A duck has a mass of 2.10 kg. As the duck paddles, a force of 0.110 N acts on it in a direction due east. In addition, the current of the water exerts a force of 0.240 N in a direction of 47.0° south of east. When these forces begin to act, the velocity of the duck is 0.120 m/s in a direction due east. Find (a) the magnitude and (b) the direction (relative to due east) of the displacement that the duck undergoes in 3.10 s while the forces are acting. (Note that the angle will be negative in the south of east direction.)

To find the displacement of the duck, we need to calculate the net force acting on it and then apply Newton's second law of motion, which relates the net force to the mass and acceleration of an object.

1. Calculate the net force:
To find the net force, we need to add the two given forces together. The force acting due east is 0.110 N, and the force acting 47.0° south of east is 0.240 N.

To resolve the force acting south of east into its east and south components, we can use trigonometry. The east component is given by the force multiplied by the cosine of the angle, and the south component is given by the force multiplied by the sine of the angle.

East component = 0.240 N * cos(47.0°)
South component = 0.240 N * sin(47.0°)

2. Calculate the net force in the east direction:
To find the net force in the east direction, we subtract the east component of the force acting south of east from the force acting due east.

Net force in the east direction = 0.110 N - East component

3. Calculate the acceleration of the duck:
Newton's second law states that the net force is equal to the mass of the object multiplied by its acceleration.

Acceleration = Net force / Mass

4. Calculate the change in velocity:
The change in velocity can be found using the formula:

Change in velocity = Acceleration * Time

5. Calculate the displacement:
The displacement of an object can be obtained by multiplying its initial velocity by the time and adding half of the acceleration times the square of the time.

Displacement = (Initial velocity * Time) + (0.5 * Acceleration * Time^2)

(a) To find the magnitude of the displacement, simply take the absolute value of the displacement obtained in step 5.

(b) To find the direction of the displacement relative to due east, we can determine the angle between the displacement vector and the east direction. This angle can be found using trigonometry:

Angle = arctan(Displacement / Initial velocity)

Plug in the given values and follow these steps to find the solution.

To find the displacement of the duck, we can use the equation:

displacement = initial velocity * time + (1/2) * net acceleration * time^2

First, let's calculate the net acceleration acting on the duck.

1. Calculate the x-component of the net force:
Net force in the x-direction = force due east - force south of east * cos(angle south of east)

Force due east = 0.110 N (given)
Force south of east = 0.240 N (given)
Angle south of east = 47.0°

Net force in the x-direction = 0.110 N - 0.240 N * cos(47.0°)

2. Calculate the y-component of the net force:
Net force in the y-direction = force south of east * sin(angle south of east)

Net force in the y-direction = 0.240 N * sin(47.0°)

3. Calculate the net acceleration using Newton's second law:
Net acceleration = net force / mass

Mass of the duck = 2.10 kg (given)

Net acceleration = √(Net force in the x-direction)^2 + (Net force in the y-direction)^2 / 2.10 kg

Now, let's calculate the displacement of the duck.

(a) The magnitude of the displacement:
Using the equation: displacement = initial velocity * time + (1/2) * net acceleration * time^2

Initial velocity = 0.120 m/s (given)
Time = 3.10 s (given)

Displacement = 0.120 m/s * 3.10 s + (1/2) * (Net acceleration) * (3.10 s)^2

(b) The direction (relative to due east) of the displacement:
To find the direction, we can use the inverse tangent function:

Direction (relative to due east) = arctan((Net force in the y-direction) / (Net force in the x-direction))

Now, let's calculate the values:

Net force in the x-direction = 0.110 N - 0.240 N * cos(47.0°)

Net force in the y-direction = 0.240 N * sin(47.0°)

Net acceleration = √(Net force in the x-direction)^2 + (Net force in the y-direction)^2 / 2.10 kg

Displacement = 0.120 m/s * 3.10 s + (1/2) * (Net acceleration) * (3.10 s)^2

Direction (relative to due east) = arctan((Net force in the y-direction) / (Net force in the x-direction))