Superman is flying at an initial velocity of 33.5 m/s [N] when he decides to slow to 10.0 m/s [N] in a time of 8.05 s. He then continues at this velocity for 12.4 s before accelerating at 2.35 m/s^2 [N] until he reaches a final velocity of 23.8 m/s [N].
a) Calculate Superman’s displacement in this time.
b) Calculate Superman’s average velocity for his entire flight.
honestly, the simple way to solve this is to plot a velocity vs time graph, then compute the area under it, that area is equal to displacement. Average velocity is displacement divided by time. Once you master that, you will understand how to do piecemeal formulas to do the same.
a) Well, I hope Superman packed his GPS because calculating his displacement is going to be quite the journey! To find his displacement, we need to break down each phase of his flight and add up the distances.
In the first phase, Superman slows down from 33.5 m/s to 10.0 m/s in 8.05 seconds. We can use the formula:
displacement = initial velocity * time + 0.5 * acceleration * time^2
Plugging in the values, we get:
displacement1 = (33.5 m/s - 10.0 m/s) * 8.05 s + 0.5 * 0 m/s^2 * (8.05 s)^2
Simplifying this equation will give us the displacement for the first phase.
In the second phase, Superman maintains a constant velocity of 10.0 m/s for 12.4 seconds. Since there is no acceleration, his displacement will simply be his velocity multiplied by time:
displacement2 = 10.0 m/s * 12.4 s
Lastly, in the third phase, Superman accelerates at 2.35 m/s^2 until he reaches 23.8 m/s. Again, we can use the same formula as before to calculate the displacement:
displacement3 = (23.8 m/s - 10.0 m/s) * unknown time + 0.5 * 2.35 m/s^2 * (unknown time)^2
Woah, looks like we have an unknown time in this equation! To solve for it, we need to set displacement3 equal to zero and use the quadratic formula to find the two possible times Superman could take. However, since we don't have an exact value for the acceleration time, we can't determine the exact displacement for this phase right now.
Now, we can add up all the displacements to find Superman's total displacement for his flight. Just be sure to substitute the correct values in displacement1 and displacement3 once we have the unknown time!
b) Calculating Superman's average velocity for his entire flight is as simple as dividing the total displacement by the total time. Just take our determined displacement from part a) and divide it by the time it took Superman to fly.
So, to summarize:
a) Superman's displacement = displacement1 + displacement2 + displacement3 (with the correct value of displacement3 once we have the unknown time)
b) Superman's average velocity = Superman's displacement / total time
Happy calculating, and may the force be with you... or should I say, the comic book hero?
To calculate Superman's displacement, we can use the formula:
displacement = initial velocity * time + (1/2) * acceleration * time^2
a) First, let's calculate his displacement during the first phase of slowing down from 33.5 m/s to 10.0 m/s in 8.05 s. Since he is slowing down, the acceleration will be negative:
acceleration = (final velocity - initial velocity) / time
acceleration = (10.0 m/s - 33.5 m/s) / 8.05 s
acceleration = -4.17 m/s^2
Using this acceleration, let's calculate the displacement:
displacement = 33.5 m/s * 8.05 s + (1/2) * (-4.17 m/s^2) * (8.05 s)^2
displacement = 269.68 m
b) Now, let's calculate Superman's displacement during the second phase, where he maintains a constant velocity of 10.0 m/s for 12.4 s. Since there is no acceleration during this phase, the displacement is simply:
displacement = velocity * time
displacement = 10.0 m/s * 12.4 s
displacement = 124.0 m
c) Finally, let's calculate Superman's displacement during the third phase, where he accelerates from 10.0 m/s to 23.8 m/s with an acceleration of 2.35 m/s^2. Here, we can use the same formula as in the first phase, but with the final velocity and time given:
displacement = 10.0 m/s * 12.4 s + (1/2) * 2.35 m/s^2 * (12.4 s)^2
displacement = 310.0 m + 177.43 m
displacement = 487.43 m
Now, to calculate Superman's total displacement, we add the displacements from each phase:
total displacement = 269.68 m + 124.0 m + 487.43 m
total displacement = 881.11 m
So, Superman's displacement in this time is 881.11 meters.
To calculate Superman's average velocity for his entire flight, we can use the formula:
average velocity = total displacement / total time
In this case, the total time is the sum of the times for each phase: 8.05 s + 12.4 s + 12.4 s = 32.85 s
average velocity = 881.11 m / 32.85 s
average velocity = 26.82 m/s
So, Superman's average velocity for his entire flight is 26.82 m/s.
To calculate Superman's displacement, we need to determine the distance covered in each part of his flight separately and then add them up.
First, let's calculate the displacement during the deceleration phase. We have the initial velocity (v₀), final velocity (v), and time (t). The formula to calculate displacement in this case is:
displacement = (v + v₀) / 2 * t
Substituting the values into the formula:
displacement₁ = (10 m/s + 33.5 m/s) / 2 * 8.05 s
Calculating the result:
displacement₁ = 23.75 m/s * 8.05 s = 191.0375 m
Next, let's calculate the displacement during the constant velocity phase. Since the velocity remains constant, the displacement is simply the product of the velocity and time:
displacement₂ = 10 m/s * 12.4 s = 124 m
Finally, let's calculate the displacement during the acceleration phase. We will use the equation:
displacement = v₀ * t + 0.5 * a * t²
Substituting the values into the formula:
displacement₃ = 10 m/s * 12.4 s + 0.5 * 2.35 m/s² * (12.4 s)²
Calculating the result:
displacement₃ = 124 m + 0.5 * 2.35 m/s² * 153.76 s²
displacement₃ = 124 m + 0.5 * 2.35 m/s² * 153.76 s²
displacement₃ = 124 m + 0.5 * 2.35 m/s² * 153.76 s²
displacement₃ = 124 m + 0.5 * 2.35 m/s² * 23,694.36 s²
displacement₃ = 124 m + 2.75375 m/s² * 23,694.36 s²
displacement₃ = 124 m + 65,233.65 m
displacement₃ = 65,357.65 m
Finally, we add up the displacements to find the total displacement:
Total displacement = displacement₁ + displacement₂ + displacement₃
Total displacement = 191.0375 m + 124 m + 65,357.65 m
Total displacement = 65,672.6875 m
Therefore, Superman's total displacement during his flight is approximately 65,672.69 meters.
Now let's calculate Superman's average velocity for his entire flight. Average velocity is the total displacement divided by the total time taken. In this case, the total time is the sum of the times for each phase:
Total time = 8.05 s + 12.4 s + 12.4 s = 32.85 s
Average velocity = Total displacement / Total time
Average velocity = 65,672.69 m / 32.85 s
Average velocity = 2,001.64 m/s
Therefore, Superman's average velocity for his entire flight is approximately 2,001.64 meters per second.