Total energy of an airplane, work done against air resistance

In summary, the conversation involved calculating the work done against air resistance during the climb of an airliner. The correct equation for this calculation is (total energy at take-off)+(work done by engines) = (W, work done against air resistance) + (total energy at 10 000 metres). The mistake in the given calculation was not including the total energy at 10 000 metres in the equation.
  • #1
furor celtica
69
0

Homework Statement



An airliner of mass 300 tonnes is powered by four engines, each developing 15 000 kW. Its speed at take-off is 75 m(s^1), and it takes 11 minutes to reach its cruising speed of 210 m(s^1) at a height of 10 000 metres. Calculate the work done against air resistance during the climb.




Homework Equations





The Attempt at a Solution



Alright so I have reasoned that (total energy at take-off)+(work done by engines)+(W, work done against air resistance)=(total energy at 10 000 metres) and that:
Total energy at take-off = PE +KE = 0 + (0.5 x 300 000 x (75^2))
Work done by engines (in 11 minutes) = (60 000 000)/(660)
Total energy at 10 000 metres = PE + KE = (10 000 x 300 000g) + (0.5 x 300 000 x (210^2))
G=10

Which makes W (work done against air resistance) = ((10 000 x 300 000 x 10) + (0.5 x 300 000 x ((210^2)) – (0.5 x 300 000 x (75^2)) – ((60 000 000)/(660))) = 3.58 x (10^10) J correct to 3 s.f.

However, the correct answer is 3.83 x (10^9) J.
What mistake(s) did I make?
 
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  • #2
furor celtica said:
(total energy at take-off)+(work done by engines)+(W, work done against air resistance)=(total energy at 10 000 metres)

I think that the above equation must be

(total energy at take-off)+(work done by engines) = (W, work done against air resistance) + (total energy at 10 000 metres)
 
  • #3
furor celtica said:
Work done by engines (in 11 minutes) = (60 000 000)/(660)

Note that Energy = Power x time
 
  • #4
thanks!
 
  • #5




It seems like you have made a small calculation error in your equation for the work done against air resistance. The correct equation should be W = (PE at 10,000m + KE at 10,000m) - (PE at take-off + KE at take-off) - (work done by engines). This will give you the correct answer of 3.83 x (10^9) J. Additionally, it is important to note that the work done by the engines should be calculated in joules per second (Watts), so you will need to multiply your result by 11 minutes (660 seconds) to get the correct value. Other than that, your approach and reasoning seem to be correct.
 

1. What is the total energy of an airplane?

The total energy of an airplane is the sum of its kinetic energy (energy of motion) and potential energy (energy of position). These energies include the energy of the engines, the energy stored in the fuel, and the energy required to overcome air resistance.

2. How is work done against air resistance?

Work is done against air resistance when an airplane moves through the air, causing air molecules to be displaced. This displacement results in a force acting opposite to the direction of the airplane's motion, which requires the airplane's engines to continually provide energy to maintain its speed.

3. Why is air resistance important for an airplane's energy?

Air resistance is an important factor in an airplane's energy because it represents the force that must be overcome to maintain flight. The higher the air resistance, the more energy is required for the airplane to maintain its speed and altitude.

4. How does air resistance affect an airplane's fuel efficiency?

Air resistance has a significant impact on an airplane's fuel efficiency. The more air resistance an airplane experiences, the more energy is required to maintain its speed, which leads to higher fuel consumption. This is why airplanes often fly at higher altitudes where the air is less dense, reducing the amount of air resistance.

5. Can the total energy of an airplane change during flight?

Yes, the total energy of an airplane can change during flight. For example, when an airplane is climbing, its potential energy increases while its kinetic energy decreases. Similarly, during descent, the potential energy decreases while the kinetic energy increases. However, the total energy of the airplane remains constant as long as there is no net work done on the airplane, meaning the total energy is conserved.

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