A person runs along a circular path of radius 5 m. If he completes half of the circle find the magnitude of the displacement vector, How far the person ran ?

Options

A. 10 m, `5pi` m
B. `5pi m`, 10 m
C. `5pi` m , 19m
D. 14 m, 17 m

Answer

To solve the problem, we need to find two things: the magnitude of the displacement vector and the distance the person ran along the circular path.

### Step-by-Step Solution:

1. **Understanding the Circular Path:**
- The radius of the circular path is given as \( r = 5 \) meters.
- The person completes half of the circle.

2. **Finding the Displacement Vector:**
- When the person runs from point A to point B (which are endpoints of a diameter of the circle), the displacement vector \( \vec{AB} \) is the straight line connecting these two points.
- The length of the displacement vector is equal to the diameter of the circle.
- The diameter \( d \) can be calculated using the formula:
\[ d = 2 \times r = 2 \times 5 \, \text{m} = 10 \, \text{m} \]
- Thus, the magnitude of the displacement vector is \( 10 \, \text{m} \).

3. **Finding the Distance Ran:**
- The distance the person ran is half of the circumference of the circle.
- The circumference \( C \) of a circle is given by:
\[ C = 2 \pi r \]
- Therefore, half of the circumference is:
\[ \text{Distance ran} = \frac{1}{2} C = \frac{1}{2} \times 2 \pi r = \pi r \]
- Substituting the value of \( r \):
\[ \text{Distance ran} = \pi \times 5 \, \text{m} = 5 \pi \, \text{m} \]

### Final Answers:
- The magnitude of the displacement vector is \( 10 \, \text{m} \).
- The distance the person ran is \( 5 \pi \, \text{m} \).