Chapter 3

The intention of this chapter is to investigate the science of flight for determining the validity of paleontology’s community claim that there is nothing odd about large flying pterosaurs. Yet this chapter also stands on its own by deriving the Power for Flight Equations and thus greatly advancing our understanding of flight. While many readers may only be interested in understanding the science of flight others readers will also be interested in the discussion concerning animal power to fully comprehend the paradox regarding flying pterosaurs. The readers who are interested in seeing the information regarding the animal power needed for flight should click on the link below while those are interested in only the flight of airplanes can skip over the link to this section.

For centuries man has dreamed of flying, then on December 17, 1903 at Kitty Hawk, North Carolina, a couple of bicycle mechanics turn scientists won the race to be the first to fly. Aviation technology has moved forward dramatically during the hundred years that followed; the jet airliners of today are themselves longer than the first three of the Wright’s first four historic flights. And so this rapid advancement in technology might fool us into believing that the aviation experts have a complete understanding of the science of flight when in fact this is not true. The past century’s great advancement in aviation technology has largely been achieved through a trial and error process such that the technology has led the science. To further explain, the Wright brothers did not necessary understanding why a particular wing design generated more lift, they just knew from their wind tunnel testing and test flying their gliders that it did. Likewise, a century later aerodynamic engineers still rely on wind tunnel testing and the feedback from test pilots to guide the design of modern airplanes.

To clarify the science community’s lack of understanding of the requirements for flight, consider the paleontologists’ latest attempt to explain how the large pterosaurs flew. Their ‘solution’ is that these large animals used all four limbs to leap into the air. That is it; no mathematics, no citing of aerodynamic or physics principles. The giant pterosaurs just leaped into the air … and … and … they flew.

The physics principles regarding flight are applicable to planes, birds, and even pterosaurs so before we laugh at the paleontologists’ poor understanding of aerodynamics we should first wonder about the aeronautic industry’s limited understanding of flight in general. The fact that the aeronautic industry has produced a wide variety of successful planes might suggest that the aeronautical industry has a sound theoretical understanding of how planes and animals fly. Yet that assumption would be incorrect; while aerodynamic engineers know far more about flying than the paleontologists, they are still unable to identify the core theoretical requirements for achieving flight.

Building flying machines is not unlike numerous other human endeavors in that inventers are often able to advance technology even through they do not have a core theory as to how something actual works. Every newly design plane started out as an experimental plane. These experimental planes were flown by brave test pilots who communicated to the engineers how each modification to the plane affected its performance. Over the past century aviation technology advanced partly as a result of the calculate designs of the aerodynamic engineers, but just as much so the technology advanced as a result of this trial and err experimentation.

Even though it makes incorrect statements and it gives no insight to designing a wing, the Equal Transit Time / Bernoulli’s Principle explanation has been the standard explanation of flight for many decades. While this explanation is finally starting to fall out favor, it is still the most popular explanation giving in physical science textbooks. Let us take a moment to review this explanation and recognize its flaws in the hopes of undoing some of the propagated misconceptions regarding flight.

The Equal Transit Time / Bernoulli’s Principle explanation of flight says that the air traveling over the wing must complete its journey in the same amount of time as the air flowing beneath the wing. The statement implies that the two air flows that are divided at the front of the wing are soul-mates of each other such that must each travel their paths in the same amount of time before uniting at the rear of the wing. However there is no conservation law of physics requiring this, and in fact the air flowing over the top wing surface reaches the back of the wing long before the air that is traveling beneath the wing.

Bernoulli’s Principle is simply a statement regarding the relationship between fluid speed and static pressure, and in fact the Equal Transit Time / Bernoulli explanation of flight has the cause and effect reversed. The pressure above the wing is not reduced because the air is flowing faster but rather the air is flowing faster because the pressure above the wing is reduced. Invoking Bernoulli’s Principles to explain flight serves no purpose other than to confuse students into believing that their question has been answered regarding how planes fly.

