An observer is travelling in east with a velocity of 2 m/s and observes that wind is blowing in north with a velocity of 2m/s. If observer doubles his velocity then find out the velocity of wind appears to observer ?

Answer

To solve the problem step by step, we will analyze the situation using vector addition and relative velocity concepts.

### Step 1: Define the velocities
- The observer is moving east with a velocity \( V_o = 2 \, \text{m/s} \) (in the positive x-direction).
- The wind is blowing north with a velocity \( V_w = 2 \, \text{m/s} \) (in the positive y-direction).

### Step 2: Represent the velocities as vectors
- The velocity of the observer can be represented as:
\[ V_o = 2 \hat{i} \]
- The velocity of the wind can be represented as:
\[ V_w = 2 \hat{j} \]

### Step 3: Calculate the velocity of the wind with respect to the ground
- The velocity of the wind with respect to the ground is simply the vector sum of the wind's velocity and the observer's velocity:
\[ V_{wg} = V_w + V_o = 2 \hat{j} + 2 \hat{i} = 2 \hat{i} + 2 \hat{j} \]

### Step 4: Observer doubles his velocity
- If the observer doubles his velocity, his new velocity becomes:
\[ V_{o'} = 2 \times 2 \hat{i} = 4 \hat{i} \]

### Step 5: Calculate the apparent velocity of the wind to the observer
- The apparent velocity of the wind with respect to the observer is given by:
\[ V_{w'} = V_w - V_{o'} \]
- Substituting the values:
\[ V_{w'} = (2 \hat{i} + 2 \hat{j}) - (4 \hat{i}) = (2 - 4) \hat{i} + 2 \hat{j} = -2 \hat{i} + 2 \hat{j} \]

### Step 6: Determine the magnitude and direction of the apparent wind velocity
- The apparent wind velocity vector is:
\[ V_{w'} = -2 \hat{i} + 2 \hat{j} \]
- To find the magnitude:
\[ |V_{w'}| = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \, \text{m/s} \]
- The direction can be determined using the angle:
\[ \tan(\theta) = \frac{2}{-2} = -1 \quad \Rightarrow \quad \theta = 135^\circ \text{ (or } 45^\circ \text{ in the second quadrant)} \]
- This indicates that the wind appears to be blowing towards the northwest.

### Final Answer
The wind appears to the observer to be blowing at a velocity of \( 2\sqrt{2} \, \text{m/s} \) towards the northwest.

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