A man is walking north with speed 4.5 km/h . Another is running west with speed 6 km/h . The velocity (mag. and direction ) of the second relative to the first is

Options

A. 7 . 5 km/h at ab angle `tan^(-1)` (3/4) south of west
B. 7 . 5 km/h at ab angle `tan^(-1)` (3/4) west of south
C. 7 . 5 km/h south west
D. none of these

Answer

To find the velocity of the second man relative to the first, we will follow these steps:

### Step 1: Define the velocities of both men
- Let the velocity of the first man (walking north) be represented as:
\[ \vec{V_1} = 0 \hat{i} + 4.5 \hat{j} \text{ km/h} \]
(where \(\hat{i}\) is the east-west direction and \(\hat{j}\) is the north-south direction).

- Let the velocity of the second man (running west) be represented as:
\[ \vec{V_2} = -6 \hat{i} + 0 \hat{j} \text{ km/h} \]

### Step 2: Calculate the relative velocity
The relative velocity of the second man with respect to the first man is given by:
\[ \vec{V_{21}} = \vec{V_2} - \vec{V_1} \]
Substituting the values:
\[ \vec{V_{21}} = (-6 \hat{i} + 0 \hat{j}) - (0 \hat{i} + 4.5 \hat{j}) = -6 \hat{i} - 4.5 \hat{j} \text{ km/h} \]

### Step 3: Calculate the magnitude of the relative velocity
The magnitude of the relative velocity can be calculated using the Pythagorean theorem:
\[ |\vec{V_{21}}| = \sqrt{(-6)^2 + (-4.5)^2} \]
Calculating the squares:
\[ = \sqrt{36 + 20.25} = \sqrt{56.25} = 7.5 \text{ km/h} \]

### Step 4: Calculate the direction of the relative velocity
To find the direction, we can use the tangent function:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4.5}{6} \]
Calculating \(\theta\):
\[ \theta = \tan^{-1}\left(\frac{4.5}{6}\right) \]
Calculating the angle:
\[ \theta \approx 36.87^\circ \]
This angle is measured from the west towards the south, so the direction of the second man's velocity relative to the first man is:
\[ \text{South of West} \]

### Final Answer
The magnitude of the velocity of the second man relative to the first is \(7.5 \text{ km/h}\) and the direction is \(36.87^\circ\) South of West.

---