Mathematics and Physics of Flight for Private Pilots

This article brings together the main mathematical and physical relationships a private pilot should understand. The goal is not to replace the POH/AFM, but to understand what is behind the performance charts, aircraft limitations, climb speeds, descent planning, turn performance, stall speed, fuel planning, and weight and balance calculations. The formulas below are tools for understanding, estimating, cross-checking and building pilot judgement.

1 kt = 1 NM/h ≈1.852 km/h ≈ 0.514 m/s

1 m ≈ 3.28 ft

The International Standard Atmosphere (ISA) is the reference atmosphere used for aircraft performance, altimetry and flight planning.

Pressure = 1013.25 hPa / 29.92 inHg

Temperature = 15°C / 288.15 K

Density = 1.225 kg/m³

Temperature decreases approximately by 2°C per 1000 ft.

ISA temperature: ISA Temp = 15 − 2 × altitude /1000

ISA deviation: ΔISA = OAT − ISA Temp

Pressure altitude can be estimated from aerodrome elevation by correcting for the difference between standard pressure and the actual QNH. Use the first formula when QNH is given in hPa, and the second one when QNH is given in inHg.

High temperature, low pressure and high humidity increase density altitude. High density altitude reduces engine power, propeller efficiency, lift production and climb performance.

For more definitions, examples and practical explanations, refer to our articles “Take-off, Climb and Landing Performance” and “Understanding Altimetry: A Practical Guide for Private Pilots.”

Good to know! The atmosphere is governed by the gas law:

Where

• • p is pressure;
  • ρ (rho) is air density;
  • R is the gas constant;
  • and T is temperature in Kelvin.

The gas law shows that density depends on:

• • Pressure;
  • Temperature.

(The lift formula will be explained later in the article.)

Lower density means the airplane behaves as if it were at a higher altitude.

Therefore, on a hot day or at high altitude:

• • Takeoff roll increases;
  • Climb performance decreases;
  • True airspeed required for the same lift increases.

That is why at constant IAS, TAS increases with altitude (We will discuss this later in this article). The instrument does not measure true speed. It measures pressure, and pressure depends on density.

A normally aspirated engine depends on air density.

Lower density means:

• • Less oxygen per intake stroke;
  • Less fuel burned;
  • Less power produced.

Climb performance decreases because excess power decreases.

Speed directly read on the airspeed indicator.

This is the most important speed for flying the aircraft because it reflects the aerodynamic forces acting on the aircraft structure and wings.

IAS is used for:

• • Stall speeds;
  • Flap operating speeds;
  • Manoeuvring speed;
  • Rotation speed;
  • Approach speed;
  • Aircraft limitations;
  • Flying the aircraft in general.

Here is an extract from a Robin Aircraft Flight Manual (AFM/POH). The speeds shown on the airspeed indicator are IAS (Indicated Airspeeds).

IAS corrected for position and instrument errors. At low speed in light aircraft, CAS is usually very close to IAS.

CAS corrected for compressibility effects. Mostly used in high-speed or high-altitude flight.

Rarely used in everyday PPL operations.

Actual speed of the aircraft through the air mass. TAS increases with altitude because air density decreases.

Calculating TAS

Quick estimation rule:

• • +1% TAS per 600 ft of altitude
  • +1% TAS per 5°C above ISA

For the same IAS, the aircraft must move faster through the air at higher altitude to produce the same dynamic pressure on the pitot tube.

TAS is mainly used for:

• • Navigation;
  • Flight planning;
  • Performance calculations;
  • Turn radius calculations;
  • Wind correction calculations.

Actual speed of the aircraft relative to the ground.

GS depends on wind:

GS is mainly used for:

• • Time calculations;
  • ETA calculations;
  • Navigation;
  • Fuel planning.

At high altitude, airliners fly at a high true airspeed and relatively close to the speed of sound.

As altitude increases air density decreases. For the same IAS, TAS increases.

Temperature generally decreases in the troposphere, so the local speed of sound decreases.

This means that an aircraft can have a moderate IAS, but still be flying at a high Mach number.

At low altitude, pilots mainly monitor IAS because aircraft handling and structural loads are linked to dynamic pressure. At high altitude, Mach number becomes increasingly important because compressibility effects may appear before reaching normal IAS limitations.

