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Groundschool – Theory of Flight

and the properties of air

Revision 97 — page content was last expanded 20 March 2014.

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Air density — a function of atmospheric pressure and temperature — decreases with increasing height. It affects the generation of lift and aircraft performance, as does the angle of attack at which the aircraft is flown. In normal, unaccelerated flight, there is a relationship between airspeed and angle of attack and, in light aircraft, the airspeed indicator instrument — which measures dynamic pressure — acts as a very limited angle of attack indicator. The minimum flight speeds are a function of aircraft weight, angle of attack and flight loads. The flight loads applied by the pilot are chiefly accomplished by varying angle of attack and engine power. From this comes the need to establish a design manoeuvring flight envelope that defines the boundaries of the maximum allowable pilot-induced and turbulence-induced loads that may be applied to the aircraft structure without risk, together with certain critical airspeeds — plus the range of performance airspeeds available to the pilot.


2.1 The atmospheric pressure gradient

The random molecular activity or internal kinetic energy within a parcel of air is known as the static pressure and is proportional to the absolute temperature.

Static pressure exerts a force on an object (for example, an aircraft wing) at right angles to all the exposed non-porous surfaces, and is measured in newtons per square metre [pascals] of surface. In Australia air pressure is reported as hectopascals [hPa] for meteorological purposes; one hectopascal equals 100 N/m² [or one millibar].

Atmospheric pressure reflects the average density (i.e. mass per cubic metre), and thus the weight, of the column of air above a given level. So, the pressure at a point on the Earth's surface must be greater than the pressure at any height above it, in that column. An increase in surface pressure denotes an increase in mass — not thickness — of the blanket of air above the surface location. Similarly, a decrease in surface pressure denotes a decrease in the mass of air. The air throughout the column is compressed by the weight of the atmosphere above it, thus the density of a column of air is greatest at the surface and decreases with increasing altitude.

However, a warmer air column will be thicker — i.e. extend further upwards — than a cooler air column with the same surface pressure. Thus a particular pressure level will be at a higher elevation in the warmer column. This means that the level in the atmosphere at which any particular pressure occurs is also dependent on the temperature (or thickness) of the air column. Meteorological offices produce 'height contour and thickness charts' to determine the locations of upper level troughs and ridges.

2.2 Atmospheric density

The average density of dry air in mid-latitude, temperate climates, is about 1.225 kg/m³ at mean sea-level; the density decreases with increasing altitude.

There are several gas laws that relate the temperature, pressure, density and volume of air. The equation most pertinent to aeronautical needs is the equation of state:

   r = P/RT    where:
  r* = air density kg/m³
  P = static air pressure in hectopascals [hPa]
  R = the specific gas constant for dry air = 2.87
  T = air temperature in kelvins [K] = °C + 273

*r is the Greek letter rho, pronounced 'row', as in 'row your boat'.

By restating the equation of state as: P = 2.87rT , it can be seen that if density remains constant; pressure increases if temperature increases.

We can calculate the ISA standard sea-level air density, knowing that standard sea-level pressure = 1013 hPa and temperature = 15 °C or 288 K

  i.e. Air density = 1013 / (2.87 × 288) = 1.225 kg/m³ = ro*

If the air temperature happened to be 30 °C or 303 K at the same pressure, then density = 1013 / (2.87 × 303) = 1.165 kg/m³, or a 5% reduction.

*ro — the symbol for the standard sea level density is pronounced 'rho zero', or 'rho nought' if you prefer.

2.3 The ICAO International Standard Atmosphere

   The International Civil Aviation Organisation's [ICAO] International Standard Atmosphere [ISA] provides a fixed standard atmospheric model that is used for many purposes, among which are the uniform assessment of aircraft performance and the calibration of some aircraft instruments. The model is based on average climatic conditions in 40–45° North latitudes, but contains the following assumptions:
  • dry air (no water vapour present) is assumed throughout the atmosphere so the effects of humidity on air density are ignored
  • the mean sea-level [msl] pressure = 1013.25 hPa
  • the msl temperature = 15 C [288 K]
  • the tropopause is at 36 090 feet [11 km] and the pressure at the tropopause = 226.3 hPa
  • the temperature lapse rate to 36 090 feet = 6.5 C per km, or very close to 2 C per 1000 feet
  • the temperature between 36 090 and 65 600 feet [20 km] remains constant at −56.5 C.
The table below shows a few values derived from the ISA. Those pressure levels noted with a flight level designator (FL) are standard pressure levels, rounded to the nearest increment of 500 feet, used for aircraft operating above the altitude transition layer and also for aviation weather purposes, particularly thickness charts.

PressureFlight levelTemperatureAir densityAltitude
hPa °C kg/m³feet
1013   15.0 1.225 msl
1000   14.3 1.212 364
977   13.0 1.190 1000
950   11.5 1.163 1773
942   11.0 1.155 2000
908   9.0 1.121 3000
900   8.6 1.113 3243
875   7.0 1.088 4000
850 A050 5.5 1.063 4781
843   5.1 1.056 5000
812   3.1 1.024 6000
800   2.3 1.012 6394
782   1.1 0.993 7000
753   – 0.9 0.963 8000
750   –1.0 0.960 8091
724   – 2.8 0.933 9000
700 A100 – 4.6 0.908 9882
696   – 4.8 0.905 10 000
650   – 8.3 0.855 11 780
600 FL140 –12.3 0.802 13 801
550   –16.6 0.747 15 962
500 FL185 – 21.2 0.692 18 289
450   – 26.2 0.635 20 812
400 FL235 – 31.7 0.577 23 574
350   – 37.7 0.518 26 631
300 FL300 – 44.5 0.457 30 065
250 FL340 – 52.3 0.395 33 999
200 FL385 – 56.5 0.322 38 662
150 FL445 – 56.5 0.241 44 647
100   – 56.5 0.161 53 083

It can be seen that (from sea level to 10 000 feet) air density decreases by about 33 grams/m³ per 1000 feet. Also, not immediately apparent from the ISA table, is that the pressure lapse rate (the rate of change with height) is exponential. It starts at about one hPa drop per 28 feet height increase, then slowing to 31 feet per hPa at 6000 feet (averaging 30 feet height increase per hPa drop up to 6500 feet), 36 feet per hPa at 10 000 feet, 50 feet at 20 000 feet and so on.

However, the following provides a useful rule of thumb:

Rule of thumb #1

    "For operations below 10 000 feet, an altitude increase (decrease) of 30 feet can be assumed for each one hPa pressure decrease (increase) and for an estimate of air density multiply the altimeter indicated altitude in 1000's of feet by 30 to find the value to deduct from 1.225 kg, e.g. at 6500 feet; 6.5 × 30 = 195 and 1.225 kg minus 195 grams = 1.03 kg." But bear in mind that the ISA model is unlikely to reflect current atmospheric conditions at a particular location, see high density altitude.

2.4 Bernoulli's principle and the continuity equation

Bernoulli's principle
Daniel Bernoulli (1700-1782) was a Swiss mathematician who propounded the principle that for a given parcel of freely flowing fluid, the sum of gravitational potential energy, kinetic energy and static pressure energy always remains constant.

For our aerodynamic purposes we can ignore the gravitational potential energy. Dynamic pressure = ½rv² and kinetic energy = ½mv² where m = mass. Air density is mass per unit volume; i.e. kg/m³ so dynamic pressure is the kinetic energy per unit volume. Static pressure is internal kinetic energy per unit volume, or pressure potential energy.

So, for our purposes (in a parcel of freely flowing air), Bernoulli's principle can be reduced to:
      ½r[dynamic pressure] + P [static pressure] = constant

The statement doesn't take into account viscosity, heat transfer or compressibility effects, but for operations below 10 000 feet and airflow velocities below 250 knots, compressibility effects can be ignored — thus no change in flow density [r] is assumed.

The statement then indicates that, in a free stream flow, if speed [v] increases static pressure [P] must decrease to maintain constant mechanical energy per unit volume; and the converse — if speed decreases, static pressure must increase. Or, turning it around, a free stream airflow will accelerate in a favourable pressure gradient and decelerate in an adverse pressure gradient.

Bernoulli's principle doesn't apply in boundary layer flow because the viscosity effects introduce loss of mechanical and thermal energy (transferred to the aircraft skin) due to the skin friction.

(Incidently Daniel Bernoulli's father Johann [born 1667] was the mathematician who first adopted the symbol 'g' for the acceleration due to gravity.)
Stagnation pressure
Another aspect of Bernoulli's principle is that the constant is the stagnation pressure — the pressure energy needed to halt the airflow — thus it can be written ½rv² + P = stagnation pressure. The stagnation pressure is the highest pressure in the system. This application of the principle is apparent in the air speed indicator, as demonstrated below.

Stagnation pressure is the basis of the ram-air parachute wing used in sport parachutes, paragliders and powered parachutes, see 'The ram-air parachute wing'. Also aircraft fuel-tank vents face into the airstream and rely on stagnation pressure to prevent inflight siphoning of fuel from the tank.
The continuity equation
There is another principle of aerodynamic interest to us — the fluid flow continuity equation — which states that, in a steadily moving airstream, the product of density, cross sectional area [s] and speed must always be a constant:

r × s × v = constant

If there is no change in density within the flow (which is the norm in the airspeed range of light aircraft; see compressibility effects) then we can state that:

s × v = constant

Thus, if air flows into a smaller cross-sectional area speed must increase to maintain the constant. Bernoulli's principle states that if speed increases, static pressure must decrease; so the velocity of a constricted airstream increases and its static pressure decreases.

Both the above principles are related to the conservation laws; Bernoulli's principle to the conservation of energy, and the continuity equation to the conservation of mass. We will examine these properties of air further in the 'Aerofoils and wings' module.

