A speedboat starts from rest and accelerates at +2.07 m/s2 for 7.00 s. At the end of this time, the boat continues for an additional 6.00 s with an acceleration of +0.523 m/s2. Following this, the boat accelerates at -1.55 m/s2 for 8.00 s. What is the velocity of the boat at t = 21.0 s?
To find the velocity of the boat at t = 21.0 s, we need to calculate the total displacement during the given time intervals and then use that information to find the final velocity.
Let's break down the problem step by step:
Step 1: Calculate the displacement for each time interval.
We have three time intervals:
- Acceleration at +2.07 m/s^2 for 7.00 s.
- Acceleration at +0.523 m/s^2 for 6.00 s.
- Acceleration at -1.55 m/s^2 for 8.00 s.
For the first interval, we can use the equation of motion: s = ut + (1/2)at^2, where u is the initial velocity, a is the acceleration, and t is the time.
The initial velocity for the first interval is 0 m/s (since the boat starts from rest), and the time is 7.00 s. So, the displacement for the first interval is:
s1 = 0 + (1/2)(2.07)(7.00)^2 = 0 + (1.035)(49) = 50.715 m
For the second interval, the initial velocity is the final velocity from the previous interval (which is unknown at this point), and the time is 6.00 s. The acceleration is +0.523 m/s^2. So, the displacement for the second interval can be found using the same equation of motion as before:
s2 = (unknown final velocity from the first interval)(6.00) + (1/2)(0.523)(6.00)^2
For the third interval, the initial velocity is the final velocity from the second interval (also unknown), and the time is 8.00 s. The acceleration is -1.55 m/s^2. So, the displacement for the third interval can be found using the same equation of motion:
s3 = (unknown final velocity from the second interval)(8.00) + (1/2)(-1.55)(8.00)^2
Step 2: Find the total displacement.
The total displacement is the sum of the three displacements:
Total displacement = s1 + s2 + s3
Step 3: Find the final velocity.
The final velocity can be calculated using the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
For the third interval, the initial velocity is the final velocity from the second interval (still unknown), the acceleration is -1.55 m/s^2, and the time is 8.00 s.
We can use this equation to find the final velocity at t = 21.0 s:
v = (unknown final velocity from the second interval) + (-1.55)(8.00)
Now that we have the equation for finding the total displacement and the equation for finding the final velocity, we can plug in the values and solve for the unknowns to find the velocity of the boat at t = 21.0 s.
To find the velocity of the boat at t = 21.0 s, we need to calculate the total displacement of the boat at that time.
Step 1: Find the displacement during the first 7.00 s.
Using the kinematic equation:
displacement = initial velocity * time + (1/2) * acceleration * time^2
Given:
initial velocity = 0 m/s
acceleration = +2.07 m/s^2
time = 7.00 s
Substituting the values into the equation:
displacement = 0 * 7.00 + (1/2) * 2.07 * (7.00)^2
displacement = 0 + (1/2) * 2.07 * 49
displacement = 0 + 1.035 * 49
displacement = 50.715 m (rounded to three decimal places)
Step 2: Find the displacement during the next 6.00 s.
Given:
initial velocity = velocity at the end of the previous interval = 2.07 * 7.00 = 14.49 m/s
acceleration = +0.523 m/s^2
time = 6.00 s
Using the same kinematic equation:
displacement = initial velocity * time + (1/2) * acceleration * time^2
displacement = 14.49 * 6.00 + (1/2) * 0.523 * (6.00)^2
displacement = 86.94 + 0.784 * 36
displacement = 86.94 + 28.104
displacement = 115.044 m (rounded to three decimal places)
Step 3: Find the displacement during the final 8.00 s.
Given:
initial velocity = velocity at the end of the previous interval = 14.49 + 0.523 * 6.00 = 17.677 m/s
acceleration = -1.55 m/s^2
time = 8.00 s
Using the same kinematic equation:
displacement = initial velocity * time + (1/2) * acceleration * time^2
displacement = 17.677 * 8.00 + (1/2) * (-1.55) * (8.00)^2
displacement = 141.416 + (-0.775) * 64
displacement = 141.416 + (-49.6)
displacement = 91.816 m (rounded to three decimal places)
Step 4: Calculate the total displacement at t = 21.0 s.
Total displacement = displacement during the first 7.00 s + displacement during the next 6.00 s + displacement during the final 8.00 s
Total displacement = 50.715 + 115.044 + 91.816
Total displacement = 257.575 m (rounded to three decimal places)
Step 5: Calculate the average velocity during the entire 21.0 s.
Average velocity = total displacement / total time
Average velocity = 257.575 / 21.0
Average velocity = 12.270 m/s (rounded to three decimal places)
Therefore, the velocity of the boat at t = 21.0 s is 12.270 m/s.