The correct short explanation of flight is that as the air flows over a wing it follows the form of the wing because air is like most other fluids in that it tends to stick to surfaces. As the air travels over and past the thickest part of wing the air is diverted downward as it holds on to the profile of the wing. In the process of the wings diverting a tremendous amount of air down an upward reactionary force is exerted on the wings of the plane that lifts the plane up.

Now that we are on the right track in understanding lift we can move forward to derive precisely how much power a flying object requires to fly according to its features such as weight, wingspan, drag coefficient, and other basic criteria such as air density. But before getting into the derivation of the Power for Flight Equations, let us begin by listing some of the key conceptual results.

1. Planes fly by constantly expending energy in the process of throwing air down. Unlike a light gas or hot air balloon that floats in the atmosphere, flying animals and flying machines use power – the expenditure of energy per unit time - in order to fly. While in flight, energy is constantly being transferred to the air in the process of throwing it down. Because air is invisible, this movement of the air is usually not directly or even indirectly observable for planes. But certainly when a helicopter hovers over a grassy field we can observed the grass being blown by the wind. Without the constant expenditure of energy these flying machines will fall from the sky.

2. Planes need power for both lift and for overcoming drag. Some of the plane’s power is used to transfer energy to the air as it throws the air down so as to lift the plane up. The remainder of the power is used to overcome the frictional drag force of the air that impedes the forward motion of the plane. Flying animals and flying machines must have the power to fulfill both of these requirements in order to fly.

3. Flying is unique from other means of travel in that the most efficient speed is not the lowest speed. With all forms of travel through a fluid, such as air or water, the power required to overcome drag goes up dramatically with speed. Consequently the best fuel efficiency is usually obtained by traveling slow. However flying has the additional power requirement of generating lift. Interesting enough, the amount of power required for lift decreases as the plane travels faster. Since the total power requirement is the summation of the power to overcome drag and the power for lift the most efficient speed for flying is not the slowest speed but rather it is a speed between the extremes of flying slow or extremely fast. Consequently nearly all planes and flying animals fly within a narrow range of medium speeds where they get the best fuel efficiency.

4. Wings generate lift by pulling air down. While it may be easier to conceptualize the wings throwing the air down by pushing it down, this is not the usual way that wings generate lift. Because air is like most fluids in having viscosity, the air flowing over the top wing surface ‘sticks’ to the surface and in doing so it is thrown down as it follows the curvature of the top wing surface. While it is possible for wings to generate lift by pushing the air down, pulling the air down generates more lift while producing less drag and so pulling the air down is far more effective than pushing the air down.

5. Galileo’s Square-Cube Law applies to flying such that the smaller objects take to the air easier than large objects. Because small objects have a greater area to weight ratio, small objects are easily tossed about in a breeze. For example, in the desert grains of sand are easily blown about while there is no chance of the wind picking up the much larger boulders. Likewise it is not much of a challenge for insects to be picked up by the wind and so it was not a great evolutionary leap for so many of these small insects to evolve the ability to fly.

6. The larger animals or planes require disproportionately greater power to fly. Without wind or updrafts to supply lift, an animal or aircraft must supply its own power to fly. Flying is more of a challenge for vertebrates than for insects because even the smallest flying vertebrate is still larger than nearly any insect. With greater size comes both greater weight and power, but the additional power is not nearly enough to compensate for lifting the greater weight. Flying birds are usually larger than most bats because birds have a higher metabolism and likewise greater relative power. Likewise the largest planes are all jet airplanes rather than being piston / propeller airplanes because the higher power output of a jet engine is required to lift the large heavy planes off the ground.

7. Flying animals such as birds and bats are restricted in size because of their relative power limitations. As the size of flying animals increase, their relative power decreases while their need for greater relative power is increasing. Ultimately the greater power requirements exceed the available power thus grounding the largest birds.

The lack of relative strength and power forces the larger birds to soar so to minimize the flapping of their wings. It is because of the difference in relative strength and power that the largest birds will seek out rising air thermals to stay airborne, while the smallest of birds, the humming bird, can hover and even fly backwards.

The wings of an airplane lift the aircraft up by throwing a tremendous amount of air down. This basic fact about flying may not be obvious to everyone because air is invisible and so its movement usually goes unnoticed. Nevertheless, enough people have made these observations to confirm the fact that wings generate lift by throwing air down. How the wings throw the air down is the key to understanding flight.

