An observer is travelling in west with velocity of 5 km/h and observes that wind is blowing in north with the velocity of `5sqrt(3)` km/h. Find out the actual velocity of wind.

Answer

To find the actual velocity of the wind with respect to the ground, we can follow these steps:

### Step 1: Understand the Problem
We have an observer moving west with a velocity of 5 km/h. The observer perceives the wind blowing north at a velocity of \(5\sqrt{3}\) km/h. We need to find the actual velocity of the wind with respect to the ground.

### Step 2: Define the Velocities
- Let the velocity of the observer \( \vec{V_o} = -5 \hat{i} \) km/h (since it is moving west, we take it as negative in the x-direction).
- The relative velocity of the wind with respect to the observer \( \vec{V_{w/o}} = 5\sqrt{3} \hat{j} \) km/h (since it is blowing north, it is positive in the y-direction).

### Step 3: Use the Relative Velocity Equation
The actual velocity of the wind \( \vec{V_w} \) with respect to the ground can be calculated using the equation:
\[ \vec{V_w} = \vec{V_{w/o}} + \vec{V_o} \]
Substituting the values we have:
\[ \vec{V_w} = 5\sqrt{3} \hat{j} - 5 \hat{i} \]

### Step 4: Calculate the Magnitude of the Actual Velocity
Now we need to find the magnitude of the actual velocity of the wind:
\[ |\vec{V_w}| = \sqrt{(-5)^2 + (5\sqrt{3})^2} \]
Calculating the squares:
\[ |\vec{V_w}| = \sqrt{25 + 75} = \sqrt{100} = 10 \text{ km/h} \]

### Step 5: Determine the Direction of the Wind
To find the direction, we can use the tangent of the angle \( \theta \):
\[ \tan(\theta) = \frac{5\sqrt{3}}{5} = \sqrt{3} \]
Thus, \( \theta = 60^\circ \).

### Step 6: Conclusion
The actual velocity of the wind is \( 10 \) km/h at an angle of \( 60^\circ \) north of west.

### Final Answer
The actual velocity of the wind is \( 10 \) km/h at \( 60^\circ \) north of west.

---