This is wrong. Planes actually fly by stretching air over their wings (creating low pressure) and the higher pressure air below pushes the plane up (to fill in the void) - So planes fly by moving air up.
PS: The reason most commercial jets cruise at ~585mph (or 85% of Mach one) is because of the huge amount of energy required to break the speed of sound.
Edit: I've you're going to downvote, you can at least comment.
Air pressure is an abstraction over movement of air particles. What a low pressure area is is fewer air molecules hitting the surface, while high pressure is more air molecules hitting the surface. So when you have more air molecules hitting the downward-facing surface, the wing pushes more air molecules down than it pushes upwards.
The air-pressure explanation and the "throwing air" explanation is basically the same thing.
The vast majority of lift for flight is achieved by pretty much throwing air down, the Bernoilli Principle is outdated as a theory behind flight (as can be demonstrated by flat-winged flight or inverted flight which shouldn't work without the pressure differential)
There's plenty of pressure differential in a lifting plate in a positive alpha. There must be in order to impart a net downward acceleration on the flow, thereby turning it. Here, have some CFD: https://www.youtube.com/watch?v=KldLqShzCRM
The problem in these discussion is "Bernoulli" is used to refer to many things, some legitimately physical (mass continuity) and some not (equal time of passage). I like FYFD's point that what we're discussing is how to explain lift to laypeople, not how to calculate and analyze it (which is done every day in order to design aircraft): http://fuckyeahfluiddynamics.tumblr.com/post/121024503205/ev.... We (mechanical/aerospace engineers) know the equations that govern fluid motion and how to solve them, whether you abstract that with Bernoulli/Newton/Kutta-Joukowsky/etc. is up to you and not that significant.
All of your references are discussing and debunking misapplication of Bernoulli.
In fact, Bernoulli applied correctly explains all of the lift. The Newton approach also explains all of the lift. They are just two different ways of looking at the same thing. Bernoulli comes from applying conservation of energy. Newton comes from applying conservation of momentum.
What's puzzling is that it is not at all uncommon in other areas of physics for there to be different ways to explain something, and this causes no problems.
If one person calculated the velocity of a dropped object by integrating the acceleration, and another person did it by using conservation of energy on the conversion from gravitational potential energy to kinetic energy, people might argue over which approach works better, but no one would argue that the other approach does not work and does not explain 100% of the velocity.
Or one person might calculate the response of a linear circuit to a sinusoidal signal by working in the frequency domain, and another person might work in the time domain. People might argue over which approach is easier, but again no one would say that only their approach works.
But with lift, people seem to like to latch onto one approach to the point of rejecting the very idea that there might be a different way to get the same result. It makes no sense to me.
The Newton approach is easier to teach reasonably correctly, so that's probably best up through, say, first year college physics and for pilot training.
Wrong. Laymen typically overestimate importance of force coming from
Bernoulli's principle. You could fly on a barn door if you had enough speed,
because attack angle is more important. Hint: why can planes fly upside down?
We still don't understand areodynamics fully, so you can't arbitrarily state
that difference between pressures above and below wings is the main force.
F=MA still holds for airplanes. If there's no downwards acceleration of air, then there's no force to push the airplane upwards.
The way that this happens comes from a number of effects, from vortexes at the tips of the wings to the angle of attack, but in the end it all comes down to F=MA. And the only thing that can be accelerated by a plane is air. Ergo, air is moving downwards so that the plane will move upwards.
It's better to just remember that energy is conserved. Bernoulli's principle is a real thing and will add to the lift a wing generates without contributing to any net motion of the air upward or downward. The work done to compress the air below the wing and expand it above helps keep the plane in the air.
But, as everyone else responding to the parent has pointed out, the forces generated in this way are not nearly enough to keep the plane in the air on their own.
> Bernoulli's principle is a real thing and will add to the lift a wing generates without contributing to any net motion of the air upward or downward.
