🧰 Quick tools & cheat sheet play with the numbers

Three small calculators for the sums pilots do most, and a one-screen list of the numbers worth knowing by heart.

📋 Cheat sheet — numbers worth knowing

📖 Overview and Definitions Chapter 1

The words, pictures and units every later chapter assumes — forces, aerofoil and wing geometry, angles, axes and load factor.

The four forces

Four forces act on every aeroplane. Lift is the aerodynamic force perpendicular to the flight path, drag is the aerodynamic force along it, weight always acts towards the centre of the Earth, and thrust is the push from the engine or propeller. Steady, level, unaccelerated flight needs both pairs to balance. If lift is more than weight the aircraft accelerates upward; if thrust exceeds drag it accelerates forward.

Words for the aerofoil

Words for the wing

Attitude, flight path and angle of attack

The three axes

An aeroplane rotates about three axes through its centre of gravity. Rotation about the longitudinal axis is roll (ailerons), about the lateral axis is pitch (elevator) and about the normal axis is yaw (rudder).

Load factor and units

Load factor n = lift ÷ weight. It is 1 in steady level flight, 2 in a 60° level turn, 0 in a push-over and −1 inverted. It is the number of “g” you feel and the number the structure feels.

🌡️ The Atmosphere Chapter 2

The air you fly in: how temperature, pressure and density change with height, and what that does to the aircraft.

The atmosphere you fly in

Pilots use the International Standard Atmosphere (ISA) as a common yardstick: 15 °C and 1013 hPa at sea level, with the temperature falling 1.98 °C for every 1,000 ft up to about 36,000 ft. Above that the air stays at −56.5 °C (the isothermal layer) up to about 65,000 ft.

The standard atmosphere at a glance

The temperature falls in a straight line with height to the tropopause, then stays constant. Pressure and density keep falling with height throughout: at FL300 pressure is 29.7 % of sea level but density is 37.4 %, because colder air is denser than the pressure ratio alone suggests.

Density altitude — how the aircraft “feels” the day

Density altitude is the altitude in the standard atmosphere that has the same density as the air you are in. A hot day raises it; a cold day lowers it.

🧪 Basic Aerodynamic Theory Chapter 3

Continuity, Bernoulli and dynamic pressure — and the chain of airspeeds from your instrument to true speed.

Continuity and Bernoulli — the two ideas behind everything

This is why the upper surface of a wing carries lower pressure: the air is squeezed over the curved top and speeds up. It is also why a pitot tube works — the air brought to rest at its nose has the full total pressure, and the difference from the static pressure is the dynamic pressure.

Dynamic pressure — the pressure of moving air

This is also how your pitot tube works. It measures total pressure minus static pressure, which equals ½ρV². A difference of 1,620 Pa at sea level means 100 kt.

The speed ladder: IAS → CAS → EAS → TAS

🌊 Subsonic Airflow Chapter 4

How air really behaves next to a wing: the thin boundary layer, separation, and the devices that keep flow attached.

Ways to keep the airflow attached

🛫 Lift Chapter 5

Where lift comes from, how it depends on angle of attack, speed, density and area, and how the planform changes it.

The lift curve — lift against angle of attack

A cambered wing makes lift even at zero angle of attack, so its zero-lift angle α₀ is negative; a symmetrical section produces none until it is tilted. Flaps, slats, wing shape and Reynolds number change the curve; weight, speed and density only change which point on it you are flying at.

Weight, wing area and speed

To fly level, lift must equal weight: W = CL × ½ρV² × S. Solving for speed shows that every speed tied to a fixed lift coefficient — stall, best glide, approach — scales with √weight.

Aspect ratio — long wings or short wings?

A low-aspect-ratio wing (short and wide) has a shallower lift-curve slope: it needs a bigger angle of attack to make the same lift, so it stalls at a higher angle of attack — and it also makes more induced drag. A high-aspect-ratio wing (long and thin, like a glider) is far more efficient.

