A person goes 10 km north and 20 km east. What will be the displacement from initial point?

Options

A. 22.36 km
B. 2 km
C. 5 km
D. 20 km

Answer

To find the displacement from the initial point after a person travels 10 km north and then 20 km east, we can follow these steps:

### Step 1: Understand the Movement
The person first travels 10 km north and then 20 km east. We can represent these movements using a coordinate system where north is along the positive y-axis and east is along the positive x-axis.

### Step 2: Represent the Movements
- The movement north (10 km) can be represented as a vector: \( \vec{A} = 0 \hat{i} + 10 \hat{j} \) (where \( \hat{i} \) is the unit vector in the east direction and \( \hat{j} \) is the unit vector in the north direction).
- The movement east (20 km) can be represented as a vector: \( \vec{B} = 20 \hat{i} + 0 \hat{j} \).

### Step 3: Determine the Final Position
To find the final position, we can add the two vectors:
\[ \vec{R} = \vec{A} + \vec{B} = (0 \hat{i} + 10 \hat{j}) + (20 \hat{i} + 0 \hat{j}) = 20 \hat{i} + 10 \hat{j} \]
This means the final position is 20 km east and 10 km north of the starting point.

### Step 4: Calculate the Displacement
Displacement is the straight-line distance from the initial point to the final point. We can use the Pythagorean theorem to calculate this:
\[ \text{Displacement} = \sqrt{(20 \text{ km})^2 + (10 \text{ km})^2} \]
Calculating the squares:
\[ = \sqrt{400 + 100} = \sqrt{500} \]
\[ = \sqrt{100 \times 5} = 10 \sqrt{5} \text{ km} \]

### Step 5: Approximate the Value
To find the numerical value, we can calculate:
\[ \sqrt{5} \approx 2.236 \]
Thus,
\[ 10 \sqrt{5} \approx 10 \times 2.236 = 22.36 \text{ km} \]

### Final Answer
The displacement from the initial point is approximately **22.36 km**.

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