The compression of the air below the wing moves it downwards, more or less. There are no ways to get around Newton's laws of motion (barring relativity, and planes aren't fast enough for that).
After the wing passes, the air is no longer compressed, and the net motion of the air is unchanged from before it encountered the wing. What then?
Anyway, I'm not saying you're wrong, I'm just saying that thinking about the individual forces involved leads to more complexity than necessary (this is usually the case). It's much simpler to just consider the energy going into the system and derive the net work produced from that.
No, there would be a small amount of lift generated, in accordance with Bernoulli's principle, despite there being no net motion of the air after the wing has passed.
Uh, by the downwash generated by the wing. Obviously. Are you even reading my posts? I am not asserting that the Bernoulli force is the dominating factor, or even a significant factor, in the lift generated by a typical wing. I've made that really clear. Your original post seems to take that even further, however, in asserting that the force does not exist at all, and this is flatly wrong. Have I misread your original post? The Bernoulli force does exist on a wing, and it is in principle possible to make a wing which will generate lift solely by Bernoulli force. Such a wing would likely not be able to lift its own weight, much less carry an airplane.
Bernoulli force is just a way to look at things. It can fully explain the lift - as well as why planes can fly upside down. Alternatively, you can operate with direct Newton laws without referring to Bernoulli effect. You can mix these two explanations :) or have some other one - these are all just different aspects of the same process.
For example, from Newton one can state that the whole lifting force F_lp acting on the plane is opposite to gravity force F_gp acting on the plane, so plane flies horizontally (in the simple case): F_lp + F_gp = 0. Next, from third law of mechanics the lifting force F_lp acting on the plane is opposite to F_da force acting on air downward: F_lp = -F_da. From the second law of mechanics force F_da acting on air causes the air momentum to change: F_da = dp/dt, which means that either the plane pushes a lot of air down or the plain pushes air down really fast, just like article says.
Looking into the picture from Bernoulli perspective, we can talk about flows around the wing. If say we have a wing which is flat underneath and curvy on top, the air is initially pushed upwards until the top point of the wing is reached. From there air continues along the wing profile, which goes down, so air has a chance to accelerate - this is effectively an empty space which becomes available to the air, and air goes from points of higher pressure to points of lower pressure. The air speed increases, the pressure remains less than in (unchanged) air below, so we're getting the Bernoulli effect - lower pressure above the wing.
If the plane is upside down, it won't fly horizontally. It has to fly somewhat upwards - in which case flows on both sides of wings change, with net effect the same - the air is pushed down, pressure under the wing is higher than above, both Newton and Bernoulli explanations work.
The way I understood this, it's impossible to generate lift without creating a net motion of air. Pressure is force per area, so saying that there is a pressure difference between the upper and lower side of the wing means that the air is exerting a force on the wing. By Newton's third law, that also means that the wing is exerting a force on the air, which has to move.
I guess you can also see it from conservation of momentum. Gravity acts on the plane, so each second there is a certain amount of downwards momentum added to the system. But the plane is not in fact moving downwards, so the all the downwards momentum has to end up in the air, so it has to move downwards. The net "lift" force is equal to the rate at which momentum is added to air.
If I understand it correctly, you can calculate the lift on the wing either in terms of the pressure on the two sides, or in terms of the downwards acceleration of air--it should be two equivalent views giving the same result. Bernoulli's principle gives a way to calculate the pressures if you only know the velocity of the air (this is a typical situation in a wind tunnel, where you make movies of moving smoke puffs), but it's not a separate effect from the air motion.
Unless I'm mistaken, ori_b is asserting that you can't generate lift without sending air off in another direction (i.e. downwash), and this is false. Again, most lift is generated this way - my posts here seem to be mistaken for arguments that Bernoulli force contributes significantly to the lift. It does not. However, the work done to reconfigure the air surrounding the wing by the Bernoulli principle will contribute to the lift, and while it does result in moving air around (not necessarily down though, actually), it does not impart any net motion to the air after the wing has passed.