The venturi effect — used in carburettors, the total energy variometer and the airframe-mounted venturi that provides suction for some flight instruments — is an application of the principles stated above.

2.5 Measuring airspeed

   The dynamic pressure of the airflow, in N/m², is represented by the expression ½r, where:
  • r is the ambient density of the air [kg/m³ ]
    ( ro is the symbol for the standard sea-level density of 1.225 kg/m³ )
  • is the aircraft (or free airstream) speed [m/s²]
   and we can deduce that the apparent speed of the airstream is related to air density and dynamic pressure.

*Note: a lower case v is the symbol for speed in physics while an upper case V is generally the symbol for the free stream speed in aerodynamics, which is why I have used the lower case v in section 2.4 but an upper case V in this section and most of this flight theory guide.)

We can measure the dynamic pressure with a simple mechanical pressure gauge. Imagine a 6 mm internal diameter aluminium tube positioned under the wing of a moving aircraft, outside the slipstream, so that the open end points forward into undisturbed airflow and the other end of the tube terminates within a spring-loaded, flexible capsule — similar to that in an aneroid barometer — thus the capsule stops the airflow within the tube.

The back pressure, applied by the capsule to stop the airflow, must be equal to the stagnation pressure. The capsule is contained within a casing which, in turn, is connected to a static vent that supplies the casing with the ambient atmospheric pressure; or, in a lower-quality system, the casing may just be open to the atmospheric pressure within the fuselage.

So, if we have stagnation (or impact pressure or ram-air pressure) — which is dynamic pressure plus static pressure — within the capsule and static pressure surrounding it, the capsule will expand or contract to reflect the changes in dynamic pressure at the mouth of the tube. (The system is a 'pitot tube' devised by Henri Pitot (1695-1771). During World War I, the airspeed indicating instruments themselves were called 'pitots'.) The capsule movement is mechanically or electrically linked to rotate a pointer on a dial. Although the dial is calibrated and marked to indicate airspeed in knots or mph rather than hPa, it is still basically a simple pressure gauge and an imperfect airspeed gauge. Because the instrument cannot determine the density component of the dynamic pressure, the calibration assumes a constant air density of 1.225 kg/m³. This cockpit instrument is then an airspeed indicator [ASI] and it displays the indicated airspeed [IAS], based on ISA conditions. The indicated airspeed is not the actual aircraft speed through the air, 'V' in the equations. A bit confusing — but brace yourself, for it gets worse!

We can calculate the dynamic pressure for the Jabiru using the scenario in section 1.4 for calculating CL; i.e. cruising at 6500 feet, true airspeed 97 knots or 50 m/s, and air density 1.0 kg/m³. The ISA atmospheric pressure at 6500 feet is about 800 hPa.

   •   static pressure = 800 hPa
   •   dynamic pressure = ½rV² = ½ × 1.0 × 50 × 50 = 1250 N/m² = 12.5 hPa

Note that the dynamic pressure at 1250 N/m², or 12.5 hPa, is less than 2% of the static pressure, but applying that dynamic pressure over the 8 m² of wing area and the lift coefficient of 0.4; i.e. 1250 × 8 × 0.4, still gives the lift force of 4000 newtons that we calculated in the 'Lift' section.

The airspeed 'V' in the equations is the true airspeed [TAS] — the free stream speed or the air distance flown over time. We know that the ASI dial is calibrated assuming a fixed air density of 1.225 kg/m³ [ro ], so a perfect ASI will only indicate the real airspeed (the true airspeed) when the actual environment density is 1.225 kg/m³; that could only occur when the aircraft is operating at low altitude.

What will be the IAS (Vis) of our example in the preceding box?

IAS = Vis = V / √(ro / r)

= 97 divided by the square root of 1.225 divided by 1.0 = 97 / 1.1 = 88 knots

From this we can deduce that a perfect ASI will generally underread. The IAS will always be less than the TAS, except in very cold conditions at low altitude where the air density may be greater than 1.225 kg/m³. For instance, using the equation of state above, if temperature was –3 °C and pressure was 1030 hPa, the density would be 1.33 kg/m³.

Density is about 1% less than (greater than) the ISA value for each 3 °C that the temperature is above (below) the ISA value.

2.6 Indicated and calibrated airspeed

The ASI as an angle of attack indicator
So, you might ask, what's the point of an ASI that really is indicating just dynamic pressure, and is unlikely to indicate your real airspeed — air distance flown over time — accurately? Well, admittedly it does mean a little more calculation to be done in navigation, but there are very significant advantages with an instrument that displays IAS rather than TAS.

This will be covered in the 'Aerofoils and wings' module. Generally, for all angles of attack in unaccelerated flight at a particular weight, there is a corresponding IAS; though the relationship between aoa and IAS does get a bit fuzzy near CLmax. So, in the absence of an angle of attack instrument, the ASI can generally* be regarded as an indication of angle of attack if the lift being produced matches aircraft mass. Also, all the performance parameters (the 'numbers') for an aircraft — best rate of climb, best angle of climb, best glide angle, etc. — require it to be flown at a particular aoa for that weight, and thus a particular IAS. Or more accurately, a particular calibrated airspeed [CAS] and that particular CAS does not change with altitude (as TAS does), but changes only with weight.

*The reason why CAS does not always correlate to aoa in level flight is that when inertia and random displacement forces — atmospheric turbulence — come into play, aoa may change momentarily without a noticeable change in CAS.

(Note: there are means of audibly conveying the angle of attack to the pilot. The simplest is a stall warning horn operated by a basic two-position vane switch incorporated in the leading edge of the wing and switched on by the airstream pressing the vane up if the aoa comes close to the stalling aoa. There are other airstream direction detectors (ADDs) that provide a range of warning tones in the pilot's headset.)
Calibration of the aircraft's airspeed indicator system
An ASI is an imperfect mechanical instrument which is subject to instrument errors; the poorer the quality of the instrument the greater the instrument error; the permissable limit for a certified ASI is ±one knot. The associated pitot/static and tubing system is also prone to pressure sensing errors due to the positioning (and design) of the the static air vent —and possibly the pitot head — relative to the airstream. Relative airflows change as aoa changes and when slipping or yawing. It is difficult to find a site on the fuselage for a static vent where the static pressure reflects the ambient static pressure. So after construction is complete the aircraft's airspeed indicating system must be 'measured' to determine the rectifications to be made to the airspeed reading indicated on the ASI.

CAS is the airspeed after you have applied corrections to the IAS for those instrument and position errors occurring at that aoa in that particular aircraft. The measured corrections should be stated on a card placed near the ASI. You should also be aware that position errors may be quite significant, possibly under-reading by 10 knots or so — particularly at high aoa or when the pilot is maintaining a significant sideslipping manoeuvre. CAS may also be known as the rectified airspeed [RAS].

Below is an airspeed correction card for a particular aircraft in balanced level flight; i.e. not slipping or skidding. The normal cruise speed for this aircraft is around 95 knots. In this particular installation the ASI significantly underreads at low speeds and overreads at high speeds.

IAS knots42526169738796104113122130
CAS knots49576471738694102110117125
Converting CAS to TAS
TAS = CAS × √(ro / r)

Using our example, from section 2.5, of the Jabiru cruising at 88 knots IAS at 6500 feet, where the air density is 1.0 kg/m²:
TAS= 88 × √(1.225 / 1.0) = 88 × 1.107 = 97 knots

We need TAS for navigation and as the density lapse rate just about follows a straight line below 10 000 feet – there is a simple mental calculation to determine TAS from CAS:

Rule of thumb #2

    "To convert CAS to TAS, multiply the (density) altitude, in 1000s of feet, by a factor of 1.5 to get the percentage increase to apply."
(For calculation of density altitude see 'High density altitude: effect on take-off/landing performance'.)

    e.g. CAS = 88 knots at 6500 feet = 6.5 x 1.5 = 10% = 97 knots. The multiplication factor increases slightly with increasing altitude, reaching 2 at 30 000 feet.

The airspeed indicator
airspeed indicator You will note the green and white peripheral arcs, and other colour marks, on the face of this instrument. These are standard markings, some of which should appear on the face of every light aircraft ASI, as they display the speed constraints applicable to the aircraft operations.

The white arc indicates the flaps operating range starting, at the lower end, from the indicated airspeed, Vso [55 knots], at which the aircraft will stall in the landing configuration with flaps fully extended, and with the throttle closed. The top end of the white arc indicates the maximum speed, Vfe [108 knots], at which the aircraft's flaps can be extended, or remain extended, without causing strain. The bottom end of the green arc indicates the stalling speed of the aircraft, Vs1 [62 knots], with flaps (and landing gear if applicable) up, throttle closed and 1g load factor. The top end of the green arc indicates the maximum structural cruise speed, Vno [173 knots]. The green arc indicates the designed range of speeds for normal operations. The yellow arc indicates a speed range in which the aircraft may be flown, but with caution and only in smooth atmospheric conditions. The red line at the top end of the yellow arc indicates the speed, Vne, that should never be exceeded because of risk of structural damage.

The other red mark at 70 knots, and the blue mark at 88 knots, are of no interest for single-engine aircraft; these markings only appear on a twin-engine aircraft ASI and relate to operations with one engine shut down.

A properly functioning ASI responds rapidly to pressure changes because there is no instrument lag. A slow response attributed to instrument lag is most likely only due to the inertia of the aircraft — when attitude in pitch is changed, an aircraft takes a little time to accelerate/decelerate to the appropriate airspeed.

Airspeed summary
  • True airspeed [TAS] = V in the dynamic pressure equation and other expressions = air distance flown over time.

  • Indicated airspeed [IAS] = Vis = airspeed displayed on the cockpit airspeed indicator [ASI] — based on a fixed air density ( ro ) of 1.225 kg/m³. The ASI only indicates true airspeed when ambient atmospheric density is actually 1.225 kg/m³ and the system error corrections are applied.