📉 Drag Chapter 6

The two kinds of drag, how each changes with speed, and where extra drag comes from — ground effect, vortices, sideslip, trim.

Two kinds of drag that pull in opposite directions

Ground effect — flying on a cushion

Within about one wingspan of the ground the ground blocks the downwash behind the wing. The result is less induced drag, more lift at the same angle of attack, and the wing stalls at a lower geometric angle.

Other sources of drag and wake

• 🌀 Trailing vortices rotate up outboard of the wingtips and down inboard, and the pair sinks behind the aircraft.
• ↔️ Sideslip adds drag because the fuselage and fin act as lifting surfaces at an angle to the flow.
• ⚖️ CG aft (within limits) means less tail download, less wing lift needed, less induced drag — and better fuel economy.

🛑 Stalling Chapter 7

The stall is not about speed. It is about angle of attack — and every other number follows from that.

The picture that ties pitch attitude, flight path and angle of attack together is in Chapter 1.

Stall speed — what moves it

Load factor — the accelerated stall

The wing can stall at any speed if you ask it for enough lift. At a given speed the maximum load factor is n = (V / Vs)².

Configuration effects on the stall

Where does the stall start?

Tools designers use to make the root stall first

• Washout — the tip is set at a lower angle of incidence (geometric), or has less camber (aerodynamic), so it is “behind” the root. With 2° of washout on a wing whose sections stall at 16°, the tip is at 14° when the root stalls: a 2° margin.
• Stall strips — small sharp wedges on the inboard leading edge to trigger an early root stall and buffet.
• Fences, vortilons and slat/Krueger layouts — see Chapters 4 and 8.

Warning, then recovery

• 🫨 Buffet — turbulent separated flow hitting the wing and tail — is the natural warning. When there is little or none, an artificial warning (shaker, horn) is required.
• 🔔 A vane or flapper on the leading edge senses the stagnation point moving down and aft as angle of attack rises. An angle-of-attack warning works at the right margin whatever the weight, bank or load factor. A fixed-speed warning does not.
• 👈 A stick pusher is fitted to aircraft with unacceptable post-stall behaviour (pitch-up, deep stall).

1. Reduce the angle of attack — nose forward. Nothing else works until the wing is flying again.
2. Apply power (maximum at low altitude) — adding power while holding attitude keeps you stalled; unstall first.
3. Level the wings with rudder for the yaw. Do not fight a dropping wing with large aileron: the down-going aileron raises the local angle of attack on the wing that is already stalling.
4. Recover gently — a big pull-up demands more than (V/Vs)² and causes a secondary stall.

Special cases

• ⚙️ Propeller aircraft at high power have a lower stall speed, a higher nose attitude, and a stronger wing-drop tendency. A twin with one engine out and the other at high power stalls asymmetrically, with the drop towards the dead engine.
• 🛬 Ground effect: the wing stalls at a lower geometric angle; leaving it after lift-off at constant attitude raises drag and cuts lift.
• 🏔️ High altitude: a 1.3 g buffet margin means about a 40° bank before low-speed buffet.
• 🔄 Hysteresis: re-attachment occurs at a lower angle than the one that stalled the wing, so a definite reduction is needed.

Spin and spiral dive

1. Power idle
2. Ailerons neutral
3. Full rudder opposite to the rotation
4. Elevator forward to unstall, then level the wings and ease out

An aft CG gives a flatter spin, a faster rotation and a harder recovery. Fin and rudder can be blanketed by the wake of the horizontal tail, which is why tail geometry keeps part of the rudder in clean air.

🪽 High Lift Devices Chapter 8

Flaps and slats let a wing built for cruise also fly slowly enough to take off and land on a runway.