Local Speed of Sound (LSS) in knots

Mach Number

If the aircraft flies too close to the speed of sound:

• • Shock waves may appear;
  • Buffeting may occur;

Aircraft controllability may be affected.

For this reason, airliners usually climb using IAS at lower levels, then switch to Mach number at higher altitude.

For this reason, airliners usually climb using IAS at lower levels, then switch to Mach number at higher altitude.

The speed factor is directly related to the aircraft ground speed (GS). It allows pilots to quickly estimate the flight time for a given distance.

And :

With:

• • Weight = force due to gravity (N);
  • Mass = quantity of matter (kg);
  • g = gravitational acceleration (≈ 9.81 m/s²).

In straight-and-level flight, the aircraft is in equilibrium:

• • Lift acts upward
  • Weight acts downward

When both forces are equal, the aircraft maintains a constant altitude.

Thrust = Drag

• • Thrust moves the aircraft forward
  • Drag opposes the motion

When both forces are equal, the aircraft maintains a constant airspeed.

This situation corresponds to:

• • straight flight;
  • constant altitude;
  • constant speed;

no acceleration.

Total drag is the sum of parasite drag and induced drag:

Drag created by the aircraft moving through the air. It mainly comes from:

• • skin friction;
  • aircraft shape;
  • landing gear;
  • antennas;
  • airflow separation.

Parasite drag increases with speed because the aircraft collides with more air molecules and aerodynamic resistance becomes stronger.

Drag created as a consequence of lift generation. When the wing produces lift, wingtip vortices and downwash appear, creating induced drag.

Induced drag is highest at low speed because the wing must operate at a higher angle of attack to generate enough lift. As speed increases, less angle of attack is required, so induced drag decreases.

At maximum L/D, the aircraft reaches its best aerodynamic efficiency: lift is high while drag is minimum.

See our article “Understanding VX and VY Climb Speeds” to better understand how drag affects climb angle and rate of climb.

A stall occurs when the critical angle of attack is exceeded. It is not directly caused by “low speed”.

A stall occurs when the critical angle of attack is exceeded. It is not directly caused by “low speed”.

You do not need to memorise this formula; it simply proves that stall speed depends on weight, air density, wing area, and maximum lift coefficient:

With:

• • W = weight;
  • ρ = air density;
  • S = wing area;
  • C_Lmax = maximum lift coefficient.

This formula shows that stall speed increases when:

• • weight increases,
  • air density decreases (true stall speed),
  • C_Lmax

A heavier aircraft requires more lift. To produce this lift, the wing must fly at a higher angle of attack. As a result, stall speed increases with weight.

For most aeroclub aircraft, the manufacturer directly provides a published stall speed, usually determined at maximum mass and under standard atmospheric conditions.

Ice, frost, or contamination disturb the airflow around the wing and reduce C_Lmax.

This causes:

• • earlier airflow separation;
  • higher drag;
  • higher stall speed.

This is why regulations require a clean aircraft for take-off and why aircraft not approved for icing conditions must avoid icing conditions.

Load factor (n) represents the ratio between lift and weight.

In straight-and-level flight:

In a coordinated level turn:

With:

• • φ= bank angle.

As bank angle increases:

• • load factor increases;
  • required lift increases;
  • angle of attack increase;
  • stall speed increases.

New stall speed under load factor:

A vertical gust instantly changes the local angle of attack. This changes lift and therefore load factor. Strong gusts can momentarily increase angle of attack beyond the critical value and trigger an accelerated stall.

Flying faster increases structural load during gusts because lift varies with speed squared.

In general:

higher speed → higher gust loads,

lower speed → lower gust loads.

For a given indicated airspeed, true airspeed increases when air density decreases. This means that altitude and temperature both matter. Higher altitude and temperature above ISA increase TAS, and this has a direct effect on turn radius.

Turn radius is proportional to the square of true airspeed. Therefore, even a moderate TAS increase can produce a significant increase in turn radius.