  • Calibrated airspeed [CAS] = IAS adjusted (mentally from an airspeed correction card) for known system errors occurring within the normal speed range.
Electronic ASI
Electronic flight instrument systems [EFIS] use solid state electronic componentry as sensors plus software to display flight data on a single screen. In such systems, the static and dynamic pressures are fed to pressure transducers which sense and convert pressures to voltages that the electronic circuitry converts to an airspeed display. See the liquid crystal primary flight display of the Dynon D10A light aircraft EFIS. The EFIS has an outside air temperature probe and, with static pressure, the software can calculate air density and thus display TAS when needed.

Electronic ASIs are also available as single panel instruments or possibly combined with an altimeter function. The electronic systems are still subject to much the same errors as a mechanical system, and the IAS has to be corrected for CAS unless there is a means for incorporating some form of compensating table into the software.

2.7 Measuring rate of ascent/descent

The vertical speed indicator
 vertical speed indicator     In flight it is important for a pilot to know the rate at which the aircraft may be ascending or descending. A simple vertical speed indicator [VSI] is a pressure gauge that measures the rate of pressure change as an aircraft is ascending or descending. The instrument display is usually calibrated in feet per minute but it may be in metres per second.

There are two pressure inputs, both from the static vent system — one to each side of a flexible diaphragm or capsule. On the open side there is a normal input that reflects the static pressure change as it occurs. On the closed side the input/output is a fine capillary tube that slows the equalising pressure change — and also the response time of the instrument. The resultant deflection of the diaphragm is magnified via a geared mechanical linkage to a dial pointer, which indicates whether the aircraft is maintaining altitude (in which case, the pressure on both sides of the diaphragm is equal), climbing or descending, and the rate, usually graduated in feet [×100] of altitude per minute. Some form of vibration damping and thermal change compensation is included within the VSI and ASI instruments.

The pressure change has two components, the most significant component is that brought about by the aircraft's rate of height change, as in climbing or descending. The other part is any vertical movement of that part of the air mass in which the aircraft is operating — rising air or sinking air. When an aircraft is climbing, rising air adds to the rate of change, sinking air reduces it. When descending, rising air decreases the rate of change, sinking air adds to it.
The variometer
In soaring flight, paraglider, hang glider, sailplane and power-assisted sailplane pilots are totally reliant on finding sources of atmospheric uplift to gain the gravitational potential energy that enables the aircraft to stay airborne for sufficient time to complete the flight plan. A variometer (usually abbreviated to vario) is a specialised vertical speed indicator that enables a pilot to derive the vertical speed of the parcel of air in which the aircraft is soaring.

For more information on varios and their uses see the article 'Basic sailplane instruments. The article only refers to fixed installation varios but very light-weight hand-held varios are available for hang glider/paraglider pilots.

2.8 Stalling airspeeds

The normal 1g stall
    One of the first questions a pilot might consider, when converting to a new aircraft type, is "What's the stall speed?" The reason for considering this is that usually, but not always, the approach speed chosen for landing is 1.3 to 1.5 times Vso — the minimum steady flight speed in the landing configuration, below which speed the aircraft will stall or at which speed the aircraft will stall if any manoeuvring is attempted.

    In aerodynamic terms, the 'stall' is the sudden widespread separation of the boundary layer from the upper wing surface that occurs when the wing exceeds a particular angle of attack. For light aircraft without high-lift devices, this is usually around 15–16° although minimum aircraft with single-surface fabric wings may have a stall aoa 2–3° lower. This critical angle of attack has no relationship with either the aircraft attitude relative to the horizon or the airspeed — it can readily be reached in a high-speed dive. But it does have a direct relationship with elevator position and thus the control column position.

The separation of the boundary layer starts at the wing trailing edge, generally near the wing root for approximately rectangular wings (and particularly for wings with 'washout'), spreading forward and outward over the upper surface until there is a significant detachment of boundary layer flow over the upper surface. There is probably little change to the under-surface boundary layer flow. Between the two remnant boundary or shear layers, a thick turbulent wake will attach to the wing and be dragged along by the aircraft. The reaction to the acceleration and energising of that wake is a sudden deceleration of the aircraft accompanied by a large increase in the nose-down pitching moment plus some loss of lift. The initial wake turbulence ('burbles') near the wing root may initiate unsteady flow over the tailplane, shaking the tailplane and thus providing a few warning buffets felt in the airframe or smaller 'nibbles' felt in the controls — aerodynamic warnings of an impending stall. There also may be 'oil-canning' noises from pressure changes on metal-skinned fuselages and wings as the thin metal flexes in response to pressure changes. On the other hand, there may be no pre-stall warning whatsoever.

Some aircraft exhibit undesirable characteristics even before boundary layer separation occurs; for example, the aircraft starts sinking excessively with increasing induced drag or wing rocking occurs. So, from a pilot's point of view, a stall is "the point following deceleration at which the pilot ceases to have full control over the aeroplane"; which adds the concept of the defined stall speed being a minimum controllable steady flight speed at which no undesirable characteristics are exhibited.

The next comments are specifically aimed at stalls induced when:
    •   flying straight and level at slower speeds
    •   in a low speed descent — such as the approach to landing
    •   in a climb — such as the initial climb after take-off
    •   in a go-around following an aborted landing approach.

The last two circumstances are sometimes referred to as full-power stalls or 'departure stalls'. In non-turbulent atmospheric conditions, and if the aircraft is in balance, all of the circumstances above can only induce a stall if the control column position is placed in, trimmed into or allowed to move into, the last half of its rearward travel. Many aircraft are designed so that the control column must be at or near the limit of its rearward travel to reach the stalling aoa. (The rearward travel range commences from the neutral position, as does the forward travel range.)

    Because of the airflow turbulence and increasing induced drag as the aoa is increasing, total drag increases and the aircraft slows as it approaches CL max. The rapid reduction in airspeed after passing the critical aoa means the wing is now unable to provide sufficient lift to totally balance weight and, in a normal stall, the aircraft starts to sink. The (possibly pronounced) nose-down pitch will occur even though the control column is near its rearward travel limit. However, some aircraft may not assume that nose-down attitude but just sink (mush down) at quite a high rate and at an extreme angle of attack. Because of the nose-up attitude, the high rate of descent may not be apparent unless the aircraft is close to the ground.

The aircraft is instantly recovered from the stall by smoothly reducing the aoa so that it is below the critical aoa; i.e. easing the control column forward and generally no further than the neutral position. If one wing stalls before the other, that wing will drop. In this case, the control column must be firmly moved sufficiently forward to unstall the dropping wing, the wings levelled with aileron then sufficient rudder applied to stop further yaw. An increase in speed is also needed (by increasing power or holding a lower nose attitude for several seconds) so that a safe flight speed is achieved quickly without wasting much altitude and the aircraft is returned to the intended flight path. See standard recovery procedure for all stall types.

If the control column movement for stall recovery is both excessive and abrupt, the result could be an aoa movement below the zero lift aoa — in which case there will be a reversed lift force on the wings, which hinders recovery. Weight-shift controlled trikes do not react well to negative g; if this is excessive, the wing spars may buckle at an outer position.

Many aircraft are designed so that the nose will drop at the stall, but the aircraft will self-recover (i.e. without pilot intervention) in a stable descent or with some oscillations which, if the control column is still held back, will result in another stall. Some aircraft may be designed so that the wing is usually not able to reach the stalling angle, but the aircraft will enter a semi-stable mushing descent — which sounds fine but can be disastrous if the pilot doesn't notice when close to the surface. A normal stall occurs when the load factor is close to normal; i.e. near 1g.

The cg position will also affect the manner of stall. If the cg is at the extreme forward limit, some aircraft may not fully stall — just mush down. If it is too far aft, the stall aoa can be reached with a much smaller rearward movement of the control column. Another factor affecting the manner of stall is the use of power. Generally, when flying slowly, the longitudinal axis of the aircraft is pitched up relative to the flight path. Consequently the thrust vector will include a vertical component — a lifting force — and the amount of lifting force provided depends on the amount of thrust. Also, for aircraft with the propeller mounted in front of the wings, the energy in the slipstream tube in slow flight increases the velocity of the airstream over part of the wing (depending also on the mounting of the wing in relation to the thrust centre line) and reduces the aoa of that part. Thus the completely stalled wing may occur at a lower speed, depending on the amount of power in use. When it occurs, the stall will be much more pronounced, possibly with a fast-acting wing drop. There are other complications because the slipstream also affects parasite drag and induced drag.

Many pilots, in suitable aircraft and atmospheric conditions, prefer to land by approaching at 1.3 to 1.5 times normal stall speed — Vso — and, after flaring with the throttle closed, holding the aircraft just above the surface; preventing it touching down by smoothly increasing CL as drag decreases V², thus maintaining constant lift until CLmax is reached. At this point, the aircraft can no longer be 'held off' and it gently sinks the short distance onto the runway, touching down in a nose-high attitude.

[ The next section in the airmanship and safety sequence is the follow-on section 'The accelerated stall' ]

The acceleration or accelerated stall
It is misleading to talk about stalling speed without further definition. The stall occurs at a particular aoa, not a particular speed. The speed — Vs — below which the stall will occur depends on the load factor. If the aircraft reaches the critical aoa under a load higher than 1g, the stalling speed will be higher than the normal 1g stall speed, at that mass. This latter stall is called an acceleration stall or accelerated stall and is usually more pronounced than a normal stall. The load factor normally increases in a turn— as we saw in section 1.10 where we calculated that, in a 45 banked turn, the load factor was 1.41 times normal. Thus, when turning, the stalling speed is higher than normal and the pilot must maintain a reasonable airspeed margin above that accelerated stall speed throughout the turn. See the table below.