The family

The numbers

Drag, glide and the approach

• Flaps raise parasite drag and usually induced drag, so (L/D)max falls and the glide steepens. L/D 12 → 7 changes the power-off glide angle from 4.8° to 8.1°.
• On a 3° path with a 600 kN aircraft, thrust required is 18.6 kN at L/D 12 but 43.6 kN at L/D 8: more drag means engines that spool from a higher, more responsive setting.
• Rate of descent on a 3° path at 140 kt: ≈ 740 ft/min (groundspeed × 5).
• With landing flap, the same speed is reached at a lower attitude — better view, better tail clearance — and the drag curve is higher and shifted to a lower speed.

Take-off or landing — which flap?

Pitch, balloon and sink

• Flaps push the wing's centre of pressure aft (a nose-down moment from the wing) and increase the downwash on the tail (a nose-up effect). The balance depends on tail height and position.
• Extending flaps at constant attitude: lift increases at once and the aircraft balloons, until the extra drag slows it and you re-trim.
• Retracting flaps at low speed and constant attitude: lift falls and the aircraft sinks unless speed or angle of attack goes up.
• Flaps lower the stalling angle, so stall-warning thresholds must change with configuration.

Limits and failures

• 🚫 VFE protects the flap structure and mechanism: air loads grow with q. Some aircraft have a load-relief system that retracts flaps one setting if speed is exceeded.
• ⚠️ Asymmetric flaps (one side failed): a roll towards the side with less flap, needing large aileron. Most systems stop movement when they detect asymmetry.
• ⚠️ Asymmetric slats at high angle of attack: roll towards the side without the slat — its wing stalls first.
• 🚫 Slats at high cruise speed would only add drag and loads, hence a maximum extension speed.
• 🔁 In a turn, flaps increase the speed margin to the stall but reduce the angle-of-attack margin.

🧊 Airframe Contamination Chapter 9

Frost, ice, rain and insects change the shape and roughness of the wing and tail — and the numbers in this guide stop applying.

Contamination — the wing no longer behaves as designed

• ❄️ Frost, ice or insect residue on the upper leading edge triggers early transition and separation, cutting CLmax and the stalling angle — the wing can stall at normal take-off rotation speed and attitude.
• 🌧️ Heavy rain roughens the surface with a water film and droplet impacts, thickening the layer: lower CLmax, more drag.
• 🔔 A stall warning triggered at a fixed angle of attack is less protective on a contaminated wing, because the real stalling angle may now be lower than the warning threshold.

Ice and the tailplane

• 🧊 Frost on take-off: lowers CLmax and the critical angle, so the wing can stall at the normal rotation attitude and speed.
• 🌧️ Heavy rain: roughness lowers CLmax and the stall margin.
• 🔕 An iced aircraft may stall without the normal warning, because the stalling angle has fallen below the trigger angle.
• 🅣 Tailplane stall: flaps increase wing downwash, so the iced tail works at a larger negative angle and stalls. Recovery is opposite to a wing stall: retract flaps to the previous setting and pull back — pushing forward makes it worse.

Rain and ice

• 🌧️ Heavy rain roughens the surface film, reducing CLmax and raising drag, plus momentum loss from the droplets. In a microburst it adds to the energy deficit.
• 🧊 Airframe icing: lower CLmax and stall angle, more drag and weight, less control effectiveness.
• ☔ Freezing rain is especially dangerous: supercooled drops spread before freezing, forming clear ice aft of the protected areas.

Where contamination bites elsewhere

• ⚖️ Stability — ice on the tailplane lowers its lift slope and stalling angle, cutting its stabilising contribution (Chapter 10).
• 🕹️ Controls — ice or water behind a control-surface hinge, or loose cables, lowers the flutter speed (Chapter 11).
• 🪽 High-lift devices — flaps increase the tail's negative angle of attack, which is what makes an iced tailplane stall (Chapter 8).

⚖️ Stability and Control Chapter 10

Why an aeroplane returns to its trimmed condition when disturbed — in pitch, and in roll and yaw — and where the CG must sit.

