A car is moving towards east with a speed of `25 km h^-1`. To the driver of the car, a bus appears to move towards north with a speed of `25 sqrt(3) km h^-1`. What is the actual velocity of the bus ?

Options

A. `50 km h^-1, 30^@ E of N`
B. `50 km h^-1, 30^@ N of E`
C. `25 km h^-1, 30^@ E of N`
D. `25 km h^-1, 30^@ N of E`

Explanation

To find the actual velocity of the bus, we can follow these steps:

### Step 1: Understand the given information
- The car is moving towards the east with a speed of \( V_c = 25 \, \text{km/h} \).
- The bus appears to the driver of the car to be moving towards the north with a speed of \( V_{bc} = 25\sqrt{3} \, \text{km/h} \).

### Step 2: Define the velocity vectors
- We can represent the velocity of the car as a vector:
\[ \vec{V_c} = 25 \hat{i} \, \text{km/h} \quad (\text{east direction}) \]
- The velocity of the bus with respect to the car is given as:
\[ \vec{V_{bc}} = 25\sqrt{3} \hat{j} \, \text{km/h} \quad (\text{north direction}) \]

### Step 3: Use the relative velocity formula
The actual velocity of the bus \( \vec{V_b} \) can be found using the formula:
\[ \vec{V_b} = \vec{V_c} + \vec{V_{bc}} \]

### Step 4: Calculate the components of the bus's velocity
Substituting the vectors into the equation:
\[ \vec{V_b} = 25 \hat{i} + 25\sqrt{3} \hat{j} \]

### Step 5: Find the magnitude of the bus's velocity
To find the actual speed of the bus, we calculate the magnitude of \( \vec{V_b} \):
\[ |\vec{V_b}| = \sqrt{(25)^2 + (25\sqrt{3})^2} \]
Calculating each term:
\[ |\vec{V_b}| = \sqrt{625 + 1875} = \sqrt{2500} = 50 \, \text{km/h} \]

### Step 6: Find the direction of the bus's velocity
To find the direction, we can use the tangent function:
\[ \tan(\theta) = \frac{V_{bc}}{V_c} = \frac{25\sqrt{3}}{25} = \sqrt{3} \]
This gives us:
\[ \theta = 60^\circ \]
However, since the bus appears to be moving north relative to the car moving east, the actual direction of the bus is \( 30^\circ \) east of north.

### Final Answer
The actual velocity of the bus is:
\[ \text{Speed} = 50 \, \text{km/h}, \quad \text{Direction} = 30^\circ \, \text{east of north} \]

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