Be aware that the airspeed at which an acceleration stall in a turn occurs is only indirectly associated with the angle of bank; it is directly brought about by the increase in load factor. Indeed, it is possible to have the aircraft banked at 60° with a stall speed less than Vs1 if the wings are 'unloaded': slight forward pressure on the control column, and the aircraft allowed to sink, produces a load less than 1g — maybe 0.8g — with a stall speed less than Vs1, even though the aircraft is steeply banked. However, once the 'unloaded' condition ceases — if the stalling angle of attack has been passed (either by the rearward movement of the control column or a gust momentarily changes the relative airflow) — the probability of a stall returns immediately.

    The speed at which an accelerated stall occurs is proportional to the square root of the load factor — in the lift equation the airspeed is squared. If that load factor is expressed relative to the normal load, e.g. 2g, then the stall speed at that load factor — Vs 2g — equals the square root of the load factor × normal 1g stall speed; e.g. square root of 2 = 1.41 × Vs.

    The aircraft's momentum may also contribute to an accelerated stall, particularly when the aircraft is diving at speed and the pilot applies a harsh rearward control column movement. This will have the initial effect of rotating the aircraft about its lateral axis while inertia momentarily maintains the aircraft on its pre-existing flight path; thus the aoa may exceed the stalling aoa (even though the control column has not been pulled back to the normal stall position) with a consequent, and rather violent, high-speed stall.

    An acceleration stall can also be produced when:
    •   the control column is jerked back whilst the aircraft is climbing or in level flight; see the flick roll
    •  an aircraft in level cruising flight encounters a strong vertical gust
    •  an abrupt change in flight path is made, which applies acceleration loads
    •  an excessive bank angle, coupled with excessive control column back pressure, is applied during a level, climbing or descending turn.

Note: The US Federal Aviation Regulations Section 23.203 airworthiness standards define accelerated stalls somewhat differently from the above, only referring to 'turning flight stalls' and 'accelerated turning stalls' for airworthiness demonstrations.

"Turning flight and accelerated turning stalls must be demonstrated in tests as follows:
(a) Establish and maintain a coordinated turn in a 30 degree bank. Reduce speed by steadily and progressively tightening the turn with the elevator until the airplane is stalled. The rate of speed reduction must be constant, and--
(1) For a turning flight stall, may not exceed one knot per second; and
(2) For an accelerated turning stall, be 3 to 5 knots per second with steadily increasing normal acceleration*."

* 'Normal acceleration' refers to the aerodynamic force parallel to the aircraft's normal axis.

[ The next section in the airmanship and safety sequence is the follow-on section 'Load factor in a turn' ]

Load factor in a turn
The table below shows the increase in stall speed at various bank angles in correctly executed level turns. The load factor or 'g' = 1/cosine of the bank angle and the Vs multiplier = the square root of the load factor. The table shows that once you reach bank angles of 30° or more, the aircraft stall speed increases rapidly; there is a 7% increase at 30°, 19% at 45° and 41% at 60°.

Thus, level turns involving bank angles exceeding 20–30° should not be made at low levels, including take-off and landing operations. Even so, the airspeed should be increased to allow an appropriate safety margin — for gentle turns, a safe speed near the ground is 1.5 × Vs.

The stall speed in a turn = Vsturn = Vs × Vs multiplier. A minimum turning speed at a safe height might be 1.2 × Vsturn. For example, if Vs is 50 knots and the bank angle is 45° then Vsturn is 50 × 1.19 = 60 knots and the minimum safe turning speed at height is 1.2 × 60 = 72 knots, or about 1.45 × Vs.

Bank angleCosineLoad factor [g]Vs multiplier
10° 0.98 1.02 1.01 [+1%]
20° 0.94 1.06 1.03 [+3%]
30° 0.87 1.15 1.07 [+7%]
40° 0.77 1.30 1.14 [+14%]
45° 0.71 1.41 1.19 [+19%]
50° 0.64 1.56 1.25 [+25%]
54° 0.59 1.7 1.3 [+30%]
60° 0.50 2.00 1.41 [+41%]
70° 0.34 2.94 1.71 [+71%]
75° 0.25 4.00 2.00 [+100%]

Note that the stall speed increases exponentially with bank angle; i.e. the 10° increase in the bank angles between 20° and 30° increases stall speed by another 4%, while the 10° increase in the bank angle between 50° and 60° increases stall speed by a further 16% (i.e. four times as much), while between 60° and 70° the stall speed is increased by a further 30%. Consequently aircraft certificated in the normal category are limited to a turning angle of bank of not more than 60°.

Note that at an approach speed of 1.3 × Vs the aircraft will stall if turning with a 54° bank. The limits on climbing and descending turns are discussed in the 'Safety: control loss in turns' module.

[The next section in the airmanship and safety sequence is the section 'Critical limiting speeds' ]

The torque stall
For high-performance aircraft, with a very high power-to-weight ratio, the possibility of a torque stall exists. The most likely scenario is a sudden application of full power in a 'go-around' following an aborted landing, where the airspeed has been allowed to decay below the safety speed. The torque of the engine and inertia of the heavy propeller tends to twist the aircraft around the propeller shaft, and the consequent roll may increase the aoa of the downgoing wing past the critical aoa. If that happens, the wing loses lift, which accelerates the roll and the aircraft loses height very rapidly. However, torque stalls are probably not applicable to light aircraft, although the torque effect may influence the characteristics of a stall in a climbing turn.

Effect of weight
If the aircraft is below its MTOW, the operating wing loading will be less than the design W/S and the stall will occur at a lower speed than that marked on the ASI.

For example, if we refer to the Jabiru, the wing area is 7.9 m², MTOW is 4200 N, Vso is 40 knots CAS and we can calculate that CL with flaps fully extended is 2.0.

We saw above in the section 'The acceleration or accelerated stall' that W/S at the stall = CL × ½rV².
We will rearrange that and say Vs² = (W/S) / (CLmax × ½r). Substituting the values, including 1.225 for density, we get:

Vs² = (4200/7.9) /(2.0 × 0.5 × 1.225) = 532/1.225 = 434 m/s and Vs = 20.8 m/s = 40 knots CAS

Now what will Vs be when the Jabiru with no passenger on board is at the low weight of 3400 N? Well, substituting that weight we get:
Vs² = (3400/7.9) /(2.0 × 0.5 × 1.225) = 430/1.225 = 351 m/s and Vs = 18.7 m/s = 36 knots CAS.

There are other, somewhat simpler, ways to calculate the reduction in Vs corresponding to a reduction in weight but what we see above is that a reduction in weight of 800 N, or about 19%, reduces Vs by 4 knots, or about 10%. This brings us to the mathematical rule of thumb that when two values are not that far apart in percentage terms, say up to 40%, their square roots are about half that distance apart in percentage; and because aerodynamic pressure is proportional to V², there are many occasions where the square root of a value is relevant. This allows a simple, but reasonably accurate, mental calculation:

Rule of thumb #3

    "The percentage reduction in Vs is half the percentage reduction in weight."

    i.e. If weight is reduced by 10% from MTOW then Vs will be reduced by 5%, and conversely, if weight is 10% over MTOW then Vs will be 5% higher — one of several reasons to avoid overloading an aircraft. (There is further discussion on weight control throughout these notes.)

Thus in the section 'The acceleration or accelerated stall' above, where we referred to unloading the wings with the aircraft banked at 60°, the load reduction from 1g down to 0.8g is 20% so the unloaded stall speed would be about 90% of Vs1. You can also see the same relationship in the preceding table; for bank angles up to 45° the percentage increase in Vs is about half the percentage increase in W/S.

It is appropriate to mention here that it is not only aircraft weight/wing loading that affects the stall speed. Some of the other critical performance values are also achieved at a particular aoa, and the associated airspeeds are also changed by a change in weight. The same rule of thumb applies to them. These critical performance values (the 'numbers') are: best rate of climb speed, best angle of climb speed, lowest power-off sink rate speed, best glide ratio speed and manoeuvring speed.

Another aspect we will look at in the 'Aerofoils and wings' module is the effect of flaps, but we will just state here that flaps provide an increased CL at all angles of attack consequently allowing a reduction in V² and the stalling speed. In some aircraft extending flaps also increases wing area, thus W/S is reduced, a handy technique for high-performance military aircraft, manoeuvring at maximum allowed wing loading — they can tighten the turn even further without breaking the aircraft.

    The final aspect of the stall is the effect of atmospheric turbulence on aoa and this affects 'manoeuvring speed'. We will look at it in the 'Wind shear and turbulence" tutorial.

2.9 V-speeds

Airspeed codes
It is important to have a simple, easily understood and universally accepted identification method for the various airspeeds at which an aircraft may be operated, but currently it's a bit messy and there is no complete, unambiguous, and universally recognised, airspeed designation system published by any regulatory authority.

Current nomenclatures are generally made up of two to six letters/numbers, with the first being V. Some of these V-speed codes — applicable to single-engine aircraft — with alternatives and definitions are shown below. These are relevant to sport and recreational aircraft including low momentum ultralight aircraft, and might appear in flight manuals, pilot's operating handbooks and even sales literature but those indicated with open bulleting   º   are probably only applicable to a few very light aircraft types.

There are two classes of airspeed codes. One is the structural design speeds (specified in national airworthiness requirements) used in determining the airframe and control surfaces strength requirements for type certification. Such speeds include the term 'design' in their description. The other class are the designer recommended operating speeds.

Please be aware that the various 'best' performance speeds mentioned below — rate of climb, angle of climb, cruising range, gliding range, etc. — merely indicate the midpoint in an airspeed range extending perhaps 1–2% either side of that point. Also, the performance speeds are very much affected by the horsepower of the particular engine fitted, plus the type of propeller and its pitch setting. If there is no pilot's operating handbook for the particular airframe/engine/propeller configuration, then the pilot must calculate the performance speeds by trial and measurement.