🔹 Longitudinal stability — pitch

The CG, the neutral point and the static margin

The neutral point is the aerodynamic centre of the whole aircraft — the CG position where stability is exactly zero. The static margin is how far ahead of it the CG sits: a bigger margin means stiffer pitch stability.

Forward or aft?

Stick force per g must be positive and not too low, so that a strong force is needed to reach limit load. With a limit of 2.5 g and 20 N per g, reaching it from 1 g takes (2.5 − 1) × 20 = 30 N.

After a disturbance: two oscillations

The phugoid is acceptable even if lightly damped or slightly divergent, because the period is so long that the pilot can correct it easily. A dynamically stable oscillation with a time to half amplitude of 2 s halves its size every 2 s.

Configuration and special cases

• 🅣 A T-tail sits above most of the wing's downwash and wake, so it is often more efficient in normal flight — the penalty is deep stall (see Chapter 7).
• 🪶 A canard aircraft gets stability by having the wing's aerodynamic centre far enough behind the CG to outweigh the foreplane's destabilising effect.
• 🖥️ Fly-by-wire aircraft can fly with relaxed or even negative static stability: the computer provides artificial stiffness and damping, giving less trim drag and a smaller tail.
• 🧊 Ice on the tailplane reduces the tail's lift slope and stalling angle, cutting its stabilising contribution.
• 🔁 A statically unstable aircraft trimmed at one speed will diverge after a small speed change.
• 🌀 A propeller ahead of the CG at high power and high angle of attack tends to be destabilising; swept wings can pitch up at high angle of attack (see Chapter 7).

🔹 Lateral and directional stability — roll and yaw

Where the “dihedral effect” comes from

Dutch roll and spiral — a balancing act

• 🛠️ A yaw damper senses yaw rate and applies rudder to oppose it. A wash-out (high-pass) filter lets it ignore the steady yaw rate of a coordinated turn.
• ⚠️ Yaw damper failed? Fly lower and slower, where natural damping is better. For a large Dutch roll use small, well-timed aileron inputs — out-of-phase rudder reinforces the oscillation.
• Fixing Dutch roll in design: a bigger fin / more yaw damping, or less dihedral effect (hence anhedral on swept high wings).
• 🧰 Spiral stability improves with more dihedral effect or a smaller fin.

Reading the ball

• 🔁 Overbanking tendency: in a steady steep turn the outer wing's higher speed produces more lift than lateral stability can resist, so a little opposite aileron holds the bank.
• 🔀 Rolling without rudder: the initial yaw is adverse — away from the turn. Also, the horizontal part of lift tilts and the aircraft sideslips towards the low wing.
• 🛞 A steady sideslip with crossed controls needs aileron against the dihedral effect and rudder against the weathercock tendency. The maximum sustainable sideslip (e.g. crosswind limit) is usually limited by rudder or aileron authority.
• 🐾 Rudder alone can turn an aircraft with strong dihedral effect: yaw → sideslip → roll.

Other effects to know

• 🛑 At very high angles of attack the fin sits in the wake of the fuselage and wing and loses effectiveness; a dorsal fin keeps the fin's side force rising at large sideslip, and a ventral fin stays in clean air. A stalled fin can cause rudder lock or directional divergence.
• 🔊 At supersonic speeds the fin's lift slope falls with Mach number, so directional stability drops.
• 🌀 Propeller slipstream striking the fin on one side shifts directional trim with power and speed. At low speed, high power, steep climb, propeller normal force and slipstream effects can reduce directional stability.
• 🛩️ After an engine failure in a twin, holding wings level with zero bank needs rudder against yaw and a sideslip towards the dead engine; a few degrees of bank towards the live engine does the job more efficiently.
• ⚙️ In fly-by-wire aircraft, turn coordination commonly adds rudder automatically to cancel sideslip during rolling.
• 🎯 Roll subsidence is a rapid, non-oscillatory convergence of roll rate after an aileron input: strong roll damping gives a short time constant.