Calculating TAS

Quick estimation rule:

• +1% TAS per 600 ft of altitude
• +1% TAS per 5°C above ISA

Calculating Turn Radius

Turn radius depends on true airspeed and bank angle:

Where:

• • R = turn radius
  • V = true airspeed
  • g = gravity = 9.81 m/s²
  • φ = bank angle

For consistency, we use SI units. Since g is expressed in m/s², speed must be converted into m/s.

Rate of turn tells you how fast your heading changes.

The rate of turn is the speed at which the aircraft changes heading. It is usually expressed in degrees per second.

Rate of turn depends on:

• • Bank angle
  • Airspeed.

With:

Rate of Turn in degrees per second;

Bank angle in degrees;

TAS in knots.

For the same bank angle, a faster aircraft turns more slowly in terms of heading change per second.

A standard rate turn is 3° per second. This means 360° = 2 minutes or 180° = 1 minute.

Quick formula for standard rate bank angle:

Turn radius tells you how much space you need.

To calculate the time required for a heading change:

In physic we talk about WORK and POWER. WORK represents the amount of energy transferred by a FORCE moving an object over a DISTANCE. POWER is the rate of doing a WORK. It represents how quickly energy is transferred or used.

This concept is important to understand where VX and VY come from. For a more detailed explanation, refer to our article: Understanding VX and VY Climb Speeds.

Kinetic energy:

Potential energy:

Total mechanical energy:

Speed and altitude are both forms of energy. A pilot is constantly managing energy.

During an approach, if you are too high but still slow, you may still have options: increase the rate of descent, allow the speed to increase if appropriate, and regain a stable approach. In this case, part of the aircraft’s potential energy (altitude) is converted into kinetic energy (speed).

However, if you are both too high and too fast, the aircraft has excess energy. You must then use the techniques taught by your instructor to regain the correct profile, such as extending the flight path, using drag, configuring earlier, or increasing descent while remaining within limitations.

Climb rate depends on excess power. More excess power means better rate of climb. For propeller aircraft, V_Y is the speed giving the maximum excess power and therefore the maximum rate of climb.

In a climb, the flight path is inclined by an angle γ. Vertical speed: ROC = TAS × sinγ Horizontal speed: Horizontal speed = TAS × cosγ For small climb angles: sinγ ≈ tanγ cosγ ≈ 1 Therefore: tanγ ≈ ROC / TAS This relationship explains why climb angle and climb rate are different.

In a climb, all forces maintain the same direction relative to the aircraft, except for weight, which is always directed vertically downward toward the center of the Earth.

During a climb, weight (W) does not only act downward; it also has a component acting backward along the flight path. This component opposes the motion and must be overcome by thrust. This component is often referred to as the rearward component of weight. It acts in the same direction as drag, but it is not an aerodynamic force.

This component is equal to:

To maintain a constant speed in a climb, thrust must balance both drag and this additional component of weight: 

Thrust = Drag + Weight x sin Θ

Rearranging:

This shows that the climb angle depends directly on the difference between thrust and drag.

Climb gradient is often misunderstood because pilots naturally look at rate of climb first. However, for obstacle clearance, terrain, or departure procedures, what really matters is not only how fast the aircraft climbs vertically, but how much altitude it gains over a given horizontal distance.

Extract from the French AIP: AD 2 LFLB SID RWY 18 RNAV INSTR 03.

For training and illustration purposes only. Do not use for flight. Always refer to the official AIP website for the latest valid version.

Climb gradient measures the altitude gained compared with the horizontal distance travelled.

It can be expressed as:

• • a percentage,
  • an angle,
  • or a value in ft/NM.

Climb gradient tells you how many feet the aircraft gains per nautical mile travelled.

Rate of climb tells you how many feet per minute the aircraft gains.

An aircraft may have a good rate of climb but a poor climb gradient if its groundspeed is high.

Do not confuse air gradient and ground gradient.

Air gradient is the climb gradient relative to the air mass. It uses TAS because TAS is the aircraft’s speed through the air.

Ground gradient is the climb gradient relative to the ground. It uses GS because GS is the aircraft’s speed over the ground.

This difference matters because wind changes groundspeed.

With a headwind:

• • GS decreases;
  • The aircraft covers less ground for the same rate of climb;
  • Ground climb gradient improves.

With a tailwind:

• • GS increases;
  • The aircraft covers more ground for the same rate of climb;
  • Ground climb gradient decreases.