Yes, both ori_b and I are asserting that you can't generate lift without sending air off in another direction. If you don't impart a net motion to the air, then by conservation of momentum you also can't generate a net lift.
A plane travelling in level flight at a constant speed has constant energy, though. Conservation of energy isn't going to explain how it keeps going in a straight line when gravity is otherwise trying to convert its potential energy into downward kinetic energy.
I think you meant to cite the third law "For every action there is an equal and opposite reaction".
In aggregate a force is applied to all the air in a way that is equal and opposite to the force applied on the plane. However "Ergo, air is moving downwards so that the plane will move upwards." is not quite complete. One need only look at an airfoil in a windtunnel to see that the action of an airfoil is not the simple "air based rocket" you would have us envision.
If what you said were true, the most efficient wing design would just be a flat plank strapped in at a 45 degree angle of attack.
What he said is an absolute truth. Momentum conservation is a fundamental law of nature, and if the air is exerting a upward force on the wing, transferring upward momentum to the wing, the wing is transferring downward momentum to the air. "Transferring downward momentum to the air" is pretty much the definition of "throwing air down". There is simply no way around that.
But it says nothing about the how a wing is throwing air down.
Yeah, I feel like the biggest fallacy when people are trying to understand "why" a wing works is to try to ascribe effect to this phenomenon or the other. All these different explanations are different ways of looking at the physical situation, but none of them are causal. A wing works because of the laws of hydrodynamics, trying to pick out one aspect as being more causal than the other is just the wrong way of looking at the problem.
There's nothing that says that two air particles that start together at the front edge of the wing and split up have to arrive at the tail edge of the wing at the same time.
So why would it be that the air on top of the wing has to go faster than the air below the wing?
The idea is that the wing is slicing through still air, so the molecules arrive back at the same spot much like you'd expect a slice of cheese to line back up with itself after cutting it. Friction on the surface of the wing means this isn't quite going to happen, but that's the idea.
Again, this assumes the air is already connected to the air around it in the same way cheese is. You can't lift a layer of air and have it connected to the air it used to be connected to. Likewise, you don't have turbulence in cheese. There's a significant difference between solids and fluids.
There's nothing that says that two air particles that start together at the front edge of the wing and split up have to arrive at the tail edge of the wing at the same time.
Exactly. This is one of the fallacies that are explained in the best reference I know about this subject: http://www.av8n.com/how/
It's even illustrated in the title picture... ;-)
"planes fly by moving air up" simply fails the sniff test - every action has an equal and opposite reaction, after all, so anything that throws air up must be pushing itself down. Since lift is a net upward force on a wing exerted by air, there has to be a net downward force on the air.
Then, to your PS, If a huge amount of energy is required to break the speed of sound, you've explained why commercial jets don't exceed mach one, not why they don't get within 100mph of it.
> Planes actually fly by stretching air over their wings (creating low pressure) and the higher pressure air below pushes the plane up (to fill in the void)
Doesn't extended upside-down flight require pointing the nose up (as in away from the ground) in order to compensate for the decrease in lift? I've yet to try this (I suppose I could; I've got an X-Plane-running flight sim rig about 10 feet away from me), but my intuition and understanding suggests that if the only difference between two planes flying in the same exact atmospheric conditions is that one of them is upside-down, the upside-down one will start losing altitude.
So I tried it in some simulated Cessna, and as I predicted, upon rotating so that my head pointed at the ground, the plane started to descend. I'm a pretty terrible pilot, though :)
I think this has a lot to do with the wing's angle of attack (most planes' wings have a slight upward tilt; when upside-down, that tilt goes downward instead). It would be more interesting to see the effect on something without an AoA bias.
Whenever I've played a flying simulator, if I've put the plane upside down, to maintain altitude I need to push the joystick forward. In an upright plane it would cause a nose dive, on an upside down plane, it creates lift. Due to it's not optimal aerodynamics, it also requires more thrust to maintain. Basically you are changing the shape of the wing to compensate for the non-conventional position.