🕹️ Controls Chapter 11

Surfaces, tabs, trim and spoilers: how the pilot moves the aircraft and removes the force from the stick.

🔹 Control surfaces

Adverse yaw — the nose that goes the wrong way

The rising wing makes more lift and more induced drag, and the down-going aileron adds profile drag, so the nose yaws away from the turn. It is greatest at low speed and high CL (the induced-drag difference is proportional to CL).

Aileron reversal at high speed

On a flexible wing, the aileron's air load twists the wing and changes its angle of attack. Above the reversal speed the lift lost by twisting exceeds the lift gained, and the roll goes the wrong way. Large transports lock out the outboard ailerons at high speed and use inboard ailerons and spoilers, which act where the wing is stiffer.

Tabs: help, feel and trim

Hinge moments, flutter and mass balance

• A free (released) elevator floats with the local flow, which reduces tail effectiveness and stability.
• At very high q on manual reversible controls, hinge moments can exceed what the pilot can supply: control blow-back limits deflection.
• Flutter is a self-excited oscillation: aerodynamic forces couple with structural bending and torsion until the amplitude grows to failure.
• Mass balance places the surface's centre of gravity on or ahead of the hinge line, removing the inertial coupling that causes flutter.
• Ice or water behind the hinge, or loose cables, lowers the flutter speed.

Pitch control and combined surfaces

• Moving the elevator's trailing edge up increases tail download and pitches the nose up.
• The forward CG limit is often set by elevator authority to flare or rotate at low speed, in ground effect.
• An all-moving tailplane (stabilator) gives more pitch moment per degree than a fixed tail plus elevator.
• Elevons = elevator + aileron (tailless aircraft). Ruddervators on a V-tail: symmetric = pitch, differential = yaw. Flaperons = flap + aileron. Tailerons roll the aircraft using differential tailplane halves — useful where wing ailerons lose effect.

Powered controls

With fully powered, irreversible controls the pilot no longer feels hinge moments, so an artificial feel system is needed. A q-feel system makes stick force increase with dynamic pressure for a given deflection, restoring natural cues and preventing overstress. Control harmony means the forces and responses in pitch, roll and yaw are balanced so that coordinated inputs feel natural.

🔹 Trim, speed brakes and spoilers

Two ways to trim

• Nose-up trim on a stabiliser means its leading edge moves down: more negative incidence, more download.
• Take-off stabiliser trim is set by CG position: a forward CG needs more nose-up trim for a normal rotation.
• ⚠️ A runaway stabiliser is dangerous because the stabiliser is more powerful than the elevator and the forces it creates can exceed what the elevator can oppose.
• On a servo-tab elevator, trimming is done by offsetting the servo tab's neutral position.

Stick force and speed

A stable aircraft trimmed at 200 kt wants to pitch down if you slow to 180 kt; to hold the lower speed you must pull. Faster than trim: push. Release the controls and it flies back toward the trim speed (with a phugoid oscillation). Re-trimming moves the zero-force speed; it does not change the stability gradient.

What forces a trim change?

Speed brakes and spoilers

• 🛑 Lift dumpers (ground spoilers) destroy lift on touchdown so weight transfers to the wheels: braking force = μ × (W − L).
• 🪟 Roll spoilers are raised on the down-going wing.
• 🚫 Symmetrical spoilers in flight give more drag, less lift at the same angle of attack, and a pitch change that may need trim.
• Fuselage or tail-cone brakes add drag near the CG line with little effect on lift or roll.
• Fuel imbalance is a rolling moment you can calculate: 1,000 kg at 6 m from the centreline = 58.9 kN·m. Aileron trim or fuel transfer removes it.

🧭 Flight Mechanics Chapter 12

Performance in the real world: climb, glide, range, turns, flying with one engine out, and the effect of wind.

🔹 Climb and glide

Climbing and gliding

🔹 Speeds for range, endurance and climb

Which speed for which job?

