Before going any further, write down this book title –

The Mechanics of Flight by A C Kermode – Pitman

What is needed to achieve flight?  Clearly we need to "overcome" gravity, in order to get off the ground, and we need to control our movements once in the air.

Let’s consider a very simple situation, a space-craft in a "weightless" condition.  It goes on doing what it is doing, with no effort from us.  It is only when we need to change the motion that we need to do something.    The usual way is to ignite a rocket motor.  This creates pressure of exhaust gases on the rocket.  The pressure applied over the area constitutes a force.  This has two effects.  Firstly it will very slightly compress the space-craft, and secondly it will accelerate it in the direction of the net pressure.

So a force creates an acceleration.  At the same time, the rocket is accelerating the exhaust gases, which stream out behind the rocket.  The total momentum, mass times velocity, of space-craft and gases does not change.  The change in momentum of the gases is exactly balanced by the change in momentum of the craft. 

A rocket does not work by pressing against air.  It can work in a vacuum.  A turbo-jet engine only needs air in order to make the compressors and turbines work, and to burn the fuel.  It does not work by the jet pressing against the air.  The engine is pushed forward by the gases pressing against parts of the engine.

See "Rocket and Spacecraft Propulsion" 

by Martin J L Turner, Springer, ISBN 85233 105 4

If we want to understand the forces which make objects fly, we can think about force or pressure, or we can think about momentum, whichever seems most helpful.

Please skip this if you don’t want to know about the physics of momentum.  To skip click Here.

The momentum of a system is calculated by multiplying the mass of each part by the velocity of each part, and adding all the products.  The word velocity implies that the direction of the speed is important.  The velocity can be resolved into three components at right angles, so that three components of momentum can be calculated, and added at the end.

Why is momentum important?  It is important because it is a conserved quantity, meaning that the momentum of an isolated system can never change.  But if a force is applied, the rate of change of momentum is equal to the force.

So when material with momentum is ejected from a rocket motor, the rocket experiences a force and gains momentum.  

Since momentum is mass times velocity, more mass or more velocity is beneficial.  But if an engine designer goes to a rocket designer or an aircraft designer and says something about carrying more mass, he or she will get a fairly unpleasant reply.  So higher velocity is a better solution for more momentum.  But the kinetic energy of an object increases as the square of the speed:  doubling the speed means four times the energy.

It doesn’t matter what is thrown out.  It can be gas, or it could be rubbish.  When a handgun is fired in a film, you see the gun kick backwards and upwards.  This is due to the momentum given it by the bullet.  The gun rotates because the barrel is not aligned with the the centre of gravity of the gun.  If you cough or sneeze, you will experience a similar effect.

But why is momentum conserved?  This is a deep question that modern physics can answer.  It turns out that any conserved quantity corresponds to a symmetry in the system.  For example, if the laws of physics are the same for a moving observer as for a stationary one, or for two people in different places, there are certain consequences.  Note that we are not saying that two observers obtain the same numerical results of measurements.  We are saying that these measurements will obey the same laws.

Let’s make a little explanation which is a bit of a cheat, but will give some idea of what might happen.

Suppose we think that some quantity like mV^P might be conserved.  To be completed . . .

We have seen that to get a lifting force on an aircraft we can apply a pressure, or we can eject material with momentum in the required direction, which is downwards.  Let’s look at the momentum option.

Let’s suppose that we have to design a ten tonne (10000kg) aircraft that can hover.  We will provide some turbo-jet engines that point downwards.  It is reasonable to suppose that the speed of sound might place a limit on the speed of the exhaust gases, so let us choose two-thirds of the speed of sound, to be conservative.  This gives about 220 m/s

We can now work out the mass of exhaust gases produced per second.  Let’s call that m.  We also need to know the acceleration due to gravity, which is 9.8 metres per second per second.  Let’s call it 10.

