After a constant acceleration of 2.2 m/s/s at 16 degrees above the +x-axis is applied by the motors of a boat for 8 seconds, the boat is moving at 61.45 m/s at 43.5 degrees above the +x-axis. What was the displacement of the boat during this time? (Give your answer in terms of a positive rotation from the +x-axis, in degrees)

Question

After a constant acceleration of 2.2 m/s/s at 16 degrees above the +x-axis is applied by the motors of a boat for 8 seconds, the boat is moving at 61.45 m/s at 43.5 degrees above the +x-axis. What was the displacement of the boat during this time? (Give your answer in terms of a positive rotation from the +x-axis, in degrees)

Answer 1

To find the displacement of the boat, we need to break down the initial velocity, acceleration, and time into their vector components.

Given:
Acceleration (a) = 2.2 m/s^2
Initial velocity (vi) = 0 m/s (since the boat starts from rest)
Time (t) = 8 s
Final velocity (vf) = 61.45 m/s
The angle with the positive x-axis at the beginning (θ_i) = 16 degrees
The angle with the positive x-axis at the end (θ_f) = 43.5 degrees

Step 1: Break down the initial velocity into x and y components.
The initial velocity can be calculated using the formula:
vi = √(vix^2 + viy^2)
Since vi = 0, both components must be zero:
vix = vi * cos(θ_i) = 0
viy = vi * sin(θ_i) = 0

Step 2: Break down the final velocity into x and y components.
The final velocity can be calculated using the formula:
vf = √(vfx^2 + vfy^2)
The x-component of the final velocity is:
vfx = vf * cos(θ_f)
Similarly, the y-component of the final velocity is:
vfy = vf * sin(θ_f)

Step 3: Calculate the displacement in the x direction (Δx).
The displacement in the x direction can be found using the kinematic equation:
Δx = vix * t + (1/2) * a * t^2
In this case, since vix = 0, the equation becomes:
Δx = (1/2) * a * t^2 = (1/2) * 2.2 m/s^2 * (8 s)^2

Step 4: Calculate the displacement in the y direction (Δy).
The displacement in the y direction can be found using the kinematic equation:
Δy = viy * t + (1/2) * a * t^2
In this case, since viy = 0, the equation becomes:
Δy = (1/2) * a * t^2 = (1/2) * 2.2 m/s^2 * (8 s)^2

Step 5: Calculate the total displacement (Δs).
The total displacement can be found using the Pythagorean theorem:
Δs = √(Δx^2 + Δy^2)

By following these steps and plugging in the values we have, you can find the displacement of the boat during this time in terms of a positive rotation from the +x-axis, in degrees.