The back of the drag curve

Weight, wind and range

• Speeds tied to a fixed CL scale with √weight. As fuel burns, the best-range speed falls — or the aircraft climbs slowly (a “cruise-climb”) to keep CL constant.
• Headwind: fly faster than still-air best range for maximum ground range, and faster than still-air best glide speed if gliding into the wind. In a tailwind, go slower.
• Specific range: 480 kt TAS burning 2,400 kg/h gives 0.200 nm/kg of air range; into a 50 kt headwind the ground range is (480 − 50)/2,400 = 0.179 nm/kg.
• Absolute ceiling: where the power available only just equals the power required — no excess power, zero rate of climb, a single possible speed.

🔹 Turning

Turning flight

🔹 Flying with one engine out

Engine failure on a twin — rudder and Vmc

After an engine failure the live engine's thrust × its distance from the centreline must be balanced by the fin and rudder. Vmc is the lowest speed at which full rudder can still do that.

• The critical engine on a twin whose propellers both turn clockwise (seen from behind) is the left: the right engine's descending blade acts further from the centreline.
• A rudder ratio changer or limiter reduces rudder travel at high speed to protect the fin.
• Rudder trim removes the sustained pedal force after engine failure; it adds no authority beyond full rudder.
• Rudder in level flight also rolls the aircraft in the direction of yaw (sideslip → dihedral effect).
• Crosswind (wing-down) landing: rudder aligns the nose with the runway; into-wind aileron holds off the drift.

🔹 Wind and performance

Wind arithmetic for every flight

Wind changes the ground distance but not the climb gradient through the air, which depends on (T − D)/W; relative to the ground a headwind steepens the path and a tailwind flattens it — important for obstacle clearance. Hot and high aerodromes degrade performance because low density cuts thrust/power and raises TAS for a given IAS.

🚀 High Speed Flight Chapter 13

Mach number, critical Mach, shock waves, Mach tuck and buffet — and why a Mach trim and a Mach limit exist.

🔹 Mach number and compressibility

Critical Mach number and how to raise it

Mcrit is the free-stream Mach number at which the local flow first reaches M 1 somewhere on the wing. Above it, shocks form; drag stays modest until drag divergence, where the shock strengthens, moves aft and triggers separation.

Climbing, cruising and the crossover altitude

• 📈 Constant CAS climb: Mach increases. Constant Mach climb or cruise: IAS/CAS falls. Constant Mach descent: CAS increases, and TAS increases once in the troposphere.
• 🔀 The crossover altitude for 300 kt CAS / M 0.80 is about 30,600 ft, where TAS ≈ 470 kt. Above it the Mach limit (MMO) governs; below it the CAS limit (VMO) does. Flying constant CAS above crossover would eventually exceed MMO.
• 🛡️ VMO protects against structural loads and flutter; MMO against compressibility effects such as shock-induced buffet, tuck and drag rise. Pushing from M 0.75 to 0.82 on a transport wing raises drag disproportionately.
• 🧮 450 kt TAS at FL400 is M 0.78 (a = 573.6 kt). Doubling TAS at constant temperature doubles the Mach number.
• 📉 At FL350 and high speed, EAS is a little lower than CAS, because the compressibility correction is subtracted.

The coffin-corner squeeze

As you climb, the lift coefficient you need rises and the buffet Mach number at which the wing buffets falls, while the high-speed limit stays roughly constant or falls. The usable speed band narrows until the low-speed buffet and high-speed buffet limits meet: the coffin corner.

Instruments and temperature

• 🔢 The Mach meter works from total and static pressure only — no temperature input needed.
• 🌡️ At constant altitude and TAS in warmer air, the Mach meter reads lower. At constant pressure altitude and constant CAS, a temperature change leaves Mach unchanged.
• 🔥 Kinetic heating of the skin is a design constraint mainly at supersonic speeds: at M 2 and 217 K, T_t ≈ 391 K.
• Below M 0.3 compressibility is neglected because stagnation density exceeds free-stream density by under 5 %.
• 🧰 Buffet-onset charts tell you the maximum altitude or bank for a given weight and Mach while keeping a load-factor margin.