First thing to consider is whether wings generate lift by the bottom wing surface pushing the air down or is it the top wing surface generating the lift by pulling the air down. Conceptually it is easier to visualize the bottom wing surface lifting the plane up by pushing the air down. Essentially this is how a kite flies; the kite is lifted up by the bottom surface of the kite deflecting the wind down.

Many of the early aviation pioneers thought that this is how a plane could fly. The wings of their aircraft were nothing more than large airtight fabrics that were intended to deflect the air downward as the propeller moved the plane forward. The idea that wing surface area equates to lift is a widely held belief even to this day. Aerodynamic engineers still calculate a plane’s wing loading – the weight of the plane divided by the area of the wing – as if the total surface area of the wings is all that mattered in determining the lift generated by the wings. But generating lift by deflecting air with a large surface is not how an airplane flies. The Wright brothers discovered that a properly shaped cambered wing produced more lift with far less forward drag than any flat surface wing. This discovery was one of the primary reasons why the Wright brothers were the first to achieve fly.

A properly cambered wing produces lift by the top wing surface pulling the air down rather than having the bottom wing surface push the air down and slightly forward. This process of pulling the air down produces very little drag in comparison to trying to generate lift by using the bottom surface to push the air down. Consequently, a properly cambered wing that pulls the air down uses far less power to achieve lift than if flight were attempted by simply using a broad surface to push the air down

Since it is desirable to use the least amount of power, in normal flight planes fly by having the top wing surface pull the air down so as to generate the required lift. The exception to this rule is the supersonic aircraft that have been giving extremely broad delta wings for the purpose of helping them land. Aviation engineers have discovered that it is easier to land supersonic aircraft by putting the plane into a ‘power stall’ rather than try to land these fast jets in the conventional way. In a power stall the pilot puts the nose of the plane extremely high so that the bottom of the broad delta wing is both pushing the air down to generate lift while it is pushing air forward. In this position the large wing surface accomplishes the goals of both slowing the plane down and providing enough lift for a gentle decent. Many birds will also land this way. Just before landing they take a more upright posture, flare their rear tail feathers, and allow their broad wings to push against the air for the last second of flight before they land. But other than these exceptions, the primary way that planes and birds generate lift is by the top wing surface pulling the air down.

For further evidence that it is the top wing surface that generates the lift please take notice that if an obstruction must be placed near the wing it is always place below the wing rather than above it. While obviously the landing gear needs to be placed below the wing it is conceivable that jet engines or military weaponry could be placed above the wing rather than below the wing. But these objects are not mounted above the wing because any object mounted this way would compromise that portion of the wing’s ability to generate lift.

It is because of a fluid’s tendency to stick to the surface that the wings are able to produce lift by pulling the air down. This phenomenon known as the Coanda effect is demonstrated by holding a smooth drinking glass horizontally so that it just touches the lightly flowing water falling from a faucet. Once the water comes in contact with the glass the stream of water follows the curvature of the glass instead of falling straight down.

Likewise the air stream adheres to the surface of a cambered wing. Just as the air flows over the thickest part of the wing it takes a downward turn so as to follow the profile of the wing. In taking this turn a force is being applied on the air pulling it down and likewise in accordance to Newton’s laws an equal and opposite force is being applied to the wing lifting it up.

During a flight it is occasionally desirable for the wings to produce more lift than what they normally produce while cruising in level flight. To produce this additional lift the rear tail elevator is adjusted to lower the tail while pointing the nose of the plane up. In this new position the front of the wing is higher than the back of the wing. This increased angle of the tilt of the wing or Angle of Attack AOA effectively increases the camber of the wing. With this increases wing camber the air flowing over the wing is more sharply pulled down as it follows the profile of the top wing surface. By increasing the tilt of the wings above the usual approximately six percent AOA the wings can generate about twice as much lift as usual but this comes at the expense of creating greater drag on the aircraft.

Besides creating greater drag a high AOA increases the risk of causing the wings to stall. A stall is the aerodynamic name given for when the air stream breaks free of its attachment to the top wing surface. We can use our water flowing from facet to observe what is happening during a stall.

