A person moves 20 m towards north-east then moves 20 m towards west and then again moves 20 m towards north-east and stops. The magnitude of displacement of the person is:

Options

A. `20 sqrt(5-2sqrt(2m))`
B. 20 m
C. `20 sqrt(5+2 sqrt(2m))`
D. None of these

Answer

To find the magnitude of the displacement of the person after moving in the specified directions, we can break down the problem step by step:

### Step 1: Understand the Movement
The person moves in three segments:
1. 20 m towards the north-east.
2. 20 m towards the west.
3. 20 m towards the north-east again.

### Step 2: Calculate the Displacement for Each Segment
1. **First Movement (North-East)**:
- The north-east direction is at a 45-degree angle to both the north and east axes.
- The components of the movement can be calculated using trigonometry:
- \( \text{North Component} = 20 \cos(45^\circ) = 20 \times \frac{1}{\sqrt{2}} = \frac{20}{\sqrt{2}} \)
- \( \text{East Component} = 20 \sin(45^\circ) = 20 \times \frac{1}{\sqrt{2}} = \frac{20}{\sqrt{2}} \)

2. **Second Movement (West)**:
- The person moves 20 m towards the west, which affects only the east-west component:
- \( \text{West Component} = -20 \) m (negative because it is in the opposite direction of east).

3. **Third Movement (North-East)**:
- Similar to the first movement:
- \( \text{North Component} = \frac{20}{\sqrt{2}} \)
- \( \text{East Component} = \frac{20}{\sqrt{2}} \)

### Step 3: Calculate Total Displacement Components
Now, we can sum up the components in both the north-south and east-west directions:
1. **Total North-South Displacement**:
- \( S_v = \frac{20}{\sqrt{2}} + \frac{20}{\sqrt{2}} = 2 \times \frac{20}{\sqrt{2}} = \frac{40}{\sqrt{2}} \)

2. **Total East-West Displacement**:
- \( S_x = \frac{20}{\sqrt{2}} - 20 + \frac{20}{\sqrt{2}} = \frac{40}{\sqrt{2}} - 20 \)

### Step 4: Simplify the Displacement Components
1. **Total East-West Displacement**:
- Convert \( 20 \) to a common denominator:
- \( S_x = \frac{40}{\sqrt{2}} - \frac{20\sqrt{2}}{\sqrt{2}} = \frac{40 - 20\sqrt{2}}{\sqrt{2}} \)

### Step 5: Calculate the Magnitude of the Displacement
Using the Pythagorean theorem:
\[ S = \sqrt{S_x^2 + S_v^2} \]
Substituting the values:
\[ S = \sqrt{\left(\frac{40 - 20\sqrt{2}}{\sqrt{2}}\right)^2 + \left(\frac{40}{\sqrt{2}}\right)^2} \]

### Step 6: Simplifying the Expression
1. Calculate \( S_x^2 \) and \( S_v^2 \):
- \( S_x^2 = \left(\frac{40 - 20\sqrt{2}}{\sqrt{2}}\right)^2 = \frac{(40 - 20\sqrt{2})^2}{2} \)
- \( S_v^2 = \left(\frac{40}{\sqrt{2}}\right)^2 = \frac{1600}{2} = 800 \)

2. Combine the two:
\[ S = \sqrt{\frac{(40 - 20\sqrt{2})^2 + 1600}{2}} \]

### Step 7: Final Calculation
After performing the calculations, we find the magnitude of the displacement.

### Final Answer
The magnitude of the displacement of the person is \( 20 \sqrt{5 - 2\sqrt{2}} \) meters.

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