Maximizing Range/Time in Air of an Airplane: Solving with Calculus

In summary, the conversation discusses maximizing the range and time of an airplane by minimizing the engine force and power, respectively. Calculus is used to find the minimum values, resulting in a maximum range of 120 km/h and a maximum time in air of 90 km/h. One minor typo is pointed out and corrected during the conversation.
  • #1
Argonaut
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Homework Statement
[ Young & Freedman - University Physics 13E, Ex 6.104]
An airplane in flight is subject to an air resistance force proportional to the square of its speed v. But there is an additional resistive force because the airplane has wings. Air flowing over the wings is pushed down and slightly forward, so from Newton's third law the air exerts a force on the wings and airplane that is up and slightly backward (Fig. P6.104). The upward force is the lift force that keeps the airplane aloft, and the backward force is called induced drag. At flying speeds, induced drag is inversely proportional to $v^2$, so the total air resistance force can be expressed by ##F_{air} = \alpha v^2 + \beta /v{^2}##, where ##\alpha## and ##\beta## are positive constants that depend on the shape and size of the airplane and the density of the air. For a Cessna 150, a small single-engine airplane, ##\alpha = 0.30 ~\rm{N} \cdot ~\rm{s^{2}/m^{2}}## and ##\beta = 3.5 \times 10^5 ~\rm{N} \cdot ~\rm{m^2/s^2}##. In steady flight, the engine must provide a forward force that exactly balances the air resistance force. (a) Calculate the speed (in km/h) at which this airplane will have the maximum range (that is, travel the greatest distance) for a given quantity of fuel. (b) Calculate the speed (in km/h) for which the airplane will have the maximum endurance (that is, remain in the air the longest time).
Relevant Equations
Work, kinetic force, power
IMG_20230423_100342__01.jpg


Is my solution correct? (I only have answers to odd-numbered exercises.)
Is it a good solution or have I overcomplicated things?

(a)

The forward force provided by the engine balances the air resistance force, so ##F_{engine}=F_{air} = \alpha v^2 + \beta /v{^2}##.

Let ##W_{engine}## be the energy content of the given quantity of fuel. Then ##W_{engine} = F_{engine}d ## where ##d## is range. So

$$ d = \frac{W_{engine}}{F_{engine}} = \frac{W_{engine}}{\alpha v^2 + \beta /v{^2}} $$

We want to maximise ##d##. We can achieve that if we minimise ##F_{engine}##, since ##W_{engine}## is a constant.

We use calculus to minimise it. Let ##f(v) = \alpha v^2 + \beta /v{^2}##. Then ##f'(v) = 2\alpha - \frac{2\beta}{v^3}##. We find the minimum value by setting ##f'(v) = 0## and rearranging it to express ##v##, we obtain

$$
v = \left(\frac{\beta}{\alpha}\right)^{1/4} = \left(\frac{3.5 \times 10^5 ~\rm{N} ~\rm{m^2/s^2}}{0.30 ~\rm{N} ~\rm{s^{2}/m^{2}}}\right)^{1/4} = 33 ~\rm{m/s} = 120 ~\rm{km/h}
$$

Thus the airplane will achieve the maximum range travelling at a speed of ##120 ~\rm{km/h}##.

(b)
##P_{av} = \frac{\Delta W}{\Delta t}## and we want to maximise ##\Delta t##. We can achieve this by minimising ##P##, since ##\Delta W = W_{engine}## is a constant. We use calculus to minimise ##P##. Let ##P = g(v)##. Then ##g'(v) = 3\alpha v^2-\frac{\beta}{v^2}##. Setting ##g'(v)=0## and rearranging it to express ##v##, we obtain

$$
v= \left(\frac{\beta}{3\alpha}\right)^{1/4} = \left(\frac{3.5 \times 10^5 ~\rm{N} ~\rm{m^2/s^2}}{3(0.30 ~\rm{N} ~\rm{s^{2}/m^{2}})}\right)^{1/4} = 25 ~\rm{m/s} = 90 ~\rm{km/h}
$$

Therefore, the airplane will achieve maximum time in air at a speed of ##90 ~\rm{km/h}##.
 
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  • #2
Looks correct to me.
 
  • Like
Likes Argonaut
  • #3
Looks fine except for a typo (dropped v) in the expression for f'.
 
  • Like
Likes Argonaut
  • #4
Thanks, both!
 
  • #5
Argonaut said:
We use calculus to minimise it. Let ##f(v) = \alpha v^2 + \beta /v{^2}##. Then ##f'(v) = 2\alpha v - \frac{2\beta}{v^3}##. We find the minimum value by setting ##f'(v) = 0## and rearranging it to express ##v##, we obtain...
Your v was missing after 2α.

Drag1_0.jpg
 

1. How can calculus be used to maximize the range/time in air of an airplane?

Calculus can be used to optimize the performance of an airplane by analyzing various factors such as air resistance, weight, and lift. By using derivatives and integrals, calculus can help determine the optimal angle of ascent and descent, as well as the most efficient speed and altitude for the airplane to travel at. This can ultimately lead to maximizing the range and time in air for the airplane.

2. What are the main variables that affect the range/time in air of an airplane?

The main variables that affect the range/time in air of an airplane include the aerodynamics of the airplane, the weather conditions, the weight of the airplane, and the engine's power and efficiency. These factors can all be analyzed and optimized using calculus to improve the airplane's performance.

3. How does air resistance impact the range/time in air of an airplane?

Air resistance, also known as drag, is one of the main forces that act upon an airplane during flight. It is caused by the friction between the airplane and the air molecules it is traveling through. This force can significantly impact the range and time in air of an airplane, as it requires more energy and fuel to overcome. Calculus can be used to determine the optimal speed and angle of ascent/descent to minimize air resistance and improve the airplane's performance.

4. Can calculus also be used to optimize the fuel efficiency of an airplane?

Yes, calculus can also be used to optimize the fuel efficiency of an airplane. By analyzing the relationship between the airplane's speed, altitude, and fuel consumption, calculus can determine the most efficient flight path for the airplane. This can help reduce fuel costs and increase the range and time in air of the airplane.

5. Are there any limitations to using calculus for optimizing the range/time in air of an airplane?

While calculus can be a powerful tool for optimizing the performance of an airplane, it does have its limitations. Calculus is based on mathematical models and assumptions, which may not always accurately reflect real-world conditions. Additionally, there are many variables and factors that can affect the performance of an airplane, making it difficult to create a perfect mathematical model. Therefore, while calculus can provide valuable insights, it should be used in conjunction with other methods and considerations to fully optimize the range and time in air of an airplane.

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