Critical limiting speeds
   •  Va — design manoeuvring speed. Design rules state that the minimum acceptable design manoeuvring speed is a fixed calculation relative to Vs1 for all aircraft within the same category. For a 'normal' category light aircraft (whose certificated load limit factor in the pitching plane is +3.8g), minimum Va = 3.8 Vs1, or 1.95 × Vs1. For a 'LSA' category aircraft (whose certificated vertical load limit factor is +4g), minimum Va = 4 Vs1, or 2 × Vs1. If the designer has opted for a design manoeuvring speed that is greater than the minimum acceptable speed then a Vo operating manoeuvring speed must also be specified.

Va is also known as the 'optimum manoeuvre speed', or the 'corner speed', to military pilots as it's at the intersection of the structural limit load factor and the maximum aerodynamic force curve (the 'A' corner) in the aircraft's manoeuvring flight envelope, i.e. Va is the speed at which an aircraft can make the tightest possible turn (the minimum radius turn) and the fastest rate of turn by applying the aerodynamic limit (maximum aoa [CL max]) and the structural load limit simultaneously. (Of course, in the military context, in such a turn the aircraft would be comparatively low in kinetic energy.) In the sailplane context the symbol Vm is used for a manoeuvring speed which is the product of the square root of the design load limit factor and the minimum flight speed.

Va is sometimes referred to as the 'speed for maximum control deflection' which has been the cause of much confusion. It is unwise to make full or abrupt applications of any one primary flight control if you are flying at a speed greater than Va, because at higher speeds it is easy to apply (see the stick force gradient) aerodynamic forces that could exceed the aircraft's structural limitations. But, even when flying below Va, it is unwise to make rapid control reversals or 'checks' such as alternating heavy applications of rudder or suddenly apply heavy asymmetric loads, e.g. heavy application of elevator and rudder or aileron; see the flick roll.

(That misleading term, 'speed for maximum control deflection', was subject to much debate in 2001 following the horrific crash of American Airlines Flight 587, an Airbus A300 which, shortly after take-off while in a climbing turn at an airspeed 20 knots below standard Va, ran into wake vortices from a Boeing 747 four miles ahead. To counter sideslip it appears the pilot flying employed four nearly full-rudder reversal movements within a seven-second period. Those pilot-commanded side forces induced a tail fin load twice the design load limit and 1.3 times the ultimate load limit, at which point the complete fin and rudder broke away. In 2010 the U.S. Federal Aviation Administation issued a 'Maneuvering Speed Limitation Statement' namely:

(i) Full application of pitch, roll, or yaw controls should be confined to speeds below the maneuvering speed; (ii) Rapid and large alternating control inputs, especially in combination with large changes in pitch, roll, or yaw, and full control inputs in more than one axis at the same time, should be avoided as they may result in structural failures at any speed, including below the maneuvering speed. )

Va is usually not marked on the ASI but there should be a placard indicating the MTOW manoeuvring speed on the instrument panel near the ASI or in the Pilot Operating Handbook or Aircraft Flight Manual; if not available, you can assume it's twice maximum weight Vs1 for non-aerobatic light aircraft and reduces as aircraft inertia (i.e. weight or, more properly, mass) reduces and thus Vs1 decreases.

Note: if a recreational pilot is foolish enough to operate an aircraft at a weight exceeding MTOW then the overloaded aircraft's stall speed will be higher; consequently Va in that condition will be higher.

So Va is not a fixed documented value, it decreases as the aircraft's weight decreases from MTOW, because the effects of the atmospheric forces become more pronounced as its inertia decreases. Sometimes the aircraft's documentation will specify the Va for weights below MTOW but it may be left up to the pilot to calculate the Va for the current aircraft weight. Using rule of thumb #3 above, the reduction in Va will be half the percentage reduction in aircraft weight; for example if, with only the pilot on board, weight is 16% below MTOW then Va is reduced by 8%. Flying at speeds below that estimated Va in turbulent conditions also enhances the possibility of stalls induced by vertical gusts — and also may reduce aileron and rudder effectiveness, so the pilot in command must be careful to select an optimum speed for the atmospheric conditions. Also note that the documented Va is calculated for the aircraft in a clean configuration — it does not apply to flight with the flaps extended, see Vfe maximum flaps extended speed.

Misuse of controls in light aircraft can generate structural loads greater than those encountered in turbulence, so Va is also useful as a 'turbulent air operating speed' and most recreational aircraft operating handbooks recommend that airspeed be reduced to Va in turbulent conditions. When flying above this speed, gust-induced loads can exceed the design limit of many structures within the aircraft, including the pilot's seat and the engine mount, possibly even the crankshaft. Gust loads in the high temperature conditions of the Australian tropical continental air mass can be extremely high. Va is the recommended indicated cruising speed (CAS) when flying in moderate turbulence — intermittent, uncomfortable jolts. At this compromise speed, the aircraft will generally produce an accelerated stall and thus alleviate the aerodynamic force (including any manoeuvring forces) on the wings and other structures, if it encounters a vertical current that imparts an acceleration sufficient to exceed the load limit factor. Read 'The speed to fly in turbulence', in the 'Decreasing your exposure to risk' guide.

   º  Vo — operating manoeuvring speed. If the aircraft designer has specified a design manoeuvring speed that is greater than the regulatory minimum (n × Vs1 where 'n' is the category limit load factor) then, when flying at Va and if a substantial nose-up pitching manoeuvre is applied, the aircraft may exceed the limit load factor before stalling. So, an operating manoeuvring speed Vo should also be established as an operating limitation speed, which is a selected speed that is not greater than n × Vs1 and is a speed where the aircraft will stall in a nose-up pitching manoeuvre before exceeding the structural load limits. Thus Va must be equal to or greater than n × Vs1 while Vo must be equal to or less than n × Vs1. The load limit factor is 3.8g for normal category sport and recreational aeroplanes and 4g for the LSA category. The square roots of 3.8 and 4 are very close — 1.95 and 2 respectively — so we can rephrase the preceding statement as 'Va must be equal to, or greater than, twice Vs1 while Vo must be equal to, or less than, twice Vs1. Remember that Vs1 stall speed reduces when aircraft weight is less than MTOW so the appropriate Va indicated air speed will decrease in proportion to the decrease in aircraft weight. Flying at speeds much below Va in turbulent conditions also enhances the possibility of stalls induced by vertical gusts — and also may reduce aileron and rudder effectiveness.

   º  Vb — design speed for maximum gust intensity. The applicable vertical gust intensities range from 25 fps (7.5 m/s) to 50 fps (15 m/s). Also known as the maximum rough air speed. It is not required to be specified for normal, utility and LSA aeroplanes (in those categories Vb would generally only differ by a few knots from Va). Vb is specified in the European Joint Airworthiness Requirements JAR-22 for sailplanes and powered sailplanes in the utility and acrobatic categories but in this case Vb is the speed at which the sailplane is able to withstand a strong vertical gust of 50 fps (15 m/s or 30 knots) without exceeding the load limit factor, i.e. it is the speed at which an encounter with a gust of the specified value produces CL max.

   •  Vfe — maximum flaps extended speed. It is indicated by the top end of the ASI white arc. Flight with flaps extended — or extending flaps — above this speed may result in distortion of the flaps or their supporting structure and extension mechanism. Various Vfe speeds may be specified according to the available flap settings. Generally speaking, the flight load limit factor is reduced by about 50% when flaps are fully extended. For example, the aircraft flight manual of a 'normal' category light aircraft will probably note that the load limit factor is reduced from 3.8g to 2g. So, extending some flap in turbulent conditions will decrease the stall speed but will reduce the load limit factor; thus it's very much a 'pilot-in-command' decision of 'when' and 'how much' to use, but certainly full flap is most unlikely in any condition.

   •  Vno — maximum structural cruise speed or 'normal operating limit', indicated by the top end of the ASI green arc. Flight above Vno should only be conducted cautiously and in smooth air, while the pilot should not apply any abnormal control inputs when cruising at, or above, Vno. Vno must be equal to or greater than Vc (below, in section 'Cruise speeds'), but in most light aircraft Vno and Vc are assumed synonymous. When cruising at, or below, Vno, the aircraft should not be damaged by a 30 feet/second vertical gust — which is at the bottom end of the moderate to strong vertical gust scale of 25–50 feet/second vertical gusts; read 'The speed to fly in turbulence'.

   •  Vne — never exceed speed, which is the IAS that should never be intentionally exceeded in a dive, or other manoeuvre in smooth air. FAR 23 requires that Vne be not more than 90% of the design diving speed Vd or the flight-demonstrated diving speed Vdf. Vd and Vdf are not included in pilot operating documentation — they are the realm of the test pilot. Vne is indicated by the red line at the top end of the ASI yellow arc. For light aircraft operating below 10 000 feet, it can usually be assumed that Vne is a fixed IAS. If aircraft have high altitude capability or particular airframe vibration characteristics, it is possible that the designer will specify Vne as a TAS, above a particular altitude and for various altitude bands of perhaps 3000 feet. If Vne varies with altitude, then FAR Part 23.1545 (c) requires a placard next to the ASI indicating the appropriate IAS limitations throughout the aircraft's operating altitude range. This particularly applies to sailplanes and powered sailplanes whose very high aspect ratio wings are candidates for flutter problems and such low drag aircraft can build up airspeed at altitude very quickly in quite shallow dives. For expanded information see 'Don't fly real fast' in the 'Decreasing your exposure to risk' guide.