🔹 Shock waves, tuck and buffet

What you feel in the cockpit

• 🔝 On a swept wing, tip-first separation moves lift forward → pitch-up. Root-first separation removes forward inboard lift and reduces tail downwash → nose-down.
• 🛠️ Mach trim counters tuck by applying nose-up stabiliser or elevator trim as a function of Mach number, restoring a positive stick-force gradient.
• 🔊 In supersonic flight the aerodynamic centre moves aft (about 25 % → 50 % chord): stability and the trim download needed both increase. A sonic boom is the continuous N-wave pressure signature sweeping over the ground, not a one-off event at M 1.

Controls near MMO

• The elevator can be less effective near MMO, because shocks on the tailplane reduce its influence — hence trimmable stabilisers and Mach trim carry the trim load.
• An all-moving tailplane stays effective, because changing the whole surface's incidence changes its pressure distribution regardless of shock position.
• Aileron effectiveness may fall through shock-induced separation and wing twist, so spoilers are preferred for roll at high speed — they spoil lift without twisting the wing.
• A certification margin between MMO and the Mach at which buffet occurs at 1 g protects the normal envelope.

Mach trim

At high Mach numbers the wing's centre of pressure moves aft and tail downwash changes, so the aircraft tends to tuck under. Mach trim applies nose-up trim as Mach rises. Without it, the aircraft develops a nose-down tendency and you pull more as speed increases; limitations usually cut the permitted Mach number when it is inoperative.

The buffet margin — a numbers game

• ⬆️ Higher altitude at constant weight and Mach reduces the margin: lower pressure forces a higher CL. The buffet-limited maximum altitude falls as weight rises.
• ⚖️ An aft CG slightly improves the envelope: less tail download means less wing lift to carry.
• 📉 The high-speed buffet boundary falls to lower CL as Mach increases beyond drag divergence.

🏗️ Limitations Chapter 14

How hard the structure can be pushed, which speeds protect it, and the weather that takes energy away: turbulence, windshear and wake.

🔹 Loads and speed limits

• Loads add: a 60° turn (n = 2) plus a vertical gust Δn = 0.5 is 2.5 g.
• Limit load is the maximum expected in service, carried without permanent deformation. Ultimate load must be carried without failure for about 3 seconds.
• Exceeding limit but staying below ultimate may cause permanent deformation, needs inspection, and should not cause failure. The margin covers uncertainty — it is not for routine use.
• Positive and negative limits both exist (typical transport, clean: +2.5 g, −1.0 g). With flaps extended the positive limit is lower (about +2.0 g) because flaps are sized only for low-speed conditions.
• 0 g gives zero lift and a ballistic arc; fuel and oil systems may be affected. −1 g means the net aerodynamic force equals weight but acts the other way relative to the air.

The V–n diagram

The speed limits

Turbulence

• The turbulence penetration speed balances stall margin (gusts raise α by about gust/V, so too slow can stall) against gust loads (which rise with EAS, so too fast overloads). It is above the best-angle-of-climb speed but below VMO.
• VB is the design speed for maximum gust intensity: a strong gust at VB produces the stall before exceeding the structural limit.
• In severe turbulence: hold attitude and wings level at the penetration speed, accepting altitude and airspeed excursions. Chasing altitude or speed adds manoeuvre loads to gust loads.
• A heavier aircraft sees smaller load-factor increments in the same gust (Δn ∝ 1/wing loading), but the absolute load is similar.

Mass limits and structure

• 🔁 Fatigue comes from repeated load cycles. A wing's life is used fastest by large cycles — low-altitude turbulent flying and many short flights. Each pressurisation is a cycle too, so cabin differential pressure is limited to keep fuselage stress (∝ Δp) within fatigue and strength limits.
• 🪢 Crosswind landing loads on the gear come from side loads if you touch down with drift (hence decrab or wing-down).

