A helicopter, starting from rest, accelerates straight up from the roof of a hospital. The lifting force does work in raising the helicopter. An 850-kg helicopter rises from rest to a speed of 6.9 m/s in a time of 3.8 s. During this time it climbs to a height of 9.4 m. What is the average power generated by the lifting force?
power= work/time= (mgh+1/2 mv^2)/time
25930.59 watts
Well, let me do some quick calculations on my invisible calculator here. *beep boop beep* Okay, so the average power generated by the lifting force can be determined by using the work-energy principle. The work done by the lifting force is equal to the change in energy (kinetic and potential).
To find the work done, we can use the formula: work = force * distance. In this case, the distance is the height the helicopter climbed, which is 9.4 m.
The force can be calculated using Newton's second law: force = mass * acceleration.
So, the force equals 850 kg * (6.9 m/s / 3.8 s).
Plugging in the values, we get: work = (850 kg * (6.9 m/s / 3.8 s)) * 9.4 m.
Now, we know that power is defined as work divided by time. So, the average power generated by the lifting force is: power = work / time.
Using the given values, we get: power = ((850 kg * (6.9 m/s / 3.8 s)) * 9.4 m) / 3.8 s.
And after all that mathematical juggling, the average power generated by the lifting force is... drumroll, please... something like 1321.51 Watts.
But hey, don't worry about memorizing all these formulas. Just remember that helicopters make everything more fun, especially when it comes to calculating average power. So, next time you're in a helicopter, don't forget to bring your calculator and a sense of humor. Happy flying!
To find the average power generated by the lifting force, we can use the formula:
Average Power = Work / Time
To find the work done by the lifting force, we can use the formula:
Work = Force x Distance
The force here is the lifting force, and the distance is the height the helicopter climbs. Therefore:
Work = Lifting force x Distance
We know that the lifting force does work in raising the helicopter, which means the work done is equal to the change in the helicopter's total mechanical energy. The change in the total mechanical energy can be calculated using the formula:
Change in Mechanical Energy = (1/2) x Mass x Velocity^2
Using this formula, we can find the initial mechanical energy (at rest) and the final mechanical energy (at a speed of 6.9 m/s) of the helicopter.
Initial Mechanical Energy = (1/2) x Mass x (Initial Velocity)^2
Final Mechanical Energy = (1/2) x Mass x (Final Velocity)^2
The difference between the final and initial mechanical energies will give us the work done by the lifting force.
Work = Final Mechanical Energy - Initial Mechanical Energy
Finally, we can substitute the values given in the problem to find the average power generated by the lifting force:
Average Power = Work / Time
Let's calculate it step by step:
1. Calculate the initial mechanical energy:
Initial Mechanical Energy = (1/2) x Mass x (Initial Velocity)^2
= (1/2) x 850 kg x (0 m/s)^2
= 0 J
2. Calculate the final mechanical energy:
Final Mechanical Energy = (1/2) x Mass x (Final Velocity)^2
= (1/2) x 850 kg x (6.9 m/s)^2
= 20385.675 J
3. Calculate the work done by the lifting force:
Work = Final Mechanical Energy - Initial Mechanical Energy
= 20385.675 J - 0 J
= 20385.675 J
4. Calculate the average power generated by the lifting force:
Average Power = Work / Time
= 20385.675 J / 3.8 s
≈ 5,361.24 W (rounded to four significant figures)
Therefore, the average power generated by the lifting force is approximately 5,361.24 Watts.
To find the average power generated by the lifting force, we need to use the equation:
Power = Work / Time
First, let's calculate the work done by the lifting force. Since the lifting force does work in raising the helicopter, the work done is equal to the change in potential energy. The potential energy change can be calculated using the formula:
ΔPE = mgh
Where m is the mass of the helicopter, g is the acceleration due to gravity, and h is the change in height.
Given:
m = 850 kg
g = 9.8 m/s² (approximate value)
h = 9.4 m
Substituting these values into the formula, we have:
ΔPE = (850 kg) * (9.8 m/s²) * (9.4 m)
ΔPE = 78,657 J (Joules)
Next, we need to find the time taken to do this work. Given in the question, the time taken is 3.8 seconds.
Finally, we can calculate the average power:
Power = Work / Time
Power = 78,657 J / 3.8 s
Power ≈ 20,678 W (Watts)
So, the average power generated by the lifting force is approximately 20,678 Watts.