An airplane in flight is subject to an air resistance force...

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 ($\textbf{Fig. P6.94}$). 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, $\alpha = 0.30 \, \mathrm{N} \cdot \mathrm{s^{2}/m^{2}}$ and $\beta = 3.5 \times 10^5 \, \mathrm{N} \cdot \mathrm{m^2/s^2}$. 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). Figure P6.94 (Figure can't copy)

Key Concepts

Induced Drag

Induced drag is directly related to the production of lift. As an aircraft generates lift to counteract its weight, the wing’s interaction with the air creates a downward flow, resulting in a reactive force that produces drag. At higher speeds, induced drag decreases and is often inversely proportional to the square of the speed, which contrasts with parasitic drag.

Optimization of Flight Performance

In aircraft performance analysis, optimization involves finding the ideal parameters (such as speed) that maximize specific objectives, like range (distance traveled per fuel unit) or endurance (time aloft). This requires balancing the opposing behaviors of different drag components, using techniques such as differentiation to identify minimum fuel consumption or power requirements.

Aerodynamic Drag

Aerodynamic drag is the force that opposes an object’s motion through a fluid. It results from the friction and pressure differences in the airflow around the object and plays a critical role in determining the performance and fuel efficiency of aircraft during flight.

Parasitic Drag

Parasitic drag, sometimes called quadratic drag, arises mainly from the friction of the air against the surface of the aircraft and the pressure drag due to the shape of the aircraft. It typically increases with the square of the speed, meaning that at higher speeds, this component of drag becomes increasingly significant.

Newton's Third Law in Aerodynamics

Newton's Third Law, which states that every action has an equal and opposite reaction, is fundamental in aerodynamics. When an aircraft wing pushes air downward to generate lift, the air exerts an upward force on the wing. This physical principle is also key to understanding phenomena like induced drag, where the reactive forces due to lift generation contribute to the overall resistance encountered by the aircraft.

Transcript

00:01 Welcome to a new problem.

00:04 This time we have a plane that's flying in a specific direction and it's being affected by multiple forces.

00:14 For example, it's being lifted up and there is also, so this is a lift.

00:23 And then there's a drag on the plane.

00:27 On top of that, the air is pulling.

00:33 Pushing up the wings.

00:35 So this is the force of the air on the wings.

00:44 And there are a couple of things that stand out in this problem.

00:47 For example, air resistance is proportional.

00:52 Air resistance is proportional to the square of the velocity of the plane.

00:58 And the wings also provide some resistance.

01:03 Remember, air flows through the wings.

01:07 So if this is a wing, you're going to have some air flowing on top of the wing and it does something.

01:16 It goes slightly forward and it goes forward and then it's pushed downwards.

01:24 So if this is the wing, this is the plane.

01:27 You know, the air flows through the wings and then it's pushed downwards.

01:35 And according to newton's third law action and reaction and reaction, forces are equal in the opposite and so this action of being pushed down it's gonna force the wings to be lifted upwards and that's what you want because you want to raise the plane but then the up the the outcome of that process the outcome of the air flowing through the wings means that you you have a backward -induced drug.

02:14 So as this one goes forward like that and then down, then you have this drug going backwards like that.

02:20 We call this an induced drug because of the air in the wing.

02:30 And it's inversely proportional to the velocity squared.

02:36 So if you look at those two aspects of the...

02:43 Air resistance and the wing resistance there is a force that comes up responsible for these two actions this action right here and this action right here and that force is modeled by alpha v squared plus beta of a v squared remember alpha for a cessna plane for example cessna 150 the alpha happens to be a 0 .30 newton second squared of a meter squared and then the beta happens to be so this is just specific for for the sessna other planes will definitely have different types of numbers constants but in general this is what happens so there are two things we're looking at in this problem the first part is that we're going to transform things in kilometers per hour we want to get the maximum range of the plane.

03:42 And then the second part is we want to get the maximum endurance of the plane...

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