> Planes actually fly by stretching air over their wings (creating low pressure) and the higher pressure air below pushes the plane up (to fill in the void) - So planes fly by moving air up.
Even if your premise is accurate (I'm not qualified to say 100%), your conclusion does not follow. Water holds up the hull of a boat, but you wouldn't say that a boat floats by moving water up.
Basically, the idea that the Bernoulli Principle is the primary factor in airplane lift is a popular lie. For some reason it continues to be broadly taught, so you can't be blamed for not knowing better. The main factor is actually angle of attack--when you hold your hand flat out a car window, tilting the leading edge up drags your hand upwards. That's the core principle of powered flight.
Most conventional planes have the wings installed with an upward angle, so they can achieve lift in level flight. Fighter and aerobatic planes have wings parallel with the fuselage, which means they have to keep the nose slightly up to fly level, but in exchange they can fly upside down almost as well as rightside up.
Yes and no. The article acknowledges that the key to improving aircraft efficiency comes in improving engine efficiency. While it says, "Engines are pretty darn efficient already," it doesn't really quantify that or do any analysis of the efficiency limits of engines.
Just because you can't gain efficiency by changing the speed of the plane, with no other changes allowed, doesn't mean you have no options :
1) reduce weight (fuselage, engine, wings, have the passengers diet first, ...)
2) increase engine fuel efficiency, displace more air for fewer hydrocarbons
3) (in the article) reduce drag on surfaces that don't generate lift
4) lower cross section (a longer, thinner tube), of course, still needs structural soundness
The lower cross section would have the additional advantage that it would make the plane fly faster.
So in a way, the article is "wrong". You can make a faster plane, assuming you can solve all the issues that the obvious modifications bring : longer, thinner and lighter fuselage and wings, combined with similar efficiency engines (of course they'd have to have that similar efficiency at higher speeds).
Another way would be to fly (much) higher. But then the cost of climbing high would start to offset the gains from flying faster.
I really like the simplified explanation of how much energy to takes to move the air out of the way (v)^3 - I've heard this stated for vehicles before which is why roof racks are so bad for mileage.
Does anyone know of a similarly simple explanation of how much energy a vehicle must use relative to it's weight?
That's actually a bit oversimplified for highly aerodynamic objects like aircraft. It is known that such shapes can approach (v)^2. A (v)^3 characteristic is more representative of shapes that actually do have to accelerate all the air up to the speed of the vehicle such as the proverbial flat plate.
This doesn't change the conclusion; you still end up with a low spot on the graph. It is just that the graph looks more like a parabola.
The best way to think of simple back-of-the-envelope calculations is this:
To go from point a to point b at a cruising speed of v, you need to speed up to v, maintain v, and brake to a complete stop. Analyze each separately:
1. Speeding up. Since this takes so little distance (compared to the total length), let's ignore resistance. This speedup takes KE = ½·m·v² of energy to do.
2. Maintain speed. This means spending energy to resist air friction (drag) and ground friction. Drag is ½·ρ·v²·Cd·A [0]; friction is (at most) μ·m·g. Energy is force × distance, and since the distance at top speed is effectively constant, it's going to be O(m + v²) in this case. [1]
3. Braking. You can theoretically recover -½·m·v² here, but most of the energy is going to be lost to heat.
So, when you sum these up, the energy it takes to drive a vehicle a certain distance is linear in mass and quadratic in speed.
[0] The drag coefficient, Cd, is really weird. At low speeds, it's actually proportional to 1/v instead of a constant (i.e., drag is linear in velocity, not quadratic).
[1] Yeah, the article says that energy is proportional to v³. It's actually wrong--the power is proportional to v³. Whether or not the total energy is proportional to v³ or v² depends on what you hold constant: if you hold total time constant, it's v³, but if you hold distance, it's v².