This is why a headwind helps obstacle clearance, while a tailwind can seriously degrade it.

A gradient can be expressed as a percentage.

For a flight path angle γ:

One nautical mile is approximately 6076 ft. So:

So:

This is useful for quickly understanding departure or obstacle requirements. This is an example of a required climb gradient on an IFR departure. At 150 kt ground speed, maintaining a minimum climb gradient of 7.6% requires at least 1,154 ft/min.

The practical formula for climb gradient in ft/NM is:

With:

• • ROC in ft/min;
  • GS in kt;
  • Gradient in ft/NM.

Why 60? Because 1 kt = 1 NM per hour, and there are 60 minutes in one hour.

So the aircraft gains 300 ft for each nautical mile travelled over the ground.

V_X is the best angle of climb speed. It gives the best climb gradient because it maximises altitude gained over horizontal distance.

This is why V_X is used when obstacle clearance is critical, while V_Y is used to reach altitude in minimum time once obstacles are no longer a factor.

For a deeper explanation, refer to our article: Understanding V_X and V_Y Climb Speeds.

Descent gradient is the opposite concept. Instead of measuring altitude gained over distance, it measures altitude lost over distance.

As with climb gradient, it can be expressed as:

• • a percentage;
  • an angle;
  • or ft/NM.

Rate of descent tells you how many feet per minute the aircraft loses.

Descent gradient tells you how many feet the aircraft loses per nautical mile.

Again, groundspeed is important. At a higher groundspeed, the aircraft covers more distance in the same time. Therefore, for the same rate of descent, the descent gradient becomes shallower.

A standard approach path is approximately 3°. This makes descent planning easier: it helps you estimate the top of descent, monitor your descent profile, and avoid arriving too high or too low near the aerodrome.

A 3° descent path corresponds to about: 318 ft/NM

For mental calculations, pilots often use: 300 ft/NM

This means that on a 3° path, the aircraft should lose about 300 ft every nautical mile.

For mental calculation:

The required rate of descent depends on groundspeed.

For a 3° path: ROD ≈ GS × 5.3

Mental rule: ROD ≈ GS × 5

This is why faster aircraft need a higher rate of descent to stay on the same descent path.

Before calculating wind correction, drift, or headings, you must clearly understand the difference between track and heading. In still air, the aircraft moves in the same direction as its nose is pointing. In that case, heading and track are the same. With wind, this is no longer true. The aircraft may be pointing in one direction, but the air mass is moving at the same time. As a result, the aircraft follows a different path over the ground.

This is why pilots must understand:

• • Heading: where the aircraft nose is pointing;
  • Track: the actual path followed over the ground;
  • Drift: the angular difference between heading and track;
  • Wind correction angle: the correction applied into wind to maintain the desired track;
  • Groundspeed: the speed of the aircraft over the ground.

• • In still air: Heading = Track
  • With wind Heading ≠ Track

The difference between heading and track is caused by drift.

Directions can be referenced to different north references.

True direction = Direction referenced to True North.

Magnetic direction = Direction referenced to Magnetic North.

Compass direction = Direction shown on the magnetic compass, affected by compass deviation.

You may therefore find:

True Track (TT) = The planned route over the ground, referenced to True North.

Magnetic Track (MT) = The planned route over the ground, referenced to Magnetic North.

True Heading (TH) = The aircraft heading referenced to True North.

Magnetic Heading (MH) = The aircraft heading referenced to Magnetic North.

Compass Heading (CH) = The heading to steer on the compass.

Variation = The angle between True North and Magnetic North.

Deviation = The error caused by the aircraft’s magnetic environment affecting the compass.

Simple memory rule:

East is least, West is best is often used in some training environments, but the safest method is to use your chart, compass correction card, and the sign convention taught by your instructor.

The wind triangle is a vector diagram showing how airspeed, wind and groundspeed combine.

• • TAS represents the aircraft speed through the air mass;
  • Wind represents the movement of the air mass over the ground;
  • GS represents the aircraft movement over the ground.

 In simple terms:

The angle to the wind is the angle between the aircraft track and the wind direction.

If the wind is directly ahead:

• • angle to wind = 0°;
  • headwind is maximum;
  • crosswind is zero.

