Final answer:
To calculate the resulting acceleration of the sailboat, we resolve the wind force into components, sum the forces, and use Newton's second law. The magnitude of the resulting acceleration is approximately 3.16 m/s², and the direction is approximately 27.3° north of east.
Step-by-step explanation:
We need to find the resulting acceleration of the sailboat by using Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma).
First, we resolve the wind force into its eastward and northward components. Since the wind force is 3.00 × 10³ N at a 45° angle, the components are:
- Eastward force (Feast) = Fwind × cos(45°) = 3000 N × 0.7071 = 2121 N
- Northward force (Fnorth) = Fwind × sin(45°) = 3000 N × 0.7071 = 2121 N
The total eastward force is the sum of the eastward component of the wind force and the force toward the east:
Total eastward force = Feast + 2000 N = 2121 N + 2000 N = 4121 N
Now we use Newton's second law to find the acceleration in the eastward (aeast) and northward (anorth) directions:
- aeast = (Total eastward force) / mass = 4121 N / 1500 kg = 2.747 m/s²
- anorth = (Northward force) / mass = 2121 N / 1500 kg = 1.414 m/s²
To find the magnitude of the resulting acceleration (a), we use the Pythagorean theorem:
a = √(aeast² + anorth²) = √(2.747² + 1.414²) m/s² ≈ 3.16 m/s²
The direction of the resulting acceleration is given by:
tan(θ) = anorth / aeast
θ = arctan(1.414 / 2.747) ≈ 27.3° north of east
Therefore, the correct option is not listed specifically in the options. The magnitude of the resulting acceleration is approximately 3.16 m/s², and the direction is approximately 27.3° north of east.