   •  Vs1 (sometimes incorrectly shown as Vsi) — stalling speed, or the minimum steady flight speed, in a specified flight configuration. For a simple aircraft, Vs1 is normally measured in level flight with flaps up, at MTOW and 1g wing loading, with engine idling following a gradual deceleration (one knot per second) — accompanied by increasing rearward movement of the control column — to that minimum flight speed. It is indicated by the bottom end of the ASI green arc, but it should be documented as both IAS and CAS; if CAS is not mentioned the quoted stall speed is probably inaccurate. Vs1 decreases as the aircraft weight decreases from MTOW, which also means that if the pilot can reduce the wing loading below 1g — by an 'unloading' manoeuvre — Vs1 is decreased. Stalling speed under a 2g wing loading, for instance, might be referred to as Vs2g.

   •  Vso — stalling speed, or the minimum steady flight speed, in the landing configuration of flaps down and engine at low or idle power as it would be just prior to touchdown. This is measured using the same method as Vs1 but with the cg at the most extreme position allowed — usually the most forward position where backward movement of the control column may be limited. It is indicated by the bottom end of the ASI white arc.In the documentation both IAS and CAS should be shown. Like Vs1, Vso decreases as the aircraft weight decreases from MTOW. The designation Vs is used as a general reference to the design stall speed.

   •  Vmin — minimum airspeed. Vs is generally specified in powered, rigid-wing recreational aircraft as the minimum speed but for other aircraft categories a 'Vmin' may be specified instead. For example for gyroplanes Vmin is the minimum controllable level flight airspeed below which there is insufficient power available to maintain altitude. For paragliders it is the minimum speed, within the wing's available trimmer range, below which the parawing loses its lift.

[The next section in the airmanship and safety sequence is the section 'The design manoeuvring flight envelope' ]

Cruise speeds
A cruising aircraft is normally flying at a moderate, fuel efficient speed and maintaining the appropriate cruising altitude. The Australian Civil Aviation Regulations hold this definition:

"cruise phase of flight" means the part of an aircraft's flight:
          (a) that starts when the aircraft reaches its first planned cruise level, ... and
         (b) that ends when the aircraft reaches the point at which the aircraft first starts its descent for the purpose of landing; and includes flight level changes made during that part of the flight.

     •  Vbr — best range, or Vmd — minimum drag, is the speed that provides maximum L/D by producing minimum drag and thus the best power-to-speed ratio. This speed might utilise about 55% power and is usually flown at the lowest altitude where the throttle is fully open to obtain that speed. Vbr/Vmd decreases as the aircraft weight decreases from MTOW. It's rather boring to fly at that speed, wind conditions have to be taken into account, and the fuel saving may not be that significant compared to flying at a speed 10% faster. Also, the engine manufacturer's operating recommendations should be followed, but mixture is usually leaned, and minimum rpm set if a constant speed propeller is fitted. Vbr/Vmd has the same basic airspeed range as Vy and Vbg [below].

power curveThere is a difference in concept between Vbr and Vmd. Pilots of low-powered aircraft are generally not interested in the best power to airspeed ratio in cruise; ground speed achieved per litre is far more significant, so in some conditions Vbr equals Vmd but in headwind conditions Vbr is increased. Look at the diagram from section 1.7 at left and note the pink line that has been drawn from the junction of the vertical and horizontal axes tangential to the power required curve. That line just touches the curve at a position corresponding to the minimum drag airspeed Vmd. Now imagine a 30 knot headwind and start the tangential line from a point along the horizontal axis that is equivalent to 30 knots; that (the blue line) will be tangent to the power required curve at a position corresponding to a higher speed — Vbr for a 30 knot headwind. The rule of thumb is to add half the head wind to the basic Vbr, which, in this case, indicates a Vbr that is about 15 knots greater than Vmd. This is the same principle used by sailplane pilots to establish their best penetration speed — see the speed polar curves for optimum glide speed in the 'Coping with emergencies guide'.

     •  Vbe — best endurance, or Vmp — minimum power, is the CAS that gives the greatest airborne time per litre (i.e. least fuel flow per hour and, of course, power is proportional to fuel flow), possibly around 80% of Vbr/Vmd, and decreases as the aircraft weight decreases from MTOW. Flight at lowest safe altitude provides best engine performance. Might utilise about 45% power at MTOW. It is the speed for minimum power required for level flight, as shown in the power required curve above.

Vbe/Vmp is the speed that might be used when flying a search pattern to allow a proper area survey, or when waiting for ground fog to disperse, but it is possibly uncomfortable to fly for long periods at such a low speed. Also the very low power setting may be inconsistent with good engine handling practice. Carburettor icing may be aggravated. The Vmp designation and speed is also used as a power-off glide speed, providing the best endurance — least rate of sink — in the glide; see 'Power-off descent speeds' below. Vbe/Vmp is in the same speed range as Vx — the best angle of climb airspeed.

   •  Vc — the design cruising speed or the optimum cruise speed — the latter being the speed that gives the most velocity (i.e. greatest distance/time) from a litre of fuel, usually utilises 75% power at MTOW and is about 20–30% greater than the maximum L/D speed — Vbr. The speed and power required both decrease as the aircraft weight decreases from MTOW. Refer to rule of thumb #3 in section 2.8 'Stalling airspeeds'.

For normal category aircraft, FAR Part 23 specifies a minimum design cruising speed (in knots) = 33 W/S. For this calculation the wing loading W/S is expressed in pounds per square foot. Many minimum ultralights are unable to comply with the FAR Part 23 design requirement for a minimum design cruising speed.

For most light aircraft, Vc is synonymous with Vno. FAR 23 Appendix A provides simplified design load criteria and allows designers of many conventional single-engine monoplanes weighing less than 2700 kg to take advantage of the simplification. That same appendix is generally duplicated in the design regulations of most other countries. One advantage is that it is not necessary to specify Vno; instead, Vc is designated in the flight manual as the maximum structural cruise speed (i.e. Vno = Vc) and that Vc may be set at 90% of Vh.

   º  Vh — the maximum level flight indicated speed (CAS) attainable at sea-level, utilising maximum continuous engine power. For most engines maximum continuous engine power at sea-level will be less than full throttle power.

Take-off and landing speeds

   º  Vle — for retractable undercarriage aircraft — the maximum indicated speed at which the landing gear can remain extended without risking gear door damage.

   º  Vlo — the maximum indicated speed at which the landing gear system can be operated. Vle and Vlo are unlikely to be applicable to most ultralights.

   º  Vlof — the lift-off indicated speed for normal take-off. Vlof is about 10% above Vmu.

   •  Vmu — minimum unstick speed. This is an indicated speed used in take-off conditions where it is advisable to lift off at the lowest possible airspeed to get the tyres off the surface (e.g. soft field or wet grass ) and safely fly in ground effect until a Vtoss is attained to allow climb-out. Acceleration after lift-off at Vmu is slow, due to the drag at the high aoa, and should not be used as an obstacle clearance technique.

     •  Vref — the threshold speed or the reference indicated approach speed. Usually about 1.3 to 1.5 times Vso plus 50% of the wind gust speed in excess of the mean wind speed; e.g. Vso = 30 knots, wind speed 10 knots gusting to 20 knots, Vref = 1.3 x 30 + 5 knots = 44 knots. Faster, heavier aircraft would tend towards the 1.3 times Vso end; lighter, slower aircraft would tend towards the 1.5 times Vso end. Normal landing procedure is to set up the approach so that an imaginary 15 metre (50 ft) high screen placed before the runway threshold is crossed at Vref and the airspeed is reduced to maybe 1.2 to 1.3 × Vso — plus the gust allowance — when rounding out prior to touchdown. The ground distance from the screen to the touch-down point can be roughly estimated, using the 1-in-60 rule, from the approach slope. For example, with a 6° slope — which is around the norm for most light 3-axis aircraft — the distance will be 60/6 × 15 = 150 m. To this must be added any float period plus the ground roll distance with normal braking, to give the total landing distance over the standard 15 m screen — in nil wind conditions.

   •  Vtoss — minimum take-off safety speed. This is an indicated speed chosen to ensure that adequate control will still exist during initial climb after lift-off if power is lost or turbulence encountered. After lift-off, the aircraft should be held down and not allowed to climb away until Vtoss is attained.

CAO 101.28, an airworthiness certification requirement for commercially supplied amateur-built kit ultralights, states in part:
"The take-off distance shall be established and shall be the distance required to reach a screen height of 50 feet from a standing start, with ... short, dry grass surface ... the aeroplane reaching the screen height at a take-off safety speed not less than 1.2 Vs1 ... take-off charts ... shall schedule distances established in accordance with the provisions of this paragraph, factored by 1.15." Sea-level ISA and nil wind conditions are implied. CAO 95.55 has much the same wording but specifies 1.3 Vs1 as the take-off safety speed.

(Similarly, CAO 101.28 states that the landing distance will be that to come to a full stop from a screen height of 50 feet, with the screen being crossed at 1.3 Vso and the same conditions as specified for the take-off distance.)

In normal take-off conditions Vtoss should be somewhere between 1.3 and 1.5 times Vs1, with 'draggy' aircraft tending to the higher value. If power is lost in the initial climb, a draggy aircraft will lose airspeed very rapidly and take some time to regain it even though the pilot reacts quickly and pushes the control column forward. See 'Engine failure after take-off'.

There is a similar code used for multi-engine aircraft — Vtos — which refers to the minimum speed for climb-out with one engine inoperative.
Climb speeds
   •  Vx — indicated speed provides best angle of climb for obstacle clearance; i.e. to attain height over the shortest ground distance using maximum thrust available. This is probably better described as the precautionary climb speed — the initial climb speed used when there are obstructions off the end of a marginal airstrip or when climbing out of an obstructed valley. Vx decreases as the aircraft weight decreases from MTOW (refer rule of thumb #3 above), but the angle of attack is maintained at around 8–10 with very high induced drag. It is the climb airspeed where the ratio of vertical speed to horizontal (ground) speed is the highest. Vx may be less than or equal to Vtoss. The aircraft's power-to-weight ratio (i.e. excess power) and L/D ratio affect the angle of climb at the designated airspeed.

