A duck has a mass of 2.80 kg. As the duck paddles, a force of 0.120 N acts on it in a direction due east. In addition, the current of the water exerts a force of 0.190 N in a direction of 48.0 ° 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 3.20 s while the forces are acting. (Note that the angle will be negative in the south of east direction.)

Add these forces as vectors.

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To find the displacement of the duck, we need to first calculate the net force acting on the duck, and then use that information to calculate the acceleration. Finally, we can use the acceleration to find the displacement.

1. Calculate the net force:
To find the net force, we need to calculate the vector sum of the two given forces. We can break the forces down into their horizontal and vertical components.

a) Horizontal components:
The force due east has no vertical component, while the force 48.0° south of east has a vertical component. We can calculate the horizontal components of the forces as follows:

Force due east (Fe) = 0.120 N (no horizontal component)
Force south of east (Fs) = 0.190 N * cos(48.0°)

b) Vertical components:
The force 48.0° south of east has a vertical component pointing south. We can calculate the vertical component of the force as follows:

Force south of east (Fs) = 0.190 N * sin(48.0°)

c) Net force:
To find the net force, we can add the horizontal components and vertical components separately:

Net horizontal force (Fnet,h) = Fe + Fs(horizontal components)
Net vertical force (Fnet,v) = Fs(vertical component)

2. Calculate the acceleration:
Using Newton's second law (F = ma), we can find the acceleration by dividing the net force by the mass of the duck:

Acceleration (a) = Fnet / mass

3. Calculate the displacement:
To calculate the displacement, we need to find the initial velocity and the time. Given that the velocity of the duck is 0.120 m/s in the east direction, and the time is 3.20 s, we can use the following equation of motion:

Displacement (d) = v0t + 0.5at^2

where v0 is the initial velocity, t is the time, a is the acceleration, and d is the displacement.

Now, let's calculate the magnitude and direction of the displacement of the duck.

a) Magnitude of displacement:
Substitute the given values into the equation d = v0t + 0.5at^2, where v0 = 0.120 m/s, t = 3.20 s, and a is the value calculated in step 2. Solve for d to find the magnitude of the displacement.

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

Angle = tan^(-1)(vertical displacement / horizontal displacement)

Substitute the vertical and horizontal displacements into the equation to find the direction of displacement.

With these steps, you should be able to find the answers to parts (a) and (b) of the question.