When you are standstill holding a flag, the flag flutters in the direction of wind. When you start running the direction of fluttering of the flag changes to the direction of the wind relative to you. In all case a flag flutters in the direction of the wind relative to the flag.
When you are standstill holding a flag the flag flutters in the north and when you run at 8 m/s due cast. theflag flutters in direction `37^(@)` north of west. Find the wind velocity.

Answer

To solve the problem, we will analyze the situation step by step, using vector components for the velocities involved.

### Step 1: Understand the initial conditions
When you are standing still, the flag flutters in the direction of the wind, which is given as north. This means that the wind velocity \( V_w \) has only a northward component:
\[ V_w = (0, V_{wn}) \quad \text{(where \( V_{wn} \) is the northward wind velocity)} \]

### Step 2: Analyze the running scenario
When you start running at a speed of 8 m/s due east, the flag flutters at an angle of 37 degrees north of west. This means that the resultant velocity of the flag (which is the wind velocity relative to you) can be represented as:
\[ V_f = (V_{fx}, V_{fy}) \]
where \( V_{fx} \) is the westward component and \( V_{fy} \) is the northward component.

### Step 3: Set up the components of the flag's velocity
Since you are running east at 8 m/s, the wind's effect on the flag can be expressed as:
\[ V_f = V_w - (8, 0) \]
This means that the wind's velocity must be adjusted by subtracting your running velocity.

### Step 4: Resolve the flag's fluttering direction
The angle of 37 degrees north of west means we can resolve the components of \( V_f \):
- The angle from the west means that the x-component (westward) is negative and the y-component (northward) is positive.
- Using trigonometry:
\[ \tan(37^\circ) = \frac{V_{fy}}{-V_{fx}} \]
From the problem, we know:
\[ \tan(37^\circ) = \frac{3}{4} \]
Let \( V_{fx} = -x \) (since it is westward) and \( V_{fy} = y \) (since it is northward). Then:
\[ \frac{y}{x} = \frac{3}{4} \implies y = \frac{3}{4}x \]

### Step 5: Relate the components to the wind velocity
From the previous step, we have:
\[ V_f = (V_{fx}, V_{fy}) = (-x, \frac{3}{4}x) \]
Substituting into the equation from Step 3:
\[ (-x, \frac{3}{4}x) = (0, V_{wn}) - (8, 0) \]
This gives us two equations:
1. \( -x = -8 \) (from the x-components)
2. \( \frac{3}{4}x = V_{wn} \) (from the y-components)

### Step 6: Solve for \( x \) and \( V_{wn} \)
From the first equation:
\[ x = 8 \]
Substituting \( x \) into the second equation:
\[ V_{wn} = \frac{3}{4}(8) = 6 \text{ m/s} \]

### Conclusion
The wind velocity is:
\[ \text{Wind Velocity } V_w = (0, 6) \text{ m/s} \]