Then 220 m = 10000 X 10, and so m = 450 kg/s, or 27000 kg/minute.  Of course, some of the weight is made up of nitrogen from the air intake, and from oxygen that has combined with the fuel.  If we assume a paraffin-like fuel with a formula C_nH2_n+2 we can estimate rate of fuel usage.  To the accuracy we need at present we can use C_nH2_n, because there is no point in doing a very accurate calculation as a first step.  Only if the result is marginal is it worth being more accurate.

The equation for the burning is

2C_nH_2n + 3nO₂ = 2nCO₂ + 2nH₂O

Using H = 1, C = 12, and O = 16, we can work out the ratio of fuel mass to exhaust mass, if we assume that the fuel is fully burned and that all the oxygen is used.  These are unlikely to be correct, but we have to start somewhere.

The fuel mass is 2n X 12 + 2n X 1 = 26n units.

The exhaust mass is 2n X 12 + 4n X 16 + 4n X 1 + 2n X 16 = 24n + 64n + 4n + 32n = 124n, which is about 4.8 times the mass of the fuel.  Better still, that part of the air that isn’t oxygen (mainly nitrogen) is in the exhaust as well.  So we can multiply the effective mass of the exhaust by adding the effect of the nitrogen.  The air contains about 21 % oxygen.  We then have (0.79 / 0.21) = 3.8 of N₂ (3.8 X 14 X 2) for every O₂.  So the nitrogen amounts to 3 X 3.8 X 14 X 2n = 319n.  The exhaust mass is now 124n + 319n = 443n, which is 17 times the mass of the fuel.

The fuel cost of hovering does not look so bad now.  Instead of 27 tonnes/minute we have 1.6 tonnes/minute.  Birds, however do not benefit like this, except indirectly, when their food is converted into energy.   The turbo-jet engine can do better than a simple engine if it is a bypass type.  In such an engine, a lot of air never reaches the flame-cans: it is thrown back straight from the intake, by large fans.

But even if we get the fuel cost down to 500 kg/minute, that’s 30 tonnes per hour, for a 10 tonne aircraft, and  we still haven’t allowed for forward flight.

So hovering flight requires a lot of energy, and it doesn’t fulfil one common requirement of flight, which is to get from one place to another.

But on an airless place like the moon, a jet of gases is the only way to generate lift.

If we look at a hovering kestrel or hummingbird, we can see how hard it is to hover.  The kestrel much prefers a windy day, so that it is in forward flight when staying in one place.  On a calm day it may hunt from a perch.

We know that there must be an easier way, because we can see buzzards, pelicans, and frigate birds gliding with little apparent effort.  They use a little energy to power the brain and nervous system in order to work out where the best lift might be, and how to recognize it as they fly into it.  And they use a little energy to control the wings and tail to optimise their use of the air, which comes from the sun.  Effectively they are solar powered.

Otherwise, they use the energy available from the air.  They are actually gliding downwards relative to the air, most of the time, but they use air that is rising faster than they are sinking.  The albatross uses some special effects of wind gradient near the surface.

We have seen how much energy is needed to hover using power.  In fact, the first sustained human flight was achieved using energy in a completely different way, namely, in a hot-air balloon.  Later, the hydrogen balloon and the helium balloon were invented.  Helium is twice as dense as hydrogen, but it is nearly as good for balloons, because what counts is the difference between the density of air and the density of the gas.

How does a balloon work?  The next diagram shows the atmospheric pressures (red) around a spherical balloon, with the variations greatly exaggerated.  The pressures inside (blue) must be at least a little greater than those outside, in order to sustain the shape of the balloon.  Actually, it does not matter much what the shape is, nor even whether the envelope is taut, but it is easier to calculate with a spherical shape.

We see that the pressures increase towards the bottom of the balloon, both inside and outside.  But the gas is less dense than air, and so the top-to-bottom pressure difference inside the balloon is less than the difference outside.  This remains true even if the pressure inside the balloon exceeds the external pressure at all points.  The variations give the lift.  In the example, the pressures have been chosen to be equal at the bottom of the balloon.  This makes it easy to see why it rises.