However, be aware that the angle of climb will also depend on the low-level wind conditions at the airfield. In a headwind, the climb angle is increased and reduced in a tailwind. Also note that aoa during climb may be only 5 or 6° below the critical aoa, thus care must be taken not to induce a 'departure stall', particularly in turbulent conditions. And remember that Vs1 increases in a turn, so that the small safety gap between Vx and Vs1 will be eroded if a climbing turn is attempted; see 'Safety: loss of control in low level turns'.

Climbing at Vx should always be regarded as a short-term precautionary procedure, and once clear of obstacles, airspeed should be increased to Vy — or any appropriate 'enroute climb speed'. The latter reduces the rate of climb but has the benefit of reducing total sector time, increasing forward visibility and increasing engine cooling — which may be beneficial to engine operation but, more importantly, provides a little more airspeed in hand should the engine falter or fail. The airspeed for Vx increases with (density) altitude and is much the same airspeed as Vbe, although engine cooling needs might require a higher airspeed.

  •  Vy — indicated speed for best rate of climb. This speed is used to attain height in the shortest time using maximum power, or possibly maximum continuous climb power. Vy decreases as the aircraft weight decreases from MTOW (refer to rule of thumb #3 above), but the angle of attack is maintained at around 6–8. After reaching a safe height airspeed may be increased to an appropriate enroute climb speed. The CAS for Vy decreases with (density) altitude — i.e. as TAS increases — and also is usually fairly close to the maximum L/D speed Vbr, taking engine cooling flows into account. Vx and Vy converge as (density) altitude increases.
Power-off descent speeds
     •  Vbg — best power-off glide This is the airspeed that provides minimum drag thus maximum L/D, or glide ratio, and thus the greatest still air glide range from the potential energy of height. It is much the same basic airspeed as Vbr/Vmd and Vy, though it may be a bit lower and decreases as the aircraft weight decreases from MTOW. However, like Vbr, wind direction and speed have to be taken into account before you can choose the Vbg speed when in a forced glide; for more information on the power-off glide speeds read the 'Know the best glide and minimum descent airspeeds' and 'Know the practical glide ratio and terrain footprint' sections in the 'Coping with emergencies guide'. In lower wind conditions, Vbg is increased in a headwind by around one quarter of the windspeed, but is decreased in a tailwind by a similar amount. In higher wind conditions, say above 25 knots, the speed changes required would be around one half of the windspeed.

     º  Vmp — minimum power. This is the speed that results in the lowest rate of sink in a power-off glide, and provides the longest duration of flight from the potential energy of height. The lowest rate of sink occurs at the minimum value of drag × velocity. It is probably around 80–85% of Vbg, and may be a similar speed to Vbe and Vx.

Vbg for an average sailplane with a wing loading of 32 kg/m² could be 50 knots, providing a glide ratio of 38:1, while Vmp would be 41 knots providing a sink rate of 0.6 m/s or 120 feet/minute. If you want further explanation of sink rates, etc. (with excellent diagrams) read this article on glider performance airspeeds.

(Note: the term Vmd meaning minimum descent, rather than minimum drag, is in common usage to designate the speed for lowest rate of sink in a power-off glide.)

Both Vbg distance and Vmp time are adversely affected by the extra drag of a windmilling propeller, which creates more drag than a stopped (but unfeathered) propeller following engine shut-down or failure. A windmilling propeller has a negative aoa and the 'thrust' direction is reversed, in effect adding to drag. Something similar might happen with some engine/propeller configurations, when simulating a glide at the specified Vbg/Vmp speed with the throttle closed but the engine still firing; the propeller drag might increase the rate of sink beyond that expected, and perhaps lead to the erroneous conclusion that the best glide speeds in the handbook are understated.

[ The next section in the airmanship and safety sequence is section 11.2 'Factors affecting safe take-off  performance']

2.10 The design manoeuvring flight envelope

The structural design manoeuvring flight envelope of a recreational aircraft describes its structural strength limitations and its maximum aerodynamic capabilities. The various envelopes have been described as "the parameters within which an aircraft can be safely operated, with average pilot ability, at varying density altitudes, airframe states, power outputs, wing loadings and atmospheric turbulence". 'Airframe states' refer to flap and spoiler extensions, undercarriage position, the gross weight and the fore-and-aft position of the centre of gravity. The flight envelope is only relevant for an aircraft within the required weight and balance conditions.

The boundary of the envelope shown below is formed by combinations of the applicable limiting positive and negative load factors and dynamic pressures (indicated airspeeds). It is a two-dimensional model that has indicated airspeed (i.e. CAS) along the horizontal axis and flight load factors expressed in 'g' accelerations along the vertical. The loads are those parallel to the aircraft's normal axis, i.e. perpendicular to the longitudinal axis — the aerodynamic loads in the aircraft's pitching plane. The symbol 'n' is generally used to identify such loads.

The parameters for a light aircraft usually are the limiting critical airspeeds — Vs, Va and Vne and the certificated limit load factors. There are other flight limitations which are not displayed in this flight envelope diagram, e.g. an angle of bank limitation of 60° for some aircraft categories. For weight-shift aircraft in particular, there are also pitch limitations; — e.g. 45° nose up or nose down from the horizontal. The manufacturers of high performance non-recreational aircraft would also provide other charts, altitude performance or turning performance for example.

The V-n [or V-g] diagram below is a simplified representation of a few aspects of the manoeuvring flight envelope for an LSA category aircraft at MTOW and low altitudes. An indicated (CAS) airspeed scale would normally be displayed along the horizontal axis and load factors (in units of 'g') along the vertical axis, between the certificated load limits for light sport aircraft of +4g to –2g. This diagram does not reflect any flaps-extended conditions. The positive and negative 'aircraft normal force coefficient curves'* are for the aircraft as a whole but can be assumed to approximate the accelerated stall speeds at loads of Vs1g × load factor.

(*Normal force: pilots consider the aerodynamic force acting on the aircraft in terms of lift and drag, lift being perpendicular to the relative free-stream airflow and drag being parallel with it. Aerodynamicists may also consider the total aerodynamic force in terms of a 'normal force' which is parallel to the aircraft's normal axis and an 'axial force' which is parallel to the longitudinal axis plus a 'side force' parallel to the lateral axis. The symbol for the aircraft normal force coefficient is Cna, however, the stated aircraft limit load factor (e.g. 3g or 4g) also applies to axial and side forces, not just the normal force.)

The stall speed at a 4g load limit factor would be Vs1g × 4 = Vs1g × 2, thus if the Vs1 stall speed was 45 knots then the stall speed where the positive curve intersects the 4g load limit factor line is 90 knots, i.e. that speed is the lowest possible speed at which the pilot can pull maximum g, i.e. 4g. That corner of the envelope is usually the position of Va — the design manoeuvring speed. You can see from the curve that at the Va airspeed the aircraft will stall when the wing loading exceeds 4g. Sustained flight is not possible in the white region to the left of the accelerated stall curves because the wings will be stalled.

(Note: the light blue area between +1g and –1g is the realm of reduced gravity, or microgravity. NASA and other organisations use C135 and DC-9 aircraft flying a parabolic trajectory to produce reduced or near-zero gravity conditions, for the aircraft occupants, for periods of 20–30 seconds. A light aircraft can be flown in that area for a brief period by 'unloading' the wings — 'bunting'.)

Except for transient turbulence loads the negative flight envelope below the 0g line mainly relates to aerobatic aircraft.

 V-n diagram

(Note: in section 2.8 we determined that a 60° banked level turn doubled the normal wing load. With Vs1 at 45 knots and Va 90 knots then visualise a horizontal line from the 2g point; the interception with the curve will equate with about 60 knots. So that would be the lowest possible speed for a 60° banked level turn.)

The white areas above and below the red lines represent structural loads beyond the positive and negative limit loads. Flight loads caused by control misuse and/or atmospheric turbulence that exceed +4g or –2g may cause temporary pilot incapacitation (greyout/blackout/redout) and airframe distortion. Flight loads 50% greater than +4g or –2g (i.e. +6g or –3g) will very likely cause airframe breakup.

In the aircraft design process the design maximum dive speed Vd is a calculated speed, but in the flight test stage the aircraft may be tested up to a speed where it still demonstrates no flutter, or other, problems. This is the flight-demonstrated dive speed Vdf which is lower than the design Vd but, possibly, it could be equal to it. The dynamic pressure that a pilot must not exceed is represented by the Vne airspeed limit, and that is required to be no more than 90% of Vd or Vdf. At Vne the aircraft is flying at a very small angle of attack, deriving most of the aerodynamic force from the dynamic pressure. If the pilot — or turbulence — suddenly increases the aoa the consequent increase in the lift coefficient CL (amplified by the aircraft's inertia momentarily maintaining the original flight path) could place an extreme load on the airframe, enough to break it. See 'Wind shear and turbulence'.

Some manoeuvring flight envelopes might have the top right corner cut off, from the Vno line intersection with the limit load factor line, to some less-than-maximum load factor along the Vne line.

For more information concerning the risks of flight at excessive speed read 'Don't fly real fast!' in the 'Decreasing your exposure to risk' tutorial.

Vne is the maximum airspeed, but full and rapid control applications are restricted to speeds at or below Va. Vno is the maximum structural cruise speed or 'normal operating limit' for flight in light to medium turbulence. Above the Vno/Vc speed flight should only be conducted cautiously and in smooth air.

So, the aircraft can be flown in the light green area without limits on smooth control use and it can be operated within the olive-green area in light to moderate turbulence, but it should not be operated in the yellow area except in a reasonably smooth atmosphere. If it is inadvertently operated in the area outside the certificated load limits, or at velocities greater than Vne, structural distortion then failure may result. The more the wings are loaded while the aircraft is operating in the region above Vne, the greater the possibility of structural failure. The potential exists to exceed both Vd and maximum load in the pullout from a spiral dive.

