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

Options

A. `40 sqrt(2)` km/hr. North-East
B. `40 sqrt(2)` km/hr. South-East
C. `40 sqrt(2)` km/hr. South -West
D. `40sqrt(2)` km/hr. North-West

Answer

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

### Step-by-Step Solution:

1. **Define the Directions:**
- Let's establish a coordinate system where:
- North is positive y-direction (j cap).
- East is positive x-direction (i cap).
- South is negative y-direction (-j cap).
- West is negative x-direction (-i cap).

2. **Identify the Velocities:**
- The bird is flying south with a velocity of 40 km/hr. In our coordinate system, this is represented as:
\[ \vec{V}_{\text{bird}} = -40 \, \hat{j} \, \text{km/hr} \]
- The train is moving east with a velocity of 40 km/hr. This is represented as:
\[ \vec{V}_{\text{train}} = 40 \, \hat{i} \, \text{km/hr} \]

3. **Calculate the Relative Velocity:**
- The velocity of the bird with respect to the train is given by the formula:
\[ \vec{V}_{\text{bird, train}} = \vec{V}_{\text{bird}} - \vec{V}_{\text{train}} \]
- Substituting the values:
\[ \vec{V}_{\text{bird, train}} = (-40 \, \hat{j}) - (40 \, \hat{i}) = -40 \, \hat{i} - 40 \, \hat{j} \]

4. **Magnitude of the Relative Velocity:**
- To find the magnitude of the relative velocity vector:
\[ |\vec{V}_{\text{bird, train}}| = \sqrt{(-40)^2 + (-40)^2} = \sqrt{1600 + 1600} = \sqrt{3200} = 40\sqrt{2} \, \text{km/hr} \]

5. **Direction of the Relative Velocity:**
- The direction of the vector \(-40 \hat{i} - 40 \hat{j}\) indicates that it is in the third quadrant (southwest direction).
- The angle with respect to the negative x-axis (west) can be calculated using:
\[ \tan(\theta) = \frac{|\text{opposite}|}{|\text{adjacent}|} = \frac{40}{40} = 1 \implies \theta = 45^\circ \]
- Therefore, the direction is \(45^\circ\) from the west towards the south, which is southwest.

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

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