A duck has a mass of 2.3 kg. As the duck paddles, a force of 0.07 N acts on it in a direction due east. In addition, the current of the water exerts a force of 0.26 N in a direction of 48° south of east. When these forces begin to act, the velocity of the duck is 0.15 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.
displacement=velocity*time
= .15m/s East * 2.2seconds
= .25*2.2 m East
To find the displacement of the duck, we need to determine its net force and then apply Newton's second law of motion. The net force acting on the duck is the vector sum of the force it exerts while paddling and the force exerted by the current.
First, let's break down the given forces into their horizontal and vertical components.
The force due to paddling is purely horizontal, so its horizontal component is 0.07 N (due east) and its vertical component is 0 N.
The force exerted by the current has two components: one in the horizontal direction and one in the vertical direction. To determine these components, we'll use trigonometry.
The horizontal component of the current force is given by: F_horizontal = F_current * cosθ
F_current = 0.26 N (given)
θ = 48° (given)
Substituting these values, we find: F_horizontal = 0.26 N * cos(48°)
To find the vertical component of the current force, we use: F_vertical = F_current * sinθ
Substituting the values: F_vertical = 0.26 N * sin(48°)
Now, we can calculate the net force in each direction by summing the horizontal and vertical forces:
Net horizontal force = Paddling force (horizontal) + Current force (horizontal)
Net vertical force = Paddling force (vertical) + Current force (vertical)
Net horizontal force = 0.07 N + 0.26 N * cos(48°)
Net vertical force = 0 N + 0.26 N * sin(48°)
Now we can use Newton's second law of motion:
F_net = m * a
In the horizontal direction:
F_net (horizontal) = m * a (horizontal)
m = 2.3 kg
a (horizontal) = change in velocity (horizontal) / time
In the vertical direction:
F_net (vertical) = m * a (vertical)
m = 2.3 kg
a (vertical) = change in velocity (vertical) / time
The change in velocity can be calculated using the initial and final velocities:
Initial velocity (horizontal) = 0.15 m/s (given)
Final velocity (horizontal) = change in velocity (horizontal) = ?
Initial velocity (vertical) = 0 m/s (given)
Final velocity (vertical) = change in velocity (vertical) = ?
We can use the equations of motion to find the final velocities:
For the horizontal direction:
Final velocity (horizontal) = Initial velocity (horizontal) + a (horizontal) * time
For the vertical direction:
Final velocity (vertical) = Initial velocity (vertical) + a (vertical) * time
Substituting the values and solving the equations, we find the final velocities.
Then, we can plug in the values of mass and acceleration into Newton's second law equations to find the net forces in each direction.
Finally, we can calculate the displacement using the formulas:
Displacement (horizontal) = Initial velocity (horizontal) * time + (1/2) * a (horizontal) * time^2
Displacement (vertical) = Initial velocity (vertical) * time + (1/2) * a (vertical) * time^2
By calculating the magnitude and direction (relative to due east) of the displacement, we can find the answer to the given question.