Vertical gusts impose loads on the wing structure by inducing rapid, but momentary, changes in aoa with consequent changes in the aerodynamic forces. The faster the aircraft is moving, the greater the gust-induced load. FAR Part 23 has requirements for designers to consider unexpected gust loads. The resulting gust envelope is often represented as the flight manoeuvring envelope with overlaid gust lines. Vb is developed by the aircraft designer as a recommended turbulence penetration speed in severe turbulence, with varying vertical gust components — up to 50 feet/second considered for a light aircraft at cruise speed. However, Vb is not specified for most light aircraft because, for such aircraft, there is probably not much difference between Va and Vb.

The flight envelope is considerably reduced if asymmetric manoeuvring loads are applied to the airframe. Such loads might be applied by an aircraft yawing (side force) or rolling (lateral force) while recovering (normal force) from a high-speed descent. The same applies to the use of flaps.

There are other attributes that define the envelope – resistance to spin and spin recovery, for example. Note that the term 'average pilot ability' doesn't imply that those who consider themselves 'above average' can push the envelope without losing control or stressing the airframe.

There is more information on the flight envelope in the safety brief document 'Don't fly real fast'.

[The next section in the airmanship and safety sequence is section 8.6 'The stall/spin phenomenon' ]

The next module in this Flight Theory Guide discusses altitude and altimeters, but first read the notes below.

Things that are handy to know

   •  Absolute temperature is expressed in kelvins [K], one K equals 1 °C. The base temperature is zero kelvin — equivalent to minus 273 °C — so 0 °C is equivalent to 273 K.

   •  In a free stream airflow, a favourable pressure gradient is one where static pressure decreases with distance downstream. An adverse pressure gradient is one where static pressure increases with distance downstream.

   •  ASI position error. The static vent is an opening, best placed at a position on the aircraft's fuselage, where atmospheric static pressure is not influenced by the shape of the fuselage or other aerodynamic disturbances. (Some aircraft may be fitted with a static vent on each side of the fuselage to counteract static pressure disturbances caused when the aircraft is slipping/skidding, and/or a switchable alternative static source within the cockpit.) The opening is a tube connected to the cockpit and supplies the ambient atmospheric pressure, or static pressure, to the three pressure sensing instruments — ASI, VSI and altimeter. The static vent is usually subject to some pressure disturbances at particular aircraft attitudes, as is the pitot tube, but probably to a lesser degree.

In addition if the airflow is not squarely aligned with the entry of the pitot head there will be a reduction in the indicated airspeed which increases as aoa increases. These disturbances result in position error: for a wing-mounted vent, the ASI may underread by 10 knots at stalling aoa. In a sideslip, a single fuselage-mounted static vent may be subject to dynamic pressure and ASI and VSI readings will consequently be completely misleading. Also, the instrument movements will have inbuilt errors, usually caused by excessive friction. Obstructions in the tubes — such as water or wasp's nests — will cause misreadings or no reading.

Position error corrections plus the instrument error corrections for the system should be noted in the Pilot's Operating Handbook and placarded on the instrument panel. The IAS corrected for instrument and position errors is called the calibrated airspeed [CAS]. Either CAS or IAS may be the reference speed in the Pilot's Operating Handbook for aircraft operations, but if the position error corrections are not shown then the ASI system has not been assessed for accuracy. In some poor ASI installations, IAS may be 20% less than CAS at low speeds, but they are usually much the same at normal cruising speeds.

Regulations for type-certificated aircraft require that the complete airspeed indicating system of pitot head, static vent, connecting tubes and instrument be tuned so that the IAS reading is within 3% of the true reading over the normal airspeed range from Vs to Vc. However, you should suspect that any non-certificated ultralight ASI system will be inaccurate at all speeds, and particularly so at high aoa. When comparing published stall speeds between different aircraft types, it is wise to determine CAS, as published IAS stall speeds may be downright misleading.

   •   Compressibility effects. The compressibility of air within the pitot tube has little effect on the accuracy of the ASI reading for aircraft operating below 10 000 feet and 200 knots; at an airspeed of 200 knots, compressibility will cause CAS to overread by only 0.5 knots or so. However, for aircraft operating at high speed or high altitude, compressibility will cause the ASI to overread significantly, so there is a need to correct CAS using a compressibility correction chart. The correction value is deducted from CAS to give the compressibility corrected CAS — otherwise known as equivalent airspeed [EAS].

For most medium-speed aircraft, it is probable that the compressibility correction value has been built into the IAS–CAS airspeed correction table.

There is no practical application for recreational pilots, but aerodynamicists use the EAS term — rather than IAS or CAS — assuming an ASI, that has no errors caused by mechanical, position, aoa or compressibility effects, would display the ISA standard condition sea-level true airspeed, which is equivalent to the dynamic pressure in the instrument at any altitude.

For more information see 'Notes: compressibility of airflow and Mach number'.

   •   Checking validity of claimed stall speeds. There is a simple method to check the validity of published stall speeds. Practically all very light aircraft (except those with single surface wings like the Wheeler Scout or weight-shift aircraft) use simple, long proven, standard camber aerofoils to form the wings. The lift coefficient attainable at maximum aoa with such wings without flaps is about 1.2 or 1.3 for faster-cruising aircraft, and 1.5 or 1.6 for the slower, higher-lift sections. If equipped with flaps over, say, half the trailing edge, then CLmax might be increased by 0.5 when the flaps are extended to at least 35°. When other high-lift devices (for example, full length leading edge slats/slots) are added to the wing, then CLmax might increase 0.6. Thus, a specialised short take-off and landing aircraft fitted with a high-lift aerofoil, full-length leading edge slats and large extended flaps would have a CLmax of (at least) 1.6 + 0.5 + 0.6 = 2.7.

The lift equation at normal stall speed is:

  Lift = CLmax × ½rV² × S = weight

or re-arranged:

CLmax = weight / (½rV² × S)

We can use that equation to check the validity of stall speed claims if we know the maximum take-off weight [MTOW] and the wing area [S]. Let's say a supplier claims that an aircraft, lacking any high-lift devices, has a stall speed of 30 knots. The MTOW is 450 kg and the wing area is 12 m.

In the equation, the weight must be expressed in newtons — so multiply kg × 10 = 4500 N; and the stall speed must be expressed in metres per second — so just halve the velocity in knots = 15 m/s: the air density used must be the ISA msl density = 1.225 kg/m.

Thus CLmax = 4500 / (0.5 × 1.225 × 15 × 15 × 12) = 2.7

A lift coefficient of 2.7 is very much higher than that achievable without high-lift devices, so you would conclude that the claimed stall speed is nonsense; a figure of 38 knots is probably closer to the mark.

   •  Estimating stall speeds. Conversely you can do a rough approximation of stall speeds using the following simplified formulae if you know the wing loading in kilograms per square metre or in pounds per square foot, and can estimate CLmax with flaps stowed or fully extended.

Stall speed [knots] = 7.8 × square root (wing loading in kg/m² divided by CLmax)

Stall speed [knots] = 17.2 × square root (wing loading in lb/ft² divided by CLmax)

Using our previous example of a lightly-loaded Jabiru with a mass of 340 kg (748 lb), wing area of 7.9 m² (85 ft²) thus wing loading = 43 kg/m² (8.8 lb/ft²) and estimating CLmax with flaps fully extended as 2.0 then:

estimated stall speed = 7.8 × square root (43/2) = 7.8 × 4.64 = 36 knots

or estimated stall speed = 17.2 × square root (8.8/2) = 17.2 × 2.1 = 36 knots

Stuff you don't need to know

   •  The tropopause marks the boundary between the two lower layers of the atmosphere — the troposphere, and above it, the stratosphere. The height of the tropopause varies daily, seasonally, and latitudinally — it is about 28 000 feet at the poles and perhaps 55 000 feet at the equator. The significant difference between the troposphere and the stratosphere is that air temperature decreases steadily with height in the troposphere, but initially remains constant then increases steadily with height in the stratosphere, until the stratopause at about 50 km. The stratosphere contains very little water vapour and is much more stable than the troposphere. The ozone layer is within the stratosphere.

   •  Boyle's law: at a constant temperature, the volume (V) of a given mass of gas is inversely proportional to the pressure (P) upon the gas; i.e. PV = constant.

   •  The pressure law: at a constant volume, the pressure is directly proportional to temperature (T) in kelvins.

   •  Charles' law: at a constant pressure, gases expand by about 1/273 of their volume, at 273 K, for each one kelvin rise in temperature; i.e. the volume of a given mass of gas at constant pressure is directly proportional to the absolute temperature.

   •  For one mole of gas, the preceding laws are combined in the gas equation PV = RT, where R = the gas constant = 2.87 when P is expressed in hectopascals. Ordinary gases do not behave exactly in accordance with the gas laws.

   •  The change in altitude for each one hPa change in pressure can be roughly calculated from the absolute temperature and the pressure at the level using the equation:=96T/P feet.

   •  The term 'burble' also refers to the atmospheric wake of an object. Skydivers refer to their wake as 'the burble' while the disturbed airflow and exhaust gases behind a the island structure of a fast-moving aircraft carrier was (and probably still is) known to pilots as 'the burble'.

Groundschool – Flight Theory Guide modules

| Flight theory contents | 1. Basic forces | 1a. Manoeuvring forces | [2. Airspeed & air properties] |

| 3. Altitude & altimeters | 4. Aerofoils & wings | 5. Engine & propeller performance | 6. Tailplane surfaces |

| 7. Stability | 8. Control | 9. Weight & balance | 10. Weight shift control | 11. Take-off considerations |

| 12. Circuit & landing | 13. Flight at excessive speed | 14. Safety: control loss in turns |

Supplementary documents

| Operations at non-controlled airfields | Safety during take-off & landing |

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