A body goes `30km` south and then `40km` east. What will be the displacement from initial point ?

Options

A. `50km, 37^(@)` South of East
B. `30km, 37^(@)` South of East
C. `40km,53^(@)` South of East
D. `70km, 53^(@)` South of East

Explanation

To find the displacement of a body that goes 30 km south and then 40 km east, we can follow these steps:

### Step 1: Understand the Movement
The body first moves 30 km in the south direction and then moves 40 km in the east direction. We can visualize this movement on a coordinate system where:
- The south direction is negative y-axis.
- The east direction is positive x-axis.

### Step 2: Represent the Movements
1. The initial point (O) can be considered as the origin (0,0).
2. After moving 30 km south, the new position (A) will be (0, -30).
3. After moving 40 km east from point A, the final position (B) will be (40, -30).

### Step 3: Calculate the Displacement
Displacement is the straight-line distance from the initial point to the final point. We can calculate this using the Pythagorean theorem.

The displacement (D) can be calculated as:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where:
- \( (x_1, y_1) \) is the initial point (0, 0).
- \( (x_2, y_2) \) is the final point (40, -30).

Substituting the values:
\[ D = \sqrt{(40 - 0)^2 + (-30 - 0)^2} \]
\[ D = \sqrt{40^2 + (-30)^2} \]
\[ D = \sqrt{1600 + 900} \]
\[ D = \sqrt{2500} \]
\[ D = 50 \text{ km} \]

### Step 4: Determine the Direction of Displacement
To find the angle of displacement with respect to the south direction, we can use trigonometry. The angle \( \theta \) can be calculated using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{40}{30} \]
Thus,
\[ \theta = \tan^{-1}\left(\frac{40}{30}\right) \]

Calculating \( \theta \):
\[ \theta \approx 53.13^\circ \]

Since this angle is measured from the east towards the south, the angle with respect to the south direction (denoted as \( \phi \)) is:
\[ \phi = 90^\circ - \theta \]
\[ \phi = 90^\circ - 53.13^\circ \]
\[ \phi \approx 36.87^\circ \]

### Final Result
The displacement from the initial point is:
- **Displacement**: 50 km
- **Direction**: 37 degrees south of east.