A duck has a mass of 2.2 kg. As the duck paddles, a force of 0.11 N acts on it in a direction due east. In addition, the current of the water exerts a force of 0.12 N in a direction of 50° south of east. When these forces begin to act, the velocity of the duck is 0.12 m/s in a direction due east. Find the magnitude and direction (relative to due east) of the displacement that the duck undergoes in 2.2 s while the forces are acting.
To find the displacement of the duck, we can use the equation of motion:
displacement = initial velocity * time + 1/2 * acceleration * time^2
First, let's find the acceleration of the duck. The net force acting on the duck is the sum of the two forces mentioned in the problem:
net force = force due to paddling + force due to water current
Given that the force due to paddling is 0.11 N in the east direction, and the force due to water current is 0.12 N at a 50° angle south of east, we can break down these forces into their x and y components.
The x-component of the force due to paddling is 0.11 N in the east direction.
The x-component of the force due to water current is 0.12 N * cos(50°) in the east direction.
The y-component of the force due to water current is 0.12 N * sin(50°) in the south direction.
Now we can calculate the net force in both the x and y directions:
net force in the x-direction = 0.11 N + (0.12 N * cos(50°))
net force in the y-direction = 0.12 N * sin(50°)
Since the mass of the duck is given as 2.2 kg, we can calculate the acceleration using Newton's second law:
acceleration = net force / mass
Next, we need to calculate the initial x and y components of the velocity of the duck. The problem states that the initial velocity of the duck is 0.12 m/s in the east direction. Therefore, the initial x-component of the velocity is 0.12 m/s and the initial y-component is 0.
Now we can use the equation of motion to calculate the displacement of the duck after 2.2 seconds.
displacement in the x-direction = initial x-velocity * time + 1/2 * acceleration in the x-direction * time^2
displacement in the y-direction = initial y-velocity * time + 1/2 * acceleration in the y-direction * time^2
Finally, we can calculate the magnitude and direction of the displacement using the Pythagorean theorem and trigonometry:
magnitude of displacement = √(displacement in the x-direction)^2 + (displacement in the y-direction)^2
direction (θ) = tan^(-1)((displacement in the y-direction) / (displacement in the x-direction))
By plugging in the values and following these steps, you can find the magnitude and direction (relative to due east) of the displacement that the duck undergoes in 2.2 seconds while the forces are acting.