the final point B is at coordinates (0, -R).

4. **Calculate the Displacement Vector**:
- The displacement vector from A to B can be calculated as:
\[ \text{Displacement} = \text{Final Position} - \text{Initial Position} = (0, -R) - (R, 0) = (-R, -R) \]

5. **Calculate the Magnitude of Displacement**:
- The magnitude of the displacement can be found using the Pythagorean theorem:
\[ \text{Magnitude of Displacement} = \sqrt{(-R)^2 + (-R)^2} = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \]

6. **Calculate Average Velocity**:
- Average velocity is defined as the total displacement divided by the total time taken:
\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{R\sqrt{2}}{t} \]

### Final Answer:
The magnitude of the average velocity of the particle is:
\[ \frac{R\sqrt{2}}{t} \]