Well then, how about charging more for faster flights, to account for fuel? I mean, domestic business class isn't that great, but people who pay others people money for it would presumably go for a faster flight.
The majority of people who fly do so in cattle class, where basically the headline ticket price is the biggest factor in choosing prices (which is why low-cost airlines offer cheap tickets but charge steep prices for doing things like bringing luggage or getting a drink on board the plane).
You can't fit enough people in first or business class to fill a plane on most routes, and you simply can't go fast enough to make speed premiums really worth it. Note that going to an airport usually requires on the order of 2 hours of hassle before accounting for flight time, which means that you've already lost a day of work in practice. No, unless it's a long-haul flight, there's little benefit to faster speeds, and the Concorde was never really profitable.
I'm not sure if I agree with the conclusion of the article; if the problem is that the effect of drag increases with speed, then the answer here seems to be to decrease drag (perhaps with a more efficient flight profile). Or would that interfere with lift?
Ah, nothing like a good old-fashioned brouhaha over angle of attack vs. the Bernoulli principle. This same argument was going on when my Dad was training as a private pilot back in the 60's. Tastes great! Less filling!
Personally, I think they're facets of the same thing, but I don't know anything about it. Beyond that the piece seemed a bit of a mess to me. The content has not aged well. I usually expect Wired to do a better job of conserving the state of what they publish.
I remember how happy I was when I firstly made this consideration myself :) --> as racing cars have a back "aileron" which forces air to push them down (and stick to the road), airplanes have the same thing but inverted, which pushes them up. This becomes exceptionally clear when thinking about how flaps change the shape of the wing.
OK, so the plane shoves air out of the way, and that takes energy. But when the plane passes, air will fill the air tube. Is there any way in theory to recover from that air filling the air tube some of the energy that was expended in making the tube?
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[ 3.0 ms ] story [ 122 ms ] threadThis is wrong. Planes actually fly by stretching air over their wings (creating low pressure) and the higher pressure air below pushes the plane up (to fill in the void) - So planes fly by moving air up.
See Bernoulli Principle (https://en.wikipedia.org/wiki/Bernoulli%27s_principle)
PS: The reason most commercial jets cruise at ~585mph (or 85% of Mach one) is because of the huge amount of energy required to break the speed of sound.
Edit: I've you're going to downvote, you can at least comment.
The air-pressure explanation and the "throwing air" explanation is basically the same thing.
See: http://home.comcast.net/~clipper-108/lift.pdf
Or:
http://physics.stackexchange.com/questions/290/what-really-a...
The problem in these discussion is "Bernoulli" is used to refer to many things, some legitimately physical (mass continuity) and some not (equal time of passage). I like FYFD's point that what we're discussing is how to explain lift to laypeople, not how to calculate and analyze it (which is done every day in order to design aircraft): http://fuckyeahfluiddynamics.tumblr.com/post/121024503205/ev.... We (mechanical/aerospace engineers) know the equations that govern fluid motion and how to solve them, whether you abstract that with Bernoulli/Newton/Kutta-Joukowsky/etc. is up to you and not that significant.
In fact, Bernoulli applied correctly explains all of the lift. The Newton approach also explains all of the lift. They are just two different ways of looking at the same thing. Bernoulli comes from applying conservation of energy. Newton comes from applying conservation of momentum.
See this discussion from NASA: https://www.grc.nasa.gov/www/K-12/airplane/bernnew.html
What's puzzling is that it is not at all uncommon in other areas of physics for there to be different ways to explain something, and this causes no problems.
If one person calculated the velocity of a dropped object by integrating the acceleration, and another person did it by using conservation of energy on the conversion from gravitational potential energy to kinetic energy, people might argue over which approach works better, but no one would argue that the other approach does not work and does not explain 100% of the velocity.
