A bird watcher travels through the forest, walking 0.50 km due east, 0.75 km due south, and 2.15 km in a direction `35.0^(@)` north of west. The time required for this trip is 2.50 h. Determine the magnitude and direction (relative to due west) of the bird watcher's average velocity. Use kilometers and hours for distance and time, respectively.

Options

A. 0.540 km/h, `21^(@)` north of west
B. 1.43 km/h, `17^(@)` south of west
C. 1.22 km/h, `18^(@)` north of west
D. 1.08 km/h, `25^(@)` north of west

Answer

To solve the problem of determining the average velocity of the bird watcher, we will break down the steps systematically.

### Step 1: Define the movements in vector form
The bird watcher makes three movements:
1. 0.50 km due east (positive x-direction).
2. 0.75 km due south (negative y-direction).
3. 2.15 km at an angle of 35° north of west.

We can represent these movements in vector notation:
- The first movement (east) can be represented as:
\[ \vec{A} = 0.50 \hat{i} \, \text{km} \]

- The second movement (south) can be represented as:
\[ \vec{B} = -0.75 \hat{j} \, \text{km} \]

- The third movement (35° north of west) can be broken into components:
- The westward (negative x-direction) component:
\[ \vec{C_x} = 2.15 \cos(35°) \]
- The northward (positive y-direction) component:
\[ \vec{C_y} = 2.15 \sin(35°) \]

Calculating these components:
- \( \cos(35°) \approx 0.8192 \) and \( \sin(35°) \approx 0.5736 \)
- Thus,
\[ \vec{C_x} = 2.15 \times 0.8192 \approx 1.76 \, \text{km} \, (\text{west}) \]
\[ \vec{C_y} = 2.15 \times 0.5736 \approx 1.23 \, \text{km} \, (\text{north}) \]

So, the vector for the third movement is:
\[ \vec{C} = -1.76 \hat{i} + 1.23 \hat{j} \]

### Step 2: Sum the vectors to find the resultant displacement
Now we can find the total displacement vector \( \vec{R} \) by summing the three vectors:
\[ \vec{R} = \vec{A} + \vec{B} + \vec{C} \]
\[ \vec{R} = (0.50 \hat{i} - 0.75 \hat{j} - 1.76 \hat{i} + 1.23 \hat{j}) \]
Combining the components:
\[ \vec{R} = (0.50 - 1.76) \hat{i} + (-0.75 + 1.23) \hat{j} \]
\[ \vec{R} = -1.26 \hat{i} + 0.48 \hat{j} \]

### Step 3: Calculate the magnitude of the displacement
The magnitude of the displacement \( R \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{(-1.26)^2 + (0.48)^2} \]
Calculating:
\[ R = \sqrt{1.5876 + 0.2304} \]
\[ R = \sqrt{1.818} \approx 1.35 \, \text{km} \]

### Step 4: Calculate the average velocity
Average velocity \( \vec{V}_{avg} \) is given by the formula:
\[ \vec{V}_{avg} = \frac{\vec{R}}{t} \]
Where \( t = 2.50 \, \text{h} \):
\[ V_{avg} = \frac{1.35 \, \text{km}}{2.50 \, \text{h}} \approx 0.54 \, \text{km/h} \]

### Step 5: Determine the direction of the average velocity
To find the direction relative to due west, we can use the tangent of the angle \( \theta \):
\[ \tan(\theta) = \frac{y}{-x} = \frac{0.48}{1.26} \]
Calculating:
\[ \theta = \tan^{-1}\left(\frac{0.48}{1.26}\right) \approx 20.0° \]

Thus, the average velocity is approximately:
- **Magnitude**: \( 0.54 \, \text{km/h} \)
- **Direction**: \( 20.0° \) north of west.