A duck has a mass of 2.8 kg. As the duck paddles, a force of 0.09 N acts on it in a direction due east. In addition, the current of the water exerts a force of 0.20 N in a direction of 47° 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 2.5 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

In this case, acceleration is the net force acting on the duck divided by its mass. Let's go step by step to solve the problem:

Step 1: Calculate the net force acting on the duck.
To find the net force, we need to calculate the vector sum of the two forces acting on the duck: the force due to paddling and the force due to the current.

Net Force = Force due to paddling + Force due to current

Given:
Force due to paddling = 0.09 N (due east)
Force due to current = 0.20 N at an angle 47° south of east

To calculate the net force, we need to find the horizontal and vertical components of the force due to the current.

Horizontal Component (Fx) = Force * cos(angle)
Horizontal Component = 0.2 N * cos(47°)

Vertical Component (Fy) = Force * sin(angle)
Vertical Component = 0.2 N * sin(47°)

Now we can find the net force:

Net Force (Fnet) = Force due to paddling (in the x-direction) + Horizontal Component of Force due to current
Fnet = 0.09 N + 0.2 N * cos(47°)

Step 2: Calculate the acceleration.
Acceleration (a) = Net Force / Mass
a = Fnet / 2.8 kg

Step 3: Calculate the final velocity.
Using the equation of motion, we have:
Final Velocity = Initial Velocity + Acceleration * Time
Since the duck is initially at rest (0.12 m/s in the east direction), the initial velocity is zero.

Final Velocity = 0 + a * 2.5 s

Step 4: Calculate the displacement.
Using the equation of motion mentioned earlier:
Displacement = Initial Velocity * Time + (1/2) * Acceleration * Time^2
Again, the initial velocity is zero because the duck starts at rest.

Displacement = 0 + (1/2) * a * (2.5 s)^2

Step 5: Find the magnitude and direction of displacement.
To find the magnitude (absolute value) of displacement, calculate the magnitude of the vector using the Pythagorean theorem:

Magnitude of Displacement = sqrt( (Displacement in x-direction)^2 + (Displacement in y-direction)^2 )

To find the direction relative to due east, use the arctan function to calculate the angle:

Direction = arctan(Displacement in y-direction / Displacement in x-direction)

Plug in the values from the calculations into these equations, and you will obtain the magnitude and direction (relative to due east) of the displacement that the duck undergoes.

To find the displacement of the duck, we can use the concept of net force and apply Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration.

Step 1: Calculate the net force acting on the duck.
Given:
Mass of the duck (m) = 2.8 kg
Force acting due east (F1) = 0.09 N
Force due to the current (F2) = 0.20 N
Angle of force due to the current (θ) = 47°

The horizontal components of the two forces are given by:

F1x = F1 * cos(0°) = F1 * cos(0) = 0.09 N * cos(0) = 0.09 N
F2x = F2 * cos(47°)

The vertical components of the two forces are given by:

F1y = F1 * sin(0°) = F1 * sin(0) = 0 N
F2y = F2 * sin(47°)

Step 2: Calculate the net x and y components of the force.
The net x-component of the force (Fx) is given by the sum of the x-components of each force:

Fx = F1x + F2x

The net y-component of the force (Fy) is given by the sum of the y-components of each force:

Fy = F1y + F2y

Step 3: Calculate the acceleration of the duck.
The net force acting on the duck is given by:

Fnet = √(Fx^2 + Fy^2)

Using Newton's second law of motion, we can write:

Fnet = m * a

Rearranging the equation, we can solve for the acceleration (a):

a = Fnet / m

Step 4: Calculate the displacement of the duck.
The displacement (d) can be calculated using the equation of motion:

d = v0t + (1/2)at^2

Where:
v0 is the initial velocity of the duck
t is the time duration the forces are acting (2.5s)

Given:
Initial velocity (v0) = 0.12 m/s
Time duration (t) = 2.5 s

Substituting the values into the equation, we can solve for the displacement:

d = (0.12 m/s)(2.5 s) + (1/2)a(2.5 s)^2

Now, follow the steps to find the magnitude and direction of the displacement of the duck.