A 16000 kg sailboat experiences an eastward
force of 12400 N due to tide pushing its hull
while the wind pushes the sails with a force of
51900 N directed toward the northwest (45�
westward of North or 45� northward ofWest).
What is the magnitude of the resultant ac-
celeration of the sailboat?
Answer in units of m/s2
To find the magnitude of the resultant acceleration of the sailboat, we can use Newton's second law of motion, which states that the acceleration of an object is equal to the net force acting on it divided by its mass.
In this case, we have two forces acting on the sailboat: the eastward force due to the tide (F1 = 12400 N) and the northwestward force due to the wind (F2 = 51900 N). The net force is the vector sum of these two forces.
To find the net force, we can break down the northwestward force into its horizontal and vertical components. Since the angle is 45º westward of North or 45º northward of West, the vertical component of the force is equal to the horizontal component. Therefore, the horizontal component of the force is F2 * cos(45º) and the vertical component is F2 * sin(45º).
The net force in the horizontal direction is the sum of the forces F1 and the horizontal component of F2, while the net force in the vertical direction is the vertical component of F2. The magnitude of the resultant acceleration is given by:
a = √(a_horizontal^2 + a_vertical^2)
Now, let's calculate the magnitude of the resultant acceleration step by step:
Horizontal component of F2: F2 * cos(45º)
= 51900 N * cos(45º)
= 51900 N * 0.707 (approximately)
≈ 36691.53 N
Vertical component of F2: F2 * sin(45º)
= 51900 N * sin(45º)
= 51900 N * 0.707 (approximately)
≈ 36691.53 N
Net force in horizontal direction: F1 + Horizontal component of F2
= 12400 N + 36691.53 N
≈ 49091.53 N
Net force in vertical direction: Vertical component of F2
≈ 36691.53 N
Magnitude of resultant acceleration: √(a_horizontal^2 + a_vertical^2)
= √(49091.53 N^2 + 36691.53 N^2)
= √(2410764600 N^2)
≈ 49198.35 N
Therefore, the magnitude of the resultant acceleration of the sailboat is approximately 49198.35 m/s^2.
To find the magnitude of the resultant acceleration of the sailboat, we need to calculate the net force acting on the sailboat and then divide it by the mass of the sailboat.
First, let's resolve the wind force into its horizontal and vertical components. The wind force is directed toward the northwest at an angle of 45 degrees westward of North or 45 degrees northward of West.
The horizontal component of the wind force can be calculated using trigonometry:
Horizontal component of wind force = Wind force * cos(angle)
Horizontal component of wind force = 51900 N * cos(45 degrees)
Next, let's resolve the tide force into its horizontal and vertical components. The tide force is directed eastward.
The horizontal component of the tide force is equal to the total tide force since it is already directed eastward.
Now, we can calculate the net horizontal force by adding the horizontal components of both forces:
Net horizontal force = Horizontal component of wind force + Horizontal component of tide force
Finally, we can calculate the resultant acceleration using Newton's second law:
Resultant acceleration = Net horizontal force / mass of the sailboat
Plug in the given values and calculate the resultant acceleration.