We place a wing profile next to the water stream so that the fluid sticks to the surface and in so doing it is diverted from flowing straight down. As we turn the wing profile we can increase how far we divert the water from flowing straight down. But if we turn our wing too far the water stream breaks free from our model wing surface. This is what happens during a stall except that it is air flowing horizontally rather than water falling vertically. The most likely time when the air stream may break from the wing is when the AOA is high and the plane is flying slowly. When a stall occurs the wings lose their lift.

While we often focus on the air stream that is flowing closes to the wings this effect that the wings have in pulling the air down actually extends far above the wings. We can think of the wings as scooping the still air above the wings and throwing it down. This scoop has an effective area A of ½ b² where b is the wingspan. This effective area of the scoop based on the wingspan assumes that the camber of the wing is of the correct proportion to that of the chord and so on so as to create an aerodynamic and effective wing.

In this equation the width of the wing is not included. Obviously the wing needs to be broad enough to create the profile that produces the Coanda effect but any width beyond this does not produce any more lift due to this effect. Thus it is not actually correct to calculate lift according to wing surface area. To give an example, if two planes have the same wing area yet one has short broad wings while the other has narrow wings that are twice as long as the first, the plane with the longer wings will generate four times as much lift as the other. It is for this reason that planes such as sailplanes that wish to maximize lift will have long narrow wings.

This is not to say that the planes with broad wings are being wasteful. Broader wings enable a plane to safely fly in rougher weather or at a slower speed without stalling. This is because the broader wing gives the air stream a greater attachment area thus making it more difficult for the air stream to break free of the top wing surface so as to produce a stall. This ability to fly slower without stalling can be important to large birds or planes when they are taking off or landing. Take note during your next flight on an airliner to observe that the pilot will expand the wings at the early part of approaching the landing.

With the correct understanding of the science of flight we can obtain tremendous insight into why the wings of birds or planes are shaped a particular way.

In this section the three key concepts to understanding flight are: 1) wings lift a plane up by throwing a tremendous amount of air down, 2) because air has viscosity it sticks to and follows the top wing surface, 3) in following the top wing surface the air is pulled down and as an equal and opposite reaction the wings are lifted up.

As a plane flies through the air the wings effectively grab the air to throw it down so as to lift the plane up. The act of throwing air down produces a reactionary force that lifts up on the wings. This upward force on the wings counterbalances the weight of the plane that is directed downward. Besides these two forces counterbalancing each other in the vertical direction, there is another pair of forces, the drag and the thrust, that counterbalance each other in the horizontal direction. While the plane is flying along at constant speed and level flight the vector summation of all of these forces is zero.

The idea that an object with zero net force acting on it will continue moving in a straight line is Newton’s first law, while the action and reaction forces acting on the wing is an example of Newton’s third law that all forces are equal and opposite. What is next needed is the application of Newton’s second law of motion for equating the weight of the plane to how much air the wings need to throw down for the purpose of lifting the plane up.

Most scholars of physical science are familiar with Newton’s second law as being F = m a; when a net force acts on an object its acceleration in the same direction will be inversely proportional to its mass. But while this equation is useful in solving the vast majority of physics problems involving objects, it is not really helpful when we are generating a force by constantly throwing down a fluid. Fortunately Newton actually stated his second law as force being equal to a change in momentum with a change in time: F = Δ(v m) / Δt. By starting with this more general statement we can write Newton’s second law applicable to a fluid as F = v dm / dt.

This is the same equation that we would use to describe rocket propulsion. It tells us that we generate the greatest lifting force by maximizing the rate of fluid mass per unit time that we thrown down and we throw that fluid down at the highest speed possible. For our airplane this force is the lifting force acting on the wings that in turn is equal in magnitude to the weight of the plane so that

W = v (dm / dt)

W is the weight of the plane or flying animal, v is the downward speed of the air, and dm / dt is the mass of the air being thrown down per unit time.