🔹 Windshear, wake and weather hazards

The microburst

• A microburst is an intense, localised downdraft (horizontal extent under about 4 km) spreading out on reaching the ground; wind changes across it can exceed 40 kt. A macroburst is the same thing on a larger scale (over about 4 km) and longer duration.
• Outflow winds are strongest close to the ground, just outside the downdraft core, a few minutes after first contact.
• ✈️ Heavy jets are especially vulnerable: large inertia (groundspeed adapts slowly) and engine spool-up time from approach idle delay recovery.

Windshear escape

1. Maximum thrust. It supplies the energy the shear is taking away.
2. Pitch up towards the recommended attitude, accepting flight at the edge of the usable angle of attack — an intermittent stick-shaker is accepted because height matters more than speed margin near the ground. Reduce pitch only enough to stop it.
3. Do not retract flaps or gear. Flaps up reduces CLmax and gear transit adds drag. Change configuration only when clear of the shear with a positive climb.

• 🛰️ Predictive windshear radar detects Doppler velocity divergence in precipitation ahead; reactive systems use the aircraft's own energy state.
• 🌩️ Shear accompanies thunderstorms and gust fronts (sometimes tens of kilometres from the cell), fronts, low-level jets, temperature inversions (calm air beneath strong wind aloft), sea-breeze fronts, and terrain. A warm front's sloping surface can give shear on climb or descent through it.
• 📉 On approach, a headwind that decreases toward the surface causes a progressive IAS loss requiring extra thrust. A headwind that increases with height after take-off briefly boosts IAS and climb.
• 🧗 Surface wind can differ from the wind at 500 ft because of friction, inversions and terrain: in the friction layer, wind speed increases with height.
• 🗻 Mountain waves and rotors: strong vertical currents, severe turbulence downwind of ridges (possibly in clear air); clear air turbulence is most often near jet streams where shear is strong.
• 🚦 A gusty-day speed additive on approach maintains a margin above the stall when gusts momentarily reduce IAS.
• 🧗 An updraft at constant attitude raises angle of attack and lift (and load factor in turbulence); a downdraft does the opposite.

Wake turbulence

🌀 Propellers Chapter 15

A propeller is a rotating wing. Its blade angle, the airspeed and the rpm decide how much thrust it makes — and how the aircraft yaws.

Fixed pitch versus constant speed

Feather, windmill and reverse

Efficiency, advance ratio and thrust

• A fixed-pitch propeller's efficiency peaks at one advance ratio and falls on either side.
• Static efficiency is zero (no speed = no thrust power), even though static thrust is high. Typical best efficiency is 0.80–0.88.
• Power available from a propeller aircraft is roughly constant with speed (η × P), so thrust available falls with speed — unlike a jet.
• That is why the best rate of climb is near the minimum-power speed, and why a prop aircraft cruising at the minimum-drag speed gets its maximum range.
• 🔧 Reduction gearboxes on turbine engines keep propeller rpm — and tip Mach number — acceptable. Propeller efficiency falls at high cruise speeds because the tip Mach number nears sonic, limiting conventional propellers to about M 0.6–0.7.
• A normally aspirated piston engine's power falls roughly in proportion to air density. A bigger diameter at constant power improves ideal efficiency but raises tip speed and needs ground clearance.

The four propeller effects (single engine, clockwise from behind)

• 🔁 Counter-rotating propellers on a twin (opposite hands) cancel net torque and P-factor effects and remove the critical-engine asymmetry. Contra-rotating (coaxial) propellers recover slipstream swirl energy.
• 💨 Slipstream over the wing root and tail increases local dynamic pressure: power-on stall speeds are lower and elevator/rudder authority improves at low speed. A pusher works in the disturbed wake and has less effect on the tail.
• For a constant-speed propeller in climb (low V, high power), the blade angle is finer than in cruise.