The force exerted by the wind on the sails of a sailboat is 500 N north. The water exerts a force of 150 N east. If the boat (including its crew) has a mass of 290 kg, what are the magnitude and direction of its acceleration?
Forces North = 500
Force East = 150
acceleration North = 500/290 = 1.72 m/s^2
acceleration East = 150/290 = .517 m/s^2
magnitude = sqrt(1.72^2+.517^2)
= 1.80 m/s^2
tan of angle north of east = 1.72/.517
so direction = 73.3 deg N of E
or
16.7 deg East of North
To find the magnitude and direction of the boat's acceleration, we need to calculate the net force acting on the boat and then use Newton's second law of motion.
Step 1: Determine the horizontal and vertical components of the wind and water forces.
The force exerted by the wind:
Horizontal component = 0 N (since it acts in the north direction)
Vertical component = 500 N (since it acts in the north direction)
The force exerted by the water:
Horizontal component = 150 N (since it acts in the east direction)
Vertical component = 0 N (since it acts horizontally)
Step 2: Calculate the net horizontal and vertical forces.
Horizontal net force = sum of horizontal forces = 150 N (force exerted by water in the east direction)
Vertical net force = sum of vertical forces = 500 N (force exerted by wind in the north direction)
Step 3: Determine the magnitude and direction of the net force.
Magnitude of net force = sqrt((horizontal net force)^2 + (vertical net force)^2)
= sqrt((150 N)^2 + (500 N)^2)
= sqrt(22500 N^2 + 250000 N^2)
= sqrt(272500 N^2)
= 522.42 N
Direction of net force = atan(vertical net force / horizontal net force)
= atan(500 N / 150 N)
= atan(3.33333)
= 71.57 degrees north of east
Step 4: Calculate the acceleration using Newton's second law.
Newton's second law states that the net force acting on an object is equal to its mass times its acceleration.
Net force = mass × acceleration
522.42 N (magnitude of net force) = 290 kg × acceleration
Solving for acceleration:
acceleration = 522.42 N / 290 kg
= 1.8 m/s^2
Therefore, the magnitude of the boat's acceleration is 1.8 m/s^2, and its direction is 71.57 degrees north of east.
To determine the magnitude and direction of the boat's acceleration, you need to find the net force acting on the boat using Newton's second law of motion.
Newton's second law states that the net force acting on an object is equal to the product of its mass and acceleration. The equation can be written as:
Net force = mass × acceleration
In this case, the net force acting on the boat is the vector sum of the wind force and water force, given as:
Net force = Force due to wind + Force due to water
Since the wind force is acting in the north direction (positive Y-axis) with a magnitude of 500 N and the water force is acting in the east direction (positive X-axis) with a magnitude of 150 N, we can represent these forces as vectors:
Force due to wind = 500 N (north)
Force due to water = 150 N (east)
Now, to find the net force, we will break down the forces into their respective components. We can use basic trigonometry to do this:
Force due to wind in the X-axis direction = 0 N (no component in the east direction)
Force due to wind in the Y-axis direction = 500 N
Force due to water in the X-axis direction = 150 N
Force due to water in the Y-axis direction = 0 N (no component in the north direction)
Next, we can calculate the net force in each axis by summing up the respective components:
Net force in the X-axis = Force due to wind in the X-axis direction + Force due to water in the X-axis direction = 150 N
Net force in the Y-axis = Force due to wind in the Y-axis direction + Force due to water in the Y-axis direction = 500 N
Now that we have the net force components in the X and Y axes, we can use these values along with the mass of the boat (290 kg) to calculate the boat's acceleration in each direction.
Using the equation Net force = mass × acceleration:
Net force in the X-axis = mass × acceleration in the X-axis
150 N = 290 kg × acceleration in the X-axis
Solving for acceleration in the X-axis, we get:
acceleration in the X-axis = 150 N / 290 kg ≈ 0.5172 m/s² (east)
Net force in the Y-axis = mass × acceleration in the Y-axis
500 N = 290 kg × acceleration in the Y-axis
Solving for acceleration in the Y-axis, we get:
acceleration in the Y-axis = 500 N / 290 kg ≈ 1.7241 m/s² (north)
Therefore, the magnitude of the boat's acceleration is approximately 1.82 m/s², and the direction is at an angle of approximately 20.5° north of east.