This flow of mass per unit time is produced by the wings grabbing the air and throwing it down. Earlier it was stated that the cross sectional area of air A grabbed by the wings is about ½ b². Thus

(dm / dt) = D dV / dt

(dm / dt) = D A (dx / dt)

(dm / dt) = D A v

(dm / dt) = D (½ b²) v

Substituting this into our main lift equation gives:

W = v D (½ b²) v

Where W is the weight of the plane, v is the speed of the air being thrown down, D is the density of the air, v is the relative speed of the plane through the air, and ½ b² is the effective area A surrounding the plane affected by the wings as they cut through the air.

This equation tells us that for a greater weight of our plane and its cargo we can either 1) throw the air down at a faster speed, 2) fly at lower altitude where the air’s density is greater, 3) fly faster, or 4) build a plane with longer wings that are capable of influencing more of the air surrounding the plane.

As the plane flies through the air, the wings are converting some of the planes forward thrust to throwing air down so as to lift the plane up. This process of throwing the air down requires energy; energy that the plane gives to the air. The kinetic energy of a moving object is calculate as

E = ½ m v².

Since the plane is continuously giving this energy to the air it is more appropriate to discuss power, the use of energy per unit time.

P = dE /dt

Substituting the energy equation into the power equation gives:

P_L = d(½ m v²) /dt

This simplifies to:

P_L = ½ (dm / dt) v².

The mass flow rate (dm / dt) was calculated earlier as

(dm / dt) = D (½ b²) v

and we can use the final equation of the last section to solve for the speed that the air is thrown down as:

v = 2 W / D b² v.

Making these substitutions gives the power needed for lift based on known values:

P_L = W² / (D b² v).

Where P_L is the power needed for lift, W is the weight of the plane, D is the density of the air, b is the wingspan, and v is the speed of the plane. This equation gives us great insight into the fundamental requirements for flight, yet we are still not finished since we must also account for power required to overcome the fluid drag.

In addition to the power being used by the wings to lift the plane up, power is also required to overcome the parasitic drag. For most large high speed objects moving through a fluid the parasitic drag can be accurately modeled as:

F_P = ½ A C D v².

Where F_P is the parasitic drag, A is the front facing or cross sectional area, C is the aerodynamic drag coefficient, D is the density of the fluid, and v is the object’s relative speed through the fluid.

The general equation for power is

P = F v.

So the power needed to overcome the parasitic drag is:

P = ½ A C D v³.

This equation shows that the power requirements go up dramatically with an increase in speed. Therefore if the intend is to travel fast then all effort should be made to streamline the object, reduce its thickest cross sectional area, and if possible travel through a low density fluid. The reason fast cars, trains, and jets look fast even while stationary is due to the necessary of streamlining the form so as to reduce the power needed to reach the fastest speeds.

Flying is the exception from most forms of travel in that a jet airliner has a means of reducing the density of fluid that it is traveling through. The density of air decreases with altitude. A primary reason that jet airliners fly at high altitude is so that they are flying through air that has a density that is about one fourth the density of air at sea level. By flying through lower density air the airliners reduce their power requirements and achieve savings by getting better fuel mileage. To make a comparison, despite a submarine’s extremely good streamline form it can not travel nearly as fast as a jet airliner because the density of sea water is two or three thousand times greater than the density of high altitude air.

Flying is also different from other forms of travel in regards to the various speeds that they can travel. While other forms of travel can move as slow as they like, an airplane is restricted to flying at speeds above its stall speed. Not only is flying well above the stall speed necessary for safe flying it is also best to always fly fast so as to produce the best fuel mileage. Eventually though because the power used to overcome drag is a function of the speed cubed, even for planes there is a limit to how fast they can fly and still be fuel efficient. For planes there is a sweet spot between flying too slow and flying too fast where the plane gets the best fuel efficiency.

Starting on the left, the power required to lift the plane goes down as the plane flies faster. In fact a plane can only fly so slowly before it will stall; this part of the graph is shown with a dash line. The center graph shows power output required to overcome the parasitic drag increases with the cube of the speed. The addition of the power required in overcoming drag and the power required for lift gives the total power needed to fly. This is shown by the graph on the right. Number 1 is the minimum power required for flight. Planes and flying animals actually need to fly faster than this speed, to the right of point 1. Number 2 marks the cruise speed.

