A car A is moving with speed `"40 km h"^(-1)` along a straight line `30^(@)` north of east and another car B is moving with same speed along a straight line `30^(@)` south of east. The relative velocity of car A as observed from the car B is

Options

A. (a)`"40 km h"^(-1)" north - east"`
B. (b)`"40 km h"^(-1)" south"`
C. (c)`"40 km h"^(-1)" north"`
D. (d)`"40 km h"^(-1)" south - east"`

Answer

To solve the problem of finding the relative velocity of car A as observed from car B, we can follow these steps:

### Step 1: Understand the Directions and Velocities
- Car A is moving at a speed of 40 km/h at an angle of 30° north of east.
- Car B is also moving at a speed of 40 km/h at an angle of 30° south of east.

### Step 2: Break Down the Velocities into Components
For car A:
- The velocity components can be calculated using trigonometric functions:
- \( V_{Ax} = 40 \cos(30^\circ) \) (eastward component)
- \( V_{Ay} = 40 \sin(30^\circ) \) (northward component)

For car B:
- The velocity components will be:
- \( V_{Bx} = 40 \cos(30^\circ) \) (eastward component)
- \( V_{By} = -40 \sin(30^\circ) \) (southward component)

### Step 3: Calculate the Components
Using the values of cosine and sine:
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(30^\circ) = \frac{1}{2} \)

Calculating the components:
- For car A:
- \( V_{Ax} = 40 \cdot \frac{\sqrt{3}}{2} = 20\sqrt{3} \) km/h
- \( V_{Ay} = 40 \cdot \frac{1}{2} = 20 \) km/h

- For car B:
- \( V_{Bx} = 40 \cdot \frac{\sqrt{3}}{2} = 20\sqrt{3} \) km/h
- \( V_{By} = -40 \cdot \frac{1}{2} = -20 \) km/h

### Step 4: Find the Relative Velocity
The relative velocity of car A with respect to car B is given by:
\[ V_{AB} = V_A - V_B \]

Calculating the components:
- In the x-direction:
\[ V_{ABx} = V_{Ax} - V_{Bx} = 20\sqrt{3} - 20\sqrt{3} = 0 \]

- In the y-direction:
\[ V_{ABy} = V_{Ay} - V_{By} = 20 - (-20) = 20 + 20 = 40 \]

### Step 5: Combine the Components
Thus, the relative velocity of car A with respect to car B is:
\[ V_{AB} = 0 \hat{i} + 40 \hat{j} \]

### Step 6: Interpret the Result
This means that the relative velocity of car A as observed from car B is 40 km/h in the north direction.

### Final Answer
The relative velocity of car A as observed from car B is **40 km/h north**.

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