The force exerted by the wind on the sails of a sailboat is 310 N north. The water exerts a force of 240 N east. If the boat (including its crew) has a mass of 330 kg, what are the magnitude and direction of its acceleration?
magnitude in m/s2
direction in ° north of east
Add the two forces vectorially. Set the resultant equal to M a, where a is the acceleration vector. Solve for a, remembering that it is a vector. I will be glad to critique your work.
Further help will be forthcoming when work is shown.
Add the two forces vectorially? meaning?
http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/vectors/u3l1b.html
To determine the magnitude and direction of the acceleration of the sailboat, we will 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 can break down the forces acting on the sailboat into their respective components.
Given:
Force exerted by the wind on the sails = 310 N north
Force exerted by the water = 240 N east
Mass of the sailboat (including crew) = 330 kg
Step 1: Resolve the forces into their horizontal (east-west) and vertical (north-south) components.
For the wind force:
Horizontal component = 0 N (as the wind force is acting only in the north direction)
Vertical component = 310 N (as the wind force is acting in the north direction)
For the water force:
Horizontal component = 240 N (as the water force is acting in the east direction)
Vertical component = 0 N (as the water force is not acting in the north direction)
Step 2: Calculate the net force acting on the sailboat in each direction.
Horizontal net force = sum of all the horizontal forces
= 240 N (as there are no other horizontal forces)
Vertical net force = sum of all the vertical forces
= 310 N (as there are no other vertical forces)
Step 3: Now that we have the net forces in each direction, we can calculate the acceleration.
Using Newton's second law:
F = m * a
For the horizontal direction:
Horizontal net force = mass * horizontal acceleration
240 N = 330 kg * horizontal acceleration
horizontal acceleration = 240 N / 330 kg
horizontal acceleration ≈ 0.727 m/s² (rounded to three decimal places)
For the vertical direction:
Vertical net force = mass * vertical acceleration
310 N = 330 kg * vertical acceleration
vertical acceleration = 310 N / 330 kg
vertical acceleration ≈ 0.939 m/s² (rounded to three decimal places)
Step 4: Calculate the magnitude and direction of the acceleration.
Using the magnitudes of the horizontal and vertical components of acceleration, we can calculate the overall acceleration magnitude using the Pythagorean theorem:
Acceleration magnitude = sqrt((horizontal acceleration)^2 + (vertical acceleration)^2)
= sqrt((0.727 m/s²)^2 + (0.939 m/s²)^2)
≈ 1.183 m/s² (rounded to three decimal places)
Using the tangential function, we can calculate the direction of the acceleration angle.
Acceleration direction = arctan(vertical acceleration / horizontal acceleration)
= arctan(0.939 m/s² / 0.727 m/s²)
≈ 51.403° north of east (rounded to three decimal places)
Therefore, the magnitude of the sailboat's acceleration is approximately 1.183 m/s², and its direction is 51.403° north of east.