We can calculate the minimum power speed by summing together the two power requirements and then taking the derivative of the total power requirement with respect to speed. The total power is:

P_T = P_P + P_L

P_T = ½ A C D v³ + W² / (D b² v)

Taking the derivative of the total power equation with respect to speed then setting the result equal to zero, we determine the speed for minimum power:

v_min = [(2 W²) / (3 A C b² D²)]^1/4

The minimum power can now be determined by substituting this minimum power speed back into our total power equation. Yet when we do this we notice that at this speed the minimum power for lift is ¾ of the total power. Thus we can simplify our minimum power equation to be:

P_T-min = 4/3 [W² / (b² D v_min)]

These last two equations can be thought of as the Holy Grail of aerodynamics in that based on the known data of any given plane we can determine its minimum take-off speed and minimum power requirements to achieve flight. These equations are general to all types of planes and flying vertebrates.

If we continued building on the concepts presented so far we can derive a general relationship for how the power requirements increase with size. The general equation is

Thus we see that as the size of our planes increase our power requirements increase dramatically. As an example of using this power equation we can compare a Cessna 172 to a 747 jetliner. A Cessna 172 is 8.25 m long and has a 120 kW engine, so in applying our equation to the 41 m 747 we get a ballpark power requirement of 42 MW. The actual power of a 747 is about 50 MW.

To produce accurate answers when applying this equation our flying objects should be of approximately the same shape and overall density. Yet as the example demonstrates the equation can produce reasonably accurate ballpark values for the require power even when our flyers are not that similar. For example, based on the power requirements of the Cessna 172 we could also use this equation to calculate the approximate power requirements of a sparrow or a fly.

Yet regardless of our fondness for theoretical work, it is all worthless if it is not verified by the evidence. To test the equations the author has selected an assortment of planes and flying animals and then used the known data regarding these flyers weight and form as input to calculate their flight performance. When tested the Power for Flight equations not only prove themselves to be accurate, but in addition the equations are able to provide insightful answers to some historical aerodynamic controversies. The reader is encouraged to select their own planes or animals to do their own testing to validate the Power for Flight equations

The Power for Flight equations verifies the flying ability of all planes and present-day flying animals. Yet as the author expected the equations also show that the animals that flew millions of years ago would not be capable of flying today. Thus we have compelling evidence that the global environment of the past was dramatically different in such a way as to allow these prehistoric animals to fly.

In the table below a power ratio of 1.0 is the minimum requirement for a plane or animal to takeoff in ideal weather conditions. While a power ratio of around 2.2 and above allows a plane or animal to fly at its best efficiency per distance along with giving it a safety margin for flying in foul weather.

Cessna 172 and Sopwith Camel- Other than the Cessna having more power and consequently greater speed, the Cessna 172 and the Sopwith Camel are similar flying aircraft. The modern day Cessna is only slightly heavier, faster, more aerodynamic, and more powerful than the world war one Sopwith Camel. These first two flyers are supplied with enough power to be safe fliers even in foul weather.

Wright Flyer - The bi-wing of the World War One Sopwith Camel is not much different from that of the Wright Flyer. Yet the performance characteristics are dramatically different because the Wright Flyer is underpowered: it had only a 12 hp engine.

Calculations based on the historical data show that the overland speed of the Wright Flyer was only about 4 or 5 m/s even though the least power speed of the Wright Flyer is 14 m/s. The explanation for the difference is that the Wright brothers flew their plane into a strong headwind as all planes do when they takeoff. The Power for Flight equations predict that at least a 10 m/s headwind was required to be added to the Wright brothers’ 4 m/s overland speed to give the Wright Flyer the 14 m/s relative speed it needed to fly, and sure enough a check of the historical weather data states that the top wind speed on December 17, 1903 was 22 mph or 10 m/s. The brothers from Dayton, Ohio specifically chose to do their flying at Kitty Hawk, North Carolina because of its record of steady, strong winds.

