A boat has a mass of 6800 kg. Its engines generate a drive force of 4100 N, due west, while the wind exerts a force of 800 N, due east, and the water exerts a resistive force of 1200 N due east. What is the magnitude and direction of the boat's acceleration ?

Options

A. `0.54" m"//"s"^(2)`, due west
B. `0.54" m"//"s"^(2)`, due east
C. `0.66" m"//"s"^(2)`, due west
D. `0.31" m"//"s"^(2)`, due west

Answer

To solve the problem, we need to determine the net force acting on the boat and then use it to find the acceleration.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Boat:**
- Drive Force (due west): \( F_{\text{drive}} = 4100 \, \text{N} \)
- Wind Force (due east): \( F_{\text{wind}} = 800 \, \text{N} \)
- Water Resistive Force (due east): \( F_{\text{water}} = 1200 \, \text{N} \)

2. **Calculate the Net Force:**
- The net force can be calculated by subtracting the forces acting in the east direction from the force acting in the west direction.
\[ F_{\text{net}} = F_{\text{drive}} - (F_{\text{wind}} + F_{\text{water}}) \]
\[ F_{\text{net}} = 4100 \, \text{N} - (800 \, \text{N} + 1200 \, \text{N}) \]
\[ F_{\text{net}} = 4100 \, \text{N} - 2000 \, \text{N} = 2100 \, \text{N} \]
- The net force is \( 2100 \, \text{N} \) due west.

3. **Use Newton's Second Law to Find Acceleration:**
- According to Newton's second law, the acceleration \( a \) can be calculated using the formula:
\[ F_{\text{net}} = m \cdot a \]
- Rearranging the formula to find acceleration gives:
\[ a = \frac{F_{\text{net}}}{m} \]
- Substituting the values:
\[ a = \frac{2100 \, \text{N}}{6800 \, \text{kg}} \approx 0.308 \, \text{m/s}^2 \]

4. **Determine the Direction of Acceleration:**
- Since the net force is acting due west, the acceleration will also be in the same direction.

### Final Answer:
- The magnitude of the boat's acceleration is approximately \( 0.31 \, \text{m/s}^2 \) and its direction is due west.