If the wind is directly from the side:

• • angle to wind = 90°;
  • crosswind is maximum;
  • headwind/tailwind is zero.

If the wind is directly behind:

• • angle to wind = 180°;
  • tailwind is maximum;
  • crosswind is zero.

Groundspeed is the speed of the aircraft over the ground.

As seen before:

Drift angle is the angle between the aircraft heading and its track over the ground.

• • If the wind comes from the right, the aircraft drifts left;
  • If the wind comes from the left, the aircraft drifts right.

To maintain the desired track, the pilot applies a Wind Correction Angle (WCA) into wind.

The planned route on the chart is a track. The heading to fly is calculated after applying wind correction.

With crosswind:

• The aircraft drifts away from the planned track;
• The pilot must point the aircraft into wind.

This correction is called the Wind Correction Angle (WCA).

So:

Heading = Track ± Wind Correction Angle

Maximum drift occurs when the wind is exactly 90° to the track.

A practical mental method is:

If the wind is not a full crosswind, first estimate the crosswind component.

Then:

Planned track: 090°

TAS: 120 kt

Wind: 060° / 20 kt

Distance: 60 NM

The wind is 30° off the nose from the left.

Headwind component:

Crosswind component:

Groundspeed:

• • GS = 120 − 17 = 103 kt
  • Speed Factor: 60 / 120 = 0.5
  • Drift: 10 × 0.5 = 5°

The wind comes from the left, so correct left.

Heading to fly: 090° − 5° = 085°

Time: Time = 60 / 103 × 60 ≈ 35 minutes

If no drift correction is applied, the aircraft may move away from the planned route.

A simple rule based on the 1-in-60 rule is:

This chart is used to quickly find the wind components for a runway or a navigation track.

How to use the chart

1. Find the angle between the wind direction and the runway direction or planned track.

 Example:

• • Runway 13 → 130°
  • Wind: 190° / 30 kt

Angle between runway and wind: 190° − 130° = 60°

2. Follow the corresponding angle line on the chart.

Crosswind component ≈ 26 kt

Headwind component ≈ 15 kt

3.Move along this line until you reach the wind speed arc (30 kt).

4.Read the crosswind component on the bottom scale (26 kt).

5.Read the headwind component on the left scale (15 kt).

This means the aircraft has approximately 26 kt of crosswind and 15 kt of headwind.

A moment represents the turning effect created by a weight placed at a certain distance from a reference point.

With:

• • Weight = mass or weight of the item;
  • Arm = distance from the reference datum;
  • Moment = weight effect around the datum.

The aircraft centre of gravity is found by comparing the total moments with the total weight.

When adding a mass to the aircraft, the new CG can be calculated with:

Where:

• • W = original aircraft weight;
  • x = original CG arm;
  • w = added weight;
  • a = arm of the added weight.

If a mass is already inside the aircraft and is moved from one position to another, the total aircraft weight does not change, but the CG moves.

Where:

• • ΔCG = CG displacement;
  • w = moved weight;
  • d = distance the weight is moved;
  • Wtotal = total aircraft weight.

If the mass is moved forward, the CG moves forward, if the mass is moved aft, the CG moves aft.

Fuel used depends on fuel flow and time.

Example:

• • Fuel flow: 30 L/h
  • Flight time: 2 h

Endurance is the time the aircraft can fly with the fuel available.

Example:

• • Fuel on board: 120 L
  • Fuel flow: 30 L/h

This is theoretical endurance. In real flight, required reserves must always be kept. For detailed fuel planning rules, refer to our dedicated article: Minimum Fuel Requirements in NCO.

Range is the distance the aircraft can fly.

Example:

• • Groundspeed: 100 kt
  • Endurance: 4 h

Fuel may be expressed as volume, such as litres or US gallons, or mass, such as kilograms or pounds.

To convert volume into mass:

To convert mass into volume:

Avgas 100LL:

• • approximately 0.72 kg/L
  • approximately 6.0 lb/US gal

Jet A-1:

• • approximately 0.80 kg/L
  • approximately 6.7 lb/US gal

These are practical values. Always use the value required by the POH/AFM, operator procedure, or local fuel documentation when accuracy is required.