Spruce Goose – How much does it actually weigh is the key to answering the “Can it really fly” question. Depending on the source its empty weight is either 300,000 or 400,000 lbs (1.3 to 1.8 MN). It seems odd that this key data is not precisely known. It is also odd that the stated cargo weight of either 100,000 or 130,000 lbs does not correspond with the calculated cargo weight. Seven hundred and fifty men at 200 lbs each gives us 150,000 lbs for the cargo. But if we are going to fly over the Atlantic then the plane also needs 12,500 gallons or 70,000 lbs of fuel for a total of 220,000 lbs. So depending on which empty weight is correct, its full weight is either 520,000 lbs or 620,000 lbs. Expressed in metrics its full fuel and cargo weight is 2,300,000 N to 2,800,000 N.

Loaded with cargo and fuel its power ratio is only 1.1 to 1.3 thus indicating that the Spruce Goose only had half as much power as what it needed. During the design of the Spruce Goose, Howard Hughes and at least one consultant disagreed on if the Spruce Goose exceeded the limitation of how large a propeller driven plane could be. But at that time there were no simple equations for determining the power to weight limitation or if a size limitation even existed; so Howard was undeterred in his goal of building an exceptionally large plane.

As a cargo plane completed too late to fulfill its war-time mission and, more important, lacking the power to carry a cargo the Spruce Goose was not a useful or safe flyer. Yet the Spruce Goose is a historical icon and the numerous engineering challenges that were solved during the design and construction of this extremely large plane did advance aviation technology.

Human Powered Flight – Two values are given for the weight; the 980 N is the total weight of both man and plane while the 610 N is the weight of just the man. The total weight is used for calculating the plane’s performance while the weight of the man is use to calculate the power available.

The performance numbers show that human flight aircraft are clearly built around the limitation of a top athlete being capable of producing only about 290 watts. With only a 1.2 power ratio it is probably safe to state that ground effect is a significant factor in enabling these planes to fly.

Human powered flight can be achieved only by keeping the weight of both the athlete and the plane to a minimum while extending the wingspan as far out as possible. Minimize the front facing area and minimize the drag coefficient is helpful yet of secondary importance to the weight, power, and wingspan criteria.

Starling – With an 6.6 power ratio flying is a not a challenge for these and other small and medium size birds. With such an abundance of relative power it is no wonder that small and medium size birds often appear playful as they fly.

Giant Bat – Giant bats have a power ratio of only 1.6. This marginal power ratio explains why the largest bats are rarely ever seen flying in the rain.

California Condor – This is another extremely marginal flyer that can only fly when the weather conditions are favorable. A 1.1 power ratio is barely enough to allow these large birds to lift off after feasting on a carcass. But once they lift off they glide themselves over the rising air thermals that act like a rising elevator to lift them higher. Once they are soaring, these large birds require only enough power to guide their flight over the rising air thermals.

Argentavis – Six million years ago there existed an extremely large bird known as the Argentavis. It flew in South America like the modern large flyer the Andean condor. However the wingspan of the Argentavis was two and a half to three times that of the Andean condor. Its mass would be proportional of the wingspan cubed and so its mass would have been 16 to 27 times greater than the Andean condor. This places it far beyond the modern crowded field of bird species competing for the title of being the largest flying birds. Consequently many scientists investigating this matter are baffled in their attempts to explain how the Argentavis flew.

It is probable safe to assume that these ancient relatives of the California condor utilized the warm rising air thermals in same way as the large soaring birds of today. Thus the power ratio of 1.1 for the California condor should be seen as the cutoff value for birds using thermal assisted flying. Yet these birds have a power ratio of only 0.3. So even with the assistance of rising air thermals this animal could not have flown in today’s atmospheric conditions.

Quetzalcoatlus – Unlike the Argentavis there are no living relatives of the Quetzalcoatlus and this makes it difficult to estimate the Quetzalcoatlus’ mass. The author produce values that fell in a range between 500 kg to two tons; thus arriving at a rough estimate of 700 kg. With a 12 m wingspan and a chest cavity larger than that of a horse there is no getting around the fact that this was a huge animal.

