For a man running with a speed `v`. Wind appears to have a velocity `v` and inclination `120^(@)` with him. Find original velocity of wind

Options

A. V
B. 2V
C. `(V)/(2)`
D. 3V

Answer

To solve the problem of finding the original velocity of the wind when a man is running with speed \( v \) and perceives the wind to have a velocity \( v \) at an inclination of \( 120^\circ \) with respect to him, we can follow these steps:

### Step 1: Understand the Problem
We have a man running with speed \( v \). The wind appears to him to have a velocity \( v \) at an angle of \( 120^\circ \). We need to find the actual velocity of the wind with respect to the ground.

### Step 2: Set Up the Coordinate System
Assume the man is running along the positive x-axis. Thus, his velocity vector can be represented as:
\[ \vec{V_m} = v \hat{i} \]

### Step 3: Determine the Wind's Apparent Velocity
The wind's apparent velocity \( \vec{V_{wm}} \) with respect to the man can be expressed in terms of its components. Given that it makes an angle of \( 120^\circ \) with the man's direction, we can break it down into components:
\[ \vec{V_{wm}} = v \cos(120^\circ) \hat{i} + v \sin(120^\circ) \hat{j} \]
Using trigonometric values:
\[ \cos(120^\circ) = -\frac{1}{2}, \quad \sin(120^\circ) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \vec{V_{wm}} = v \left(-\frac{1}{2}\right) \hat{i} + v \left(\frac{\sqrt{3}}{2}\right) \hat{j} = -\frac{v}{2} \hat{i} + \frac{v\sqrt{3}}{2} \hat{j} \]

### Step 4: Apply the Relative Velocity Formula
The relative velocity of the wind with respect to the man can be expressed as:
\[ \vec{V_{wm}} = \vec{V_w} - \vec{V_m} \]
Where \( \vec{V_w} \) is the velocity of the wind with respect to the ground. Rearranging gives:
\[ \vec{V_w} = \vec{V_{wm}} + \vec{V_m} \]

### Step 5: Substitute the Known Values
Substituting the values we have:
\[ \vec{V_w} = \left(-\frac{v}{2} \hat{i} + \frac{v\sqrt{3}}{2} \hat{j}\right) + v \hat{i} \]
This simplifies to:
\[ \vec{V_w} = \left(v - \frac{v}{2}\right) \hat{i} + \frac{v\sqrt{3}}{2} \hat{j} = \frac{v}{2} \hat{i} + \frac{v\sqrt{3}}{2} \hat{j} \]

### Step 6: Calculate the Magnitude of the Wind's Velocity
The magnitude of the wind's velocity \( V_w \) can be calculated using the Pythagorean theorem:
\[ V_w = \sqrt{\left(\frac{v}{2}\right)^2 + \left(\frac{v\sqrt{3}}{2}\right)^2} \]
Calculating this gives:
\[ V_w = \sqrt{\frac{v^2}{4} + \frac{3v^2}{4}} = \sqrt{\frac{4v^2}{4}} = \sqrt{v^2} = v \]

### Conclusion
The original velocity of the wind with respect to the ground is:
\[ V_w = v \]