A bird is flying towards south with a velocity 40km/h and a train is moving with a velocity 40 km/h towards east. What is the velocity of the bird w.r.t. an obserber in the train ?

Options

A. `40sqrt2"km/h. N-E"`
B. `40sqrt2 "km/h. S-E"`
C. `40sqrt2" km/h. S-W"`
D. `40sqrt2" km/h. N-W"`

Explanation

To solve the problem of finding the velocity of the bird with respect to an observer in the train, we can follow these steps:

### Step-by-Step Solution:

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

2. **Set Up a Coordinate System**:
- Define a coordinate system where:
- North is in the positive y-direction,
- East is in the positive x-direction,
- South is in the negative y-direction,
- West is in the negative x-direction.

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

4. **Calculate the Velocity of the Bird with Respect to the Train**:
- The velocity of the bird with respect to the train \( \vec{V_{b/t}} \) is given by:
\[ \vec{V_{b/t}} = \vec{V_b} - \vec{V_t} \]
- Substituting the values:
\[ \vec{V_{b/t}} = (0 \, \hat{i}, -40 \, \hat{j}) - (40 \, \hat{i}, 0 \, \hat{j}) = (-40 \, \hat{i}, -40 \, \hat{j}) \]

5. **Calculate the Magnitude of the Resultant Velocity**:
- The magnitude of the resultant 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 Resultant Velocity**:
- The direction of the velocity vector \( \vec{V_{b/t}} = (-40, -40) \) indicates that the bird is moving in the south-west direction.

### Final Answer:
The velocity of the bird with respect to an observer in the train is:
\[ 40\sqrt{2} \, \text{km/h} \text{ in the south-west direction.} \]

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