At the mouth of a hot-air balloon the pressure must be the same "inside" and "outside", otherwise there would be a flow of air.  The pressure difference between inside and outside must increase from zero at the bottom to a maximum value at the top.  Why is the ideal shape for a hot-air balloon not spherical?

If a helium balloon is intended to reach a very high altitude, where the density of the air is low, the balloon is almost empty at low altitude.  The higher air pressure near the bottom of the balloon squeezes the helium to the top, where it swells out the balloon into a smooth almost hemispherical shape.  As the balloon rises, the helium slowly expands to fill out the balloon.

A parachute is not exactly a device for hovering, but it does decrease the rate of sink enough to achieve a safe landing.  It also provides time to choose a suitable landing area.  A dandelion seed also sinks relative to the air, but in a wind it can drift a long way from the plant.  In turbulent air near the ground, it can find itself in air which is rising faster than it is falling, and it can go up.  

A pilot whose aircraft breaks up in a cumulo-nimbus cloud can be in serious trouble, as the rising air can carry him or her up to great altitude.  In fact, a pilot under a parachute could be carried up or down many times in the convection currents.  The pilot will be buffeted by the turbulence, and will become very cold.  So don’t fly into a cumulo-nimbus cloud.

Some plants, such as sycamores, produce seeds with wings.  When they fall, these seeds rotate.  The process is called auto-rotation.  It was used in the auto-gyro, which had a rotor rather like that of a helicopter, but without an engine.  The engine was connected only to the propeller at the front, except when starting.  If you repeatedly drop the same sycamore seed, does it always rotate in the same direction?

Another form of auto-rotation in aircraft is the spin.  This proved fatal in the early days of flying until someone worked out how to recover from it.  The problem is that the recovery process is counter-intuitive.  In a spin, all or part of one wing is in a stalled condition.  In order to recover, the stick must be pushed forward to lower the angle of attack and get it flying again.  This is not a natural reaction when the aircraft is pointing at the ground.  In addition, opposite rudder may be needed to counter the rotation.  Only when the aircraft is flying again can the stick be pulled back to recover from the dive.  It is apparent that below a certain height, recovery is impossible.

If we watch a glider passing effortlessly overhead, we might wonder how it works.  We might even wish we would do it ourselves.  Why not try?  Inside the cockpit it may not be as peaceful as you think.  The pilot may be grumbling that there is not enough rising air.

Let’s take a practical example.  Suppose a buzzard sinks at a rate of 0.5 m/s when gliding in stationary air.  In air that rises at 0.5 m/s it can break even.  In a thermal which is rising faster than that, the buzzard can rise relative to the ground while sinking relative to the air.  Buzzards, like many other large birds, are skilled at finding rising air, whether it be in a narrow thermal or in a long street.  When a reaches the point where the lift is equal to the sink, it can set off across country.  Glider pilots can do the same.

If we sit behind the wings in a large airliner, we can get a clue about lift when it takes off, and even more so when it lands.  Flaps are moved outwards and downwards, giving a very strong clue that air is being directed downwards, as in the case of the hovering jet.  In other words, downward momentum is being created.  The same effect can be felt under the rotor of a helicopter.

How does this work?  Another clue comes from golf and table-tennis.  Every golfer knows that if the ball spins about a vertical axis, it will fly in a horizontal curve instead of in a vertical plane.  This can happen if the club-head is not perpendicular to its velocity.  It can also happen if the wrong part of the head hits the ball.  The head then rotates very slightly as the ball is struck.  The correct place is called the "sweet-spot", or the centre of percussion.  

If the shaft of the club is very light compared with the head, the centre of percussion is close to the centre of gravity of the head.  Cricket bats and baseball bats are tapered towards the handle to move the centre of percussion further from the handle.  The effective length of the bat is increased over that of a flat paddle.

