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 1: Understand the Motion
The particle travels along a circular path and covers 3/4 of the circumference of the circle. The initial position and final position of the particle will help us determine the displacement.

### Step 2: Determine the Displacement
The displacement is defined as the straight-line distance between the initial and final positions of the particle.

- The initial position can be considered at point A (0 degrees) on the circle.
- After traversing 3/4 of the circle, the particle reaches point B (270 degrees).

### Step 3: Visualize the Geometry
In a circle of radius R:
- The coordinates of point A (initial position) are (R, 0).
- The coordinates of point B (final position) are (0, -R).

### Step 4: Calculate the Displacement
The displacement vector can be calculated using the distance formula between points A and B:

\[ \text{Displacement} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \]

Substituting the coordinates:
\[ \text{Displacement} = \sqrt{(0 - R)^2 + (-R - 0)^2} = \sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2} \]

### Step 5: Calculate the Average Velocity
The average velocity is defined as the total displacement divided by the total time taken:

\[ \text{Average Velocity} = \frac{\text{Displacement}}{\text{Time}} = \frac{R\sqrt{2}}{t} \]

### Final Answer
Thus, the magnitude of the average velocity of the particle in this time interval is:

\[ \frac{R\sqrt{2}}{t} \]