A body moves along a quadrant of a circle of radius (r). The displacement and distance traveled are.

Options

A. ` r sqrt 2 , pi r //2`
B. ` r _2 pi r`
C. ` 2 sqrt 2 r , pi r`
D. ` r_1 pi r//2`

Explanation

To solve the problem of finding the displacement and distance traveled by a body moving along a quadrant of a circle of radius \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Movement**:
- The body is moving along a quadrant of a circle, which means it travels along a quarter of the circular path.

2. **Drawing the Circle**:
- Draw a circle with radius \( r \). Mark the starting point (Point A) at the bottom of the quadrant (0 degrees) and the endpoint (Point B) at the top of the quadrant (90 degrees).

3. **Calculating Displacement**:
- Displacement is defined as the shortest distance between the initial and final positions.
- The initial position (Point A) is at the coordinates (0, 0) and the final position (Point B) is at the coordinates (r, r) in a Cartesian coordinate system.
- To find the displacement, we use the distance formula:
\[ \text{Displacement} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Substituting the coordinates:
\[ \text{Displacement} = \sqrt{(r - 0)^2 + (r - 0)^2} = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} \]

4. **Calculating Distance Traveled**:
- The distance traveled is the length of the path covered by the body along the quadrant.
- The circumference of a full circle is \( 2\pi r \). Since the body moves along a quadrant (which is one-fourth of the circle), the distance traveled is:
\[ \text{Distance} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2} \]

5. **Final Answers**:
- **Displacement**: \( r\sqrt{2} \)
- **Distance Traveled**: \( \frac{\pi r}{2} \)

### Summary:
- The displacement of the body moving along a quadrant of a circle of radius \( r \) is \( r\sqrt{2} \).
- The distance traveled by the body is \( \frac{\pi r}{2} \).