By moving some of the mass of a club-head to the outside, the maker can increase the moment of inertia of the head.  This will reduce the amount of rotation of the head for an off-centre strike.  The size of the sweet-spot is effectively bigger.

If a ball is struck at the wrong place on the implement, the ball flies badly, and energy is transmitted to the players hand.

How does a rotating ball cause a swerve? We don’t need to know that.  We could just build an aircraft with large rotating balls on each side.  In fact it turns out that cylinders would be better.  A few ships have actually been built with tall cylinders, rotating on a vertical axis, instead of sails.  At least you don’t have to adjust anything when the wind changes, because the cylinder looks the same from all directions.  A lot of rigging and deck-hands are no longer needed.  This idea has never caught on.  What do you think the disadvantages are?

But for an aircraft, the idea of rotating cylindrical wings is not attractive.  The drag would be huge, and great power would be needed.  if such an aircraft turned, the gyroscopic effect would complicate manoeuvres, though perhaps not so badly as with a rotary propulsion engine.

Other examples of curving by rotation are seen in table-tennis and soccer.

What makes the path of a ball curve?  By rotating a ball about an axis which is not parallel with the flight, we have introduced an asymmetry into the situation.  So it is not surprising if the flight is asymmetric too.  But we need to see how this happens.

The previous section suggested that asymmetric causes are likely to have asymmetrical effects.  A ball can be made to swerve if it is not a perfect sphere.  The seams where parts of the ball are joined together, for example by stitching, can be used as a source of asymmetry.  The seam of a baseball or a cricket ball can be positioned by the pitcher or bowler in such a way that the flow of the air is slightly different on different sides of the ball.  In skilled hands a considerable swerve can be produced.

From cricket we are reminded of another fact about flying.  It is the fast bowlers who swing the ball, not the slow ones.  So the mechanics of lift are related to speed.  An aircraft has to attain a certain speed before the lift becomes equal to its weight.

Let’s summarize what we have so far.

A difference in pressure creates a force.

Ejection of material with momentum creates a force.

A moving asymmetrical object can experience a force.

When a rotating ball moves along, the air-flow on one side is not quite the same as on the other side.  As the air passes the ball, it acquires a different velocity on the two sides, and therefore different kinetic energy.  The total energy of any part of the air must be conserved, so the energy due to pressure goes must change.  

This can only happen from a change in pressure.  And that is how the hook and slice operate in golf, how a curling free kick operates in soccer, how spin operates in tennis or table tennis.  Bjorn Borg was a noted user of heavy top spin, enabling him to execute shots late and low, with little telegraphy to the opponent.

A wing works like the rotating ball.  Its asymmetry results in a different speed of flow over the two sides, resulting in a difference in pressure.  This difference is not at all caused by the air over the curved side having to catch up with the air on the other side.  There is no reason whatsoever for the volumes of air passing the two sides of a wing to match up afterwards, and they don’t, except in the case of a symmetrical wing section with no angle of attack, giving no lift.  The "matching" explanation is plumb wrong.

Often the air on the faster side moves faster than is required by the matching explanation.  The matching explanation cannot explain lift from a tilted flat plate, because you can easily make a drawing in which both paths are the same.

Any explanation must also explain how some aircraft can fly upside down, even though they have a strongly asymmetrical wing section.  The wing section is obviously designed to provide lift, yet such a wing can also provide lift when inverted.

But it only does so when it has a large angle of attack, that is, its leading edge is much higher than its trailing edge.  A photograph of an inverted aircraft will generally show it pointing quite steeply upwards.  A fast fighter aircraft with a fairly symmetrical wing section might not show this effect.  And an aircraft designed for aerobatics might also have a symmetrical wing, so that it can fly equally well normally and inverted.

The vertical asymmetry we need for flight can come from asymmetry of wing section, or from angle of attack, or both.  In fact, if we increase engine power, the aircraft will go faster, and the wings will generate more lift.  The  aircraft will start to climb.  To hold the same altitude will require the stick to go forward to lower the nose.