Or one person might calculate the response of a linear circuit to a sinusoidal signal by working in the frequency domain, and another person might work in the time domain. People might argue over which approach is easier, but again no one would say that only their approach works.
But with lift, people seem to like to latch onto one approach to the point of rejecting the very idea that there might be a different way to get the same result. It makes no sense to me.
The Newton approach is easier to teach reasonably correctly, so that's probably best up through, say, first year college physics and for pilot training.
We still don't understand areodynamics fully, so you can't arbitrarily state that difference between pressures above and below wings is the main force.
And no, I didn't downvote you.
The way that this happens comes from a number of effects, from vortexes at the tips of the wings to the angle of attack, but in the end it all comes down to F=MA. And the only thing that can be accelerated by a plane is air. Ergo, air is moving downwards so that the plane will move upwards.
But, as everyone else responding to the parent has pointed out, the forces generated in this way are not nearly enough to keep the plane in the air on their own.
The compression of the air below the wing moves it downwards, more or less. There are no ways to get around Newton's laws of motion (barring relativity, and planes aren't fast enough for that).
Anyway, I'm not saying you're wrong, I'm just saying that thinking about the individual forces involved leads to more complexity than necessary (this is usually the case). It's much simpler to just consider the energy going into the system and derive the net work produced from that.
Then, there would be no lift generated, since there is no acceleration of mass.
For example, from Newton one can state that the whole lifting force F_lp acting on the plane is opposite to gravity force F_gp acting on the plane, so plane flies horizontally (in the simple case): F_lp + F_gp = 0. Next, from third law of mechanics the lifting force F_lp acting on the plane is opposite to F_da force acting on air downward: F_lp = -F_da. From the second law of mechanics force F_da acting on air causes the air momentum to change: F_da = dp/dt, which means that either the plane pushes a lot of air down or the plain pushes air down really fast, just like article says.
Looking into the picture from Bernoulli perspective, we can talk about flows around the wing. If say we have a wing which is flat underneath and curvy on top, the air is initially pushed upwards until the top point of the wing is reached. From there air continues along the wing profile, which goes down, so air has a chance to accelerate - this is effectively an empty space which becomes available to the air, and air goes from points of higher pressure to points of lower pressure. The air speed increases, the pressure remains less than in (unchanged) air below, so we're getting the Bernoulli effect - lower pressure above the wing.
If the plane is upside down, it won't fly horizontally. It has to fly somewhat upwards - in which case flows on both sides of wings change, with net effect the same - the air is pushed down, pressure under the wing is higher than above, both Newton and Bernoulli explanations work.
I guess you can also see it from conservation of momentum. Gravity acts on the plane, so each second there is a certain amount of downwards momentum added to the system. But the plane is not in fact moving downwards, so the all the downwards momentum has to end up in the air, so it has to move downwards. The net "lift" force is equal to the rate at which momentum is added to air.
If I understand it correctly, you can calculate the lift on the wing either in terms of the pressure on the two sides, or in terms of the downwards acceleration of air--it should be two equivalent views giving the same result. Bernoulli's principle gives a way to calculate the pressures if you only know the velocity of the air (this is a typical situation in a wind tunnel, where you make movies of moving smoke puffs), but it's not a separate effect from the air motion.
In aggregate a force is applied to all the air in a way that is equal and opposite to the force applied on the plane. However "Ergo, air is moving downwards so that the plane will move upwards." is not quite complete. One need only look at an airfoil in a windtunnel to see that the action of an airfoil is not the simple "air based rocket" you would have us envision.
If what you said were true, the most efficient wing design would just be a flat plank strapped in at a 45 degree angle of attack.
But it says nothing about the how a wing is throwing air down.
So why would it be that the air on top of the wing has to go faster than the air below the wing?
Exactly. This is one of the fallacies that are explained in the best reference I know about this subject: http://www.av8n.com/how/ It's even illustrated in the title picture... ;-)
Then, to your PS, If a huge amount of energy is required to break the speed of sound, you've explained why commercial jets don't exceed mach one, not why they don't get within 100mph of it.
