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 use the concept of relative velocity. Here’s a step-by-step solution:

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

### Step 2: Define the velocities in vector form
- Let the velocity of the car \( \vec{v}_c \) be represented as:
\[ \vec{v}_c = 25 \, \hat{i} \, \text{km/h} \]
(where \( \hat{i} \) is the unit vector in the east direction).

- The velocity of the bus relative to the car \( \vec{v}_{bc} \) is:
\[ \vec{v}_{bc} = 25\sqrt{3} \, \hat{j} \, \text{km/h} \]
(where \( \hat{j} \) is the unit vector in the north direction).

### Step 3: Use the relative velocity formula
The actual velocity of the bus \( \vec{v}_b \) can be found using the formula:
\[ \vec{v}_{bc} = \vec{v}_b - \vec{v}_c \]
Rearranging this gives:
\[ \vec{v}_b = \vec{v}_{bc} + \vec{v}_c \]

### Step 4: Substitute the values
Substituting the values we defined:
\[ \vec{v}_b = 25\sqrt{3} \, \hat{j} + 25 \, \hat{i} \]

### Step 5: Calculate the magnitude of the velocity of the bus
To find the magnitude of \( \vec{v}_b \):
\[ |\vec{v}_b| = \sqrt{(25)^2 + (25\sqrt{3})^2} \]
Calculating each term:
\[ = \sqrt{625 + 1875} = \sqrt{2500} = 50 \, \text{km/h} \]

### Step 6: Calculate the direction of the velocity of the bus
To find the angle \( \theta \) that \( \vec{v}_b \) makes with the east direction, we can use:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{25\sqrt{3}}{25} = \sqrt{3} \]
Thus, \( \theta = 60^\circ \).

### Step 7: Determine the final direction
Since the bus is moving north and the angle is measured from the east, the direction of the bus's velocity can be described as:
\[ 30^\circ \, \text{east of north} \]

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

---