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?
Magnitude = pythagorean theorem (net force)
F^2=(500^2)+(150^2)=272500
F=square root(272500)
F=522 N *answer*
Acceleration:
a (m/s^2)= F/m= (522N/290Kg)
a=1.8 m/s^2 *answer*
To find the magnitude and direction of the boat's acceleration, we need to consider the forces acting on the boat and apply Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
First, let's write down the given information:
- Force exerted by the wind on the sails = 500 N north
- Force exerted by the water = 150 N east
- Mass of the boat (including crew) = 290 kg
The net force acting on the boat can be calculated by adding the force vectors together:
Net force (F_net) = Force exerted by the wind + Force exerted by the water
To add these vectors, we can break them down into their horizontal (x) and vertical (y) components. Since the wind force is in the north direction and the water force is in the east direction, we need to find the north and east components of each force.
Given:
Wind force (F_wind) = 500 N north
Water force (F_water) = 150 N east
The north component of the wind force is:
F_wind_north = F_wind * sin(θ)
where θ is the angle between the force vector and the vertical axis.
The east component of the water force is:
F_water_east = F_water * cos(θ)
where θ is the angle between the force vector and the horizontal axis.
Assuming θ = 90 degrees for both forces (since they are perpendicular to the respective axes), we can calculate the components:
F_wind_north = 500 N * sin(90 degrees) = 500 N * 1 = 500 N
F_water_east = 150 N * cos(90 degrees) = 150 N * 0 = 0 N
Now, we can calculate the net force acting on the boat:
F_net = F_wind_north + F_water_east = 500 N + 0 N = 500 N north
Now, we can calculate the acceleration of the boat using Newton's second law:
F_net = m * a
where F_net is the net force acting on the boat, m is the mass of the boat, and a is the acceleration.
Rearranging the formula, we get:
a = F_net / m
Plugging in the values, we get:
a = 500 N / 290 kg = 1.724 m/s^2 north
Therefore, the magnitude of the boat's acceleration is 1.724 m/s^2, and it's directed north.