The force exerted by the wind on the sails of a sailboat is 340 N north. The water exerts a force of 150 N east. If the boat (including its crew) has a mass of 340 kg, what are the magnitude and direction of its acceleration?

To find the magnitude and direction of the boat's acceleration, we need to use 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.

We are given the force exerted by the wind on the sails (340 N north) and the force exerted by the water (150 N east). These forces act in different directions, so we need to break them down into their horizontal and vertical components.

The force exerted by the wind on the sails can be broken down into north and east components. Since the force is northward, the north component is 340 N. The east component is zero since the force is in the north direction only.

The force exerted by the water can be broken down into north and east components as well. Since the force is eastward, the east component is 150 N. The north component is zero since the force is in the east direction only.

Now, we can calculate the net force acting on the boat by summing the horizontal and vertical components of the forces:

Net force in the north direction = north component of wind force + north component of water force = 340 N + 0 N = 340 N
Net force in the east direction = east component of wind force + east component of water force = 0 N + 150 N = 150 N

Next, we can calculate the net force magnitude using the Pythagorean theorem:

Net force magnitude = √(Net force in the north direction)^2 + (Net force in the east direction)^2
= √(340 N)^2 + (150 N)^2
= √(115600 N^2 + 22500 N^2)
= √138100 N^2
≈ 371.96 N

Finally, we can calculate the acceleration using Newton's second law of motion:

Net force = mass * acceleration

Therefore, acceleration = Net force / mass = 371.96 N / 340 kg ≈ 1.093 m/s^2

The magnitude of the boat's acceleration is approximately 1.093 m/s^2.

To find the direction of the acceleration, we need to find the angle that the net force makes with the north direction. We can use trigonometry to find this angle:

θ = arctan((Net force in the east direction) / (Net force in the north direction)) = arctan(150 N / 340 N) ≈ 23.057 degrees

Therefore, the boat's acceleration is approximately 1.093 m/s^2 at an angle of 23.057 degrees east of north.