So why can planes fly upside down then?
I think this has a lot to do with the wing's angle of attack (most planes' wings have a slight upward tilt; when upside-down, that tilt goes downward instead). It would be more interesting to see the effect on something without an AoA bias.
Even if your premise is accurate (I'm not qualified to say 100%), your conclusion does not follow. Water holds up the hull of a boat, but you wouldn't say that a boat floats by moving water up.
Basically, the idea that the Bernoulli Principle is the primary factor in airplane lift is a popular lie. For some reason it continues to be broadly taught, so you can't be blamed for not knowing better. The main factor is actually angle of attack--when you hold your hand flat out a car window, tilting the leading edge up drags your hand upwards. That's the core principle of powered flight.
Most conventional planes have the wings installed with an upward angle, so they can achieve lift in level flight. Fighter and aerobatic planes have wings parallel with the fuselage, which means they have to keep the nose slightly up to fly level, but in exchange they can fly upside down almost as well as rightside up.
1) reduce weight (fuselage, engine, wings, have the passengers diet first, ...) 2) increase engine fuel efficiency, displace more air for fewer hydrocarbons 3) (in the article) reduce drag on surfaces that don't generate lift 4) lower cross section (a longer, thinner tube), of course, still needs structural soundness
The lower cross section would have the additional advantage that it would make the plane fly faster.
So in a way, the article is "wrong". You can make a faster plane, assuming you can solve all the issues that the obvious modifications bring : longer, thinner and lighter fuselage and wings, combined with similar efficiency engines (of course they'd have to have that similar efficiency at higher speeds).
Another way would be to fly (much) higher. But then the cost of climbing high would start to offset the gains from flying faster.
Does anyone know of a similarly simple explanation of how much energy a vehicle must use relative to it's weight?
This doesn't change the conclusion; you still end up with a low spot on the graph. It is just that the graph looks more like a parabola.
To go from point a to point b at a cruising speed of v, you need to speed up to v, maintain v, and brake to a complete stop. Analyze each separately:
1. Speeding up. Since this takes so little distance (compared to the total length), let's ignore resistance. This speedup takes KE = ½·m·v² of energy to do.
2. Maintain speed. This means spending energy to resist air friction (drag) and ground friction. Drag is ½·ρ·v²·Cd·A [0]; friction is (at most) μ·m·g. Energy is force × distance, and since the distance at top speed is effectively constant, it's going to be O(m + v²) in this case. [1]
3. Braking. You can theoretically recover -½·m·v² here, but most of the energy is going to be lost to heat.
So, when you sum these up, the energy it takes to drive a vehicle a certain distance is linear in mass and quadratic in speed.
[0] The drag coefficient, Cd, is really weird. At low speeds, it's actually proportional to 1/v instead of a constant (i.e., drag is linear in velocity, not quadratic).
[1] Yeah, the article says that energy is proportional to v³. It's actually wrong--the power is proportional to v³. Whether or not the total energy is proportional to v³ or v² depends on what you hold constant: if you hold total time constant, it's v³, but if you hold distance, it's v².
They did with limited success. https://en.wikipedia.org/wiki/Concorde
You can't fit enough people in first or business class to fill a plane on most routes, and you simply can't go fast enough to make speed premiums really worth it. Note that going to an airport usually requires on the order of 2 hours of hassle before accounting for flight time, which means that you've already lost a day of work in practice. No, unless it's a long-haul flight, there's little benefit to faster speeds, and the Concorde was never really profitable.
Personally, I think they're facets of the same thing, but I don't know anything about it. Beyond that the piece seemed a bit of a mess to me. The content has not aged well. I usually expect Wired to do a better job of conserving the state of what they publish.
Should be something like "Can We Build a More Efficient Airplane? Not Really, Says Physics", not "Why Cant Commercial Airplanes Go Faster?"