A car is going in south with a speed of `5m//s`. To a man sitting in car a bus appears to move towards west with a speed of `2sqrt(6)m//s`. What is the actual speed of the bus?

Explanation

To solve the problem, we need to determine the actual speed of the bus given the speed of the car and the apparent speed of the bus as observed from the car. Let's break down the solution step by step.

### Step 1: Define the velocities
- The car is moving south with a speed of \( v_c = 5 \, \text{m/s} \).
- The bus appears to move west with a speed of \( v_{b/c} = 2\sqrt{6} \, \text{m/s} \) as observed from the car.

### Step 2: Set up the coordinate system
- We can define a coordinate system where:
- South is in the negative y-direction.
- West is in the negative x-direction.

### Step 3: Express the velocities in vector form
- The velocity of the car can be expressed as:
\[ \vec{v_c} = 0 \hat{i} - 5 \hat{j} \quad (\text{where } \hat{i} \text{ is east-west direction and } \hat{j} \text{ is north-south direction}) \]
- The velocity of the bus with respect to the car can be expressed as:
\[ \vec{v_{b/c}} = -2\sqrt{6} \hat{i} + 0 \hat{j} \]

### Step 4: Use the relative velocity equation
- The actual velocity of the bus \( \vec{v_b} \) can be found using the equation:
\[ \vec{v_b} = \vec{v_{b/c}} + \vec{v_c} \]
- Substituting the values:
\[ \vec{v_b} = (-2\sqrt{6} \hat{i} + 0 \hat{j}) + (0 \hat{i} - 5 \hat{j}) \]
- This simplifies to:
\[ \vec{v_b} = -2\sqrt{6} \hat{i} - 5 \hat{j} \]

### Step 5: Calculate the magnitude of the velocity of the bus
- The magnitude of the velocity vector \( \vec{v_b} \) is given by:
\[ |\vec{v_b}| = \sqrt{(-2\sqrt{6})^2 + (-5)^2} \]
- Calculating the squares:
\[ (-2\sqrt{6})^2 = 4 \times 6 = 24 \]
\[ (-5)^2 = 25 \]
- Therefore, the magnitude becomes:
\[ |\vec{v_b}| = \sqrt{24 + 25} = \sqrt{49} = 7 \, \text{m/s} \]

### Final Answer
The actual speed of the bus is \( 7 \, \text{m/s} \).

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