A car 'A' moves due north at a speed of `40km//hr`, while another 'B' moves due east at a speed of `30km//hr`. Find the velocity of car B relative to car A (both in magnitude and direction).

Options

A. `40km//hr`, at an angle `tan^(-1)(3/5)` east of south
B. `50km//hr`, at an angle `tan^(-1)(3/5)` east of south
C. `40km//hr`, at an angle `tan^(-1)(3/4)` east of south
D. `50km//hr`, at an angle `tan^(-1)(3/4)` east of south

Explanation

To find the velocity of car B relative to car A, we can follow these steps:

### Step 1: Define the velocities of the cars
- Let the velocity of car A (V_A) be represented as a vector:
\[ \vec{V_A} = 40 \, \text{km/h} \, \hat{j} \]
(since it moves due north, we can represent north as the positive y-direction).

- Let the velocity of car B (V_B) be represented as a vector:
\[ \vec{V_B} = 30 \, \text{km/h} \, \hat{i} \]
(since it moves due east, we can represent east as the positive x-direction).

### Step 2: Calculate the relative velocity of car B with respect to car A
The relative velocity of B with respect to A (V_BA) can be calculated using the formula:
\[ \vec{V_{BA}} = \vec{V_B} - \vec{V_A} \]

Substituting the values:
\[ \vec{V_{BA}} = (30 \, \hat{i}) - (40 \, \hat{j}) \]
This gives us:
\[ \vec{V_{BA}} = 30 \, \hat{i} - 40 \, \hat{j} \]

### Step 3: Calculate the magnitude of the relative velocity
The magnitude of the relative velocity vector can be calculated using the Pythagorean theorem:
\[ |\vec{V_{BA}}| = \sqrt{(30)^2 + (-40)^2} \]
Calculating this:
\[ |\vec{V_{BA}}| = \sqrt{900 + 1600} = \sqrt{2500} = 50 \, \text{km/h} \]

### Step 4: Calculate the direction of the relative velocity
To find the direction, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-40}{30} \]
Calculating the angle:
\[ \theta = \tan^{-1}\left(\frac{-40}{30}\right) \]
This gives:
\[ \theta \approx -53.13^\circ \]
This angle is measured from the positive x-axis (east direction) towards the negative y-axis (south direction).

### Step 5: Conclusion
The velocity of car B relative to car A is:
- Magnitude: \(50 \, \text{km/h}\)
- Direction: \(53.13^\circ\) south of east.