A car is moving with a speed of 25 km/hour in the East. A bus is moving with a speed of `25sqrt(3)` km/hour in the North. What will be the relative velocity of the bus for the car's driver ?

Answer

To find the relative velocity of the bus as observed by the car's driver, we can follow these steps:

### Step 1: Define the velocities of the car and the bus
- The car is moving east with a speed of 25 km/h. We can represent this velocity as:
\[ \vec{V}_{\text{car}} = 25 \hat{i} \text{ km/h} \]
- The bus is moving north with a speed of \( 25\sqrt{3} \) km/h. We can represent this velocity as:
\[ \vec{V}_{\text{bus}} = 25\sqrt{3} \hat{j} \text{ km/h} \]

### Step 2: Calculate the relative velocity of the bus with respect to the car
The relative velocity of the bus with respect to the car is given by the formula:
\[ \vec{V}_{\text{relative}} = \vec{V}_{\text{bus}} - \vec{V}_{\text{car}} \]
Substituting the values we have:
\[ \vec{V}_{\text{relative}} = (25\sqrt{3} \hat{j}) - (25 \hat{i}) \]
This can be rewritten as:
\[ \vec{V}_{\text{relative}} = -25 \hat{i} + 25\sqrt{3} \hat{j} \]

### Step 3: Find the magnitude of the relative velocity
To find the magnitude of the relative velocity, we use the Pythagorean theorem:
\[ |\vec{V}_{\text{relative}}| = \sqrt{(-25)^2 + (25\sqrt{3})^2} \]
Calculating this gives:
\[ |\vec{V}_{\text{relative}}| = \sqrt{625 + 1875} = \sqrt{2500} = 50 \text{ km/h} \]

### Step 4: Determine the direction of the relative velocity
To find the angle \( \theta \) that the relative velocity makes with the negative x-axis (west direction), we can use the tangent function:
\[ \tan(\theta) = \frac{25\sqrt{3}}{25} = \sqrt{3} \]
Thus, \( \theta = 60^\circ \).

### Conclusion
The relative velocity of the bus as observed by the car's driver is:
\[ 50 \text{ km/h at an angle of } 60^\circ \text{ from the west towards the north.} \]

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