However some paleontologists tell a different story in estimating the Quetzalcoatlus mass to be between 90 and 250 kg. Thus they are claiming that the Quetzalcoatlus had a body density that was about seven times less than any animal presently flying. These paleontologists need to do some explaining as to how the muscle, bone and other bodily parts of a Quetzalcoatlus could have a density seven times less than what is found in present day birds.

Some paleontologists have suggested that despite being reptiles the pterosaurs may have been warm-blooded. This is a feasible hypothesis and so the available power for the Quetzalcoatlus is calculated using the power-to-weight equation for a warm-blooded mammal. Even so, the power ratio of the Quetzalcoatlus is still only nine percent of the minimum requirement for flight.

So how did the Argentavis and Quetzalcoatlus fly? The answer is in the equations:

If the answer is still not clear then just read on to the next chapter, the chapter that searches and finds the solution to the paradox of gigantic birds, pterosaurs, and dinosaurs.

Pterosaurs

• Introduction to the Pterosauria - U. C. MUSEUM OF PALEONTOLOGY
  
• Ancient Flyers - NATIONAL BUSINESS AVIATION ASSOCIATION
  
• New Discovery of Pterosaurs with 18m Wingspan - BBC NEWS
  
• Quetzalcoatlus Northropi - BIG BEND NATIONAL PARK
  
• Pterodactyls were too heavy to fly - Katsufumi Sato
  
• Could Pterosaurs Actually Fly? - SCIENCE
  

How Wings Generate Lift

• Airfoils and Airflow - John S. Denker
  
• How do airplanes fly, really? - THE STRAIGHT DOPE
  
• Bernoulli Verse Newton Explanation of Lift - NASA
  
• How Airplanes Fly - David Anderson and Scott Eberhardt
  
• Airfoil Lifting Force Misconception - Bill Beaty
  
• Airfoils - Chris Heintz
  

Science of Flight

• Aerodynamics for Students - D.J. Auld & K.Srinivas
  
• Forces on a Plane in Flight - NASA
  
• The Simple Science of Flight - Henk Tennekes
  
• Aerodynamics - WIKIPEDIA
  
• Aircraft Flight Mechanics - WIKIPEDIA
  
• Jet Propulsion - Laurence K. Loftin
  
• Theory of Flight - John Brandon
  
• Swept Wing - WIKIPEDIA
  
• Aerodynamic Theory - Kas Kasravi
  
• Dynamics of Airplane Flight - ABOUT.COM
  

Drag Coefficient

• Drag Force in a Medium - PHYSICS NET
  
• Drag Coefficient - NASA
  
• Drag Coefficient - ENGINEERING TOOLBOX
  
• Aerodynamic Drag - The Physics Hypertextbook
  

Data on Airplanes

• Airliner Takeoff Speeds - Jeff Scott
  
• Concorde Supersonic Flight - Bruce Graham
  
• Data on Jet Airliners - TURKISH AIRLINES
  
• Boeing 747-400 Jumbo Jet
  
• Data on Sailplanes
  
• Supersonic Transport - WIKIPEDIA
  
• Sopwith Camel - WIKIPEDIA
  
• Cessna 172 - WIKIPEDIA
  
• Cessna R172K
  

Spruce Goose

• Spruce Goose - Mike Neely
  
• Spruce Goose - Michael King Smith
  
• How come the Spruce Goose flew only once? - Cecil Adams
  
• Spruce Goose - EVERGREEN AVIATION MUSEUM
  

Wright Flyer

• Wright Flyer - PBS
• Wright Flyer - WIKIPEDIA
  
• The Wright Brothers: The Invention of the Aerial Age - SMITHSONIAN
  

Human Powered Flight

• Human Powered Flight - SKYTEC ENGINEERING
  
• Human Powered Flight - Chris Roper
  

Science of Flight Applied to Birds

• Science of Flight Applied to Birds - Colin Pennycuick
  
• Bird Flight - WIKIPEDIA
  
• Bird Flight - Gary Ritchison
  
• Taking Off Bird Flight - Paul and Bernice Noll
  
• Birds - NATIONAL BUSINESS AVIATION ASSOCIATION
  
• Amazing Bird Records - David M. Bird