A bird is flying towards north with a velocity `40 km h^-1` and a train is moving with velocity `40 km h^-1` towards east. What is the velocity of the bird boted by a man in the train ?

Options

A. `40 sqrt(2) km h^-1 N - E`
B. `40 sqrt(2) km h^-1 S - E`
C. `40 sqrt(2) km h^-1 N - W`
D. `40 sqrt(2) km h^-1 S - W`

Explanation

To solve the problem of finding the velocity of the bird as observed by a man in the train, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Velocities**:
- The bird is flying north with a velocity of \( V_B = 40 \, \text{km/h} \).
- The train is moving east with a velocity of \( V_T = 40 \, \text{km/h} \).

2. **Set Up the Coordinate System**:
- Let the north direction be along the positive y-axis.
- Let the east direction be along the positive x-axis.

3. **Express Velocities in Vector Form**:
- The velocity of the bird can be represented as a vector:
\[ \vec{V_B} = 0 \hat{i} + 40 \hat{j} \, \text{km/h} \]
- The velocity of the train can be represented as a vector:
\[ \vec{V_T} = 40 \hat{i} + 0 \hat{j} \, \text{km/h} \]

4. **Calculate the Relative Velocity**:
- The velocity of the bird with respect to the train (or the man in the train) is given by:
\[ \vec{V_{B/T}} = \vec{V_B} - \vec{V_T} \]
- Substituting the vectors:
\[ \vec{V_{B/T}} = (0 \hat{i} + 40 \hat{j}) - (40 \hat{i} + 0 \hat{j}) = -40 \hat{i} + 40 \hat{j} \]

5. **Calculate the Magnitude of the Relative Velocity**:
- The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V_{B/T}}| = \sqrt{(-40)^2 + (40)^2} = \sqrt{1600 + 1600} = \sqrt{3200} = 40\sqrt{2} \, \text{km/h} \]

6. **Determine the Direction of the Relative Velocity**:
- To find the angle \( \phi \) that the velocity vector makes with the east direction, we can use:
\[ \tan \phi = \frac{V_B}{|V_T|} = \frac{40}{40} = 1 \]
- Thus, \( \phi = 45^\circ \).

7. **Identify the Direction**:
- Since the bird's velocity relative to the train is in the negative x-direction (west) and positive y-direction (north), the direction of the relative velocity is northwest.

### Final Answer:
The velocity of the bird as observed by a man in the train is:
- Magnitude: \( 40\sqrt{2} \, \text{km/h} \)
- Direction: Northwest.