Aircraft Performance

Range, endurance, climb and takeoff: the standard atmosphere and density altitude, the drag polar and best lift-to-drag ratio, thrust and power required, rate of climb, gliding flight, the Breguet range and endurance equations, takeoff and landing distances, turns and the V-n diagram, and how electric, hybrid and sustainable-fuel aircraft change the numbers, with six labs. Free, self-paced, and tied to NASA and DOE work on cleaner flight.

NASA (ISS066-E-080762, Nov. 8, 2021)NASA (STS102-357-031, March 18, 2001)NASA/Kim Shiflett (KSC-20210927-PH-KLS04_0013, Sept. 27, 2021)NASA Blue Marble

Courses › AERO 320

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About this course

This is a free self-study course outline written by Saving Planets. It is not a university course, it carries no academic credit, and no school is involved.

Its scope follows a one-semester junior aircraft performance course. It is modeled on the topics and depth of the performance parts of MIT 16.01–16.04 Unified Engineering and MIT 16.885J Aircraft Systems Engineering (free on MIT OpenCourseWare), on courses such as Purdue AAE 251 and Georgia Tech AE 3310, and on the FAA Pilot’s Handbook of Aeronautical Knowledge, Chapter 11, which we link where they help. None of those authors, schools or agencies endorses this page.

What we add: every unit ties to real work with a cited public source (laminar-flow wings, truss-braced wings, hybrid engines, sustainable aviation fuel and transportation energy), every number on the page is re-computed by a checking script before it goes live, and every lab has a free simulation, a spreadsheet or a safe home version.

Work through it in order. Each unit has goals, key ideas, equations, step-by-step examples, practice with full answers, common mistakes, flashcards and a quiz. Your progress is saved only in this browser.

How this course helps save the planet

Performance is where aircraft design meets fuel burn. Every percent of drag removed, every kilogram saved and every better climb or cruise profile shows up directly in the Breguet range equation, and so in fuel use and emissions.

NASA lists three main ways to make aviation sustainable: efficient aircraft technology, sustainable aviation fuel, and operations and infrastructure. Each unit ends with a short, sourced section. Values that are not from a source are labeled as sample values.

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Before you start

Prerequisites

  • Aerodynamics I: lift and drag coefficients, the drag polar and induced drag.
  • Dynamics: Newton’s laws, work and energy.
  • Calculus II: integrals and natural logarithms (for range and endurance).
  • Thermodynamics helps with engines and the atmosphere but is not required.

How to study

  • Start every problem with a free-body diagram: lift, weight, thrust and drag, and the flight-path angle.
  • Write down the atmosphere first: altitude, density, temperature and speed of sound.
  • Keep weight in newtons and mass in kilograms; most errors in performance come from mixing them.
  • Know the key speeds by heart: stall, minimum drag, minimum power, best climb and approach.
  • Check limits: zero excess thrust means zero climb; heavier aircraft need more runway, roughly as weight squared.
  • Do every practice problem on paper before opening the answer, and retake the quizzes a few days later.

Lab equipment (for the wet labs)

  • A computer or tablet for the free NASA FoilSim and PhET Projectile Motion simulations.
  • Spreadsheet software for the atmosphere, drag-polar, range and takeoff labs.
  • Printer paper, a ruler or tape measure, paper clips and a stopwatch or phone timer.
  • A safe open indoor space or a calm outdoor space for glider flights.

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Suggested 15-week schedule

About 6 to 9 hours a week: reading, practice problems and, in lab weeks, the lab.

WeekUnitTopicsReading
1The Standard Atmosphere, Airspeeds and Density AltitudeStandard atmosphere, airspeeds, density altitude; Lab AFAA PHAK Ch. 11
2Forces in Flight and the Drag PolarForces in flight and the drag polar; Lab CNASA Glenn; MIT 16.01
3Thrust and Power RequiredThrust and power requiredMIT 16.01
4Climb PerformanceClimb angle and rateFAA PHAK Ch. 11
5Climb PerformanceSpecific excess power, ceilings, time to climbMIT 16.01
6Gliding FlightGliding flight; Lab BNASA Glenn
7Range and Endurance: the Breguet EquationsFuel consumption and the Breguet range equationMIT 16.885J
8Range and Endurance: the Breguet EquationsEndurance, payload–range; Lab D; midterm practiceMIT 16.885J
9Takeoff PerformanceStall speed, high lift, ground rollFAA Airplane Flying Handbook
10Takeoff PerformanceDensity altitude, wind, balanced field; Lab EFAA PHAK Ch. 11
11Landing PerformanceApproach and landing distanceFAA Airplane Flying Handbook
12Turning Flight and the V-n DiagramTurns, load factor, turn radiusFAA Airplane Flying Handbook
13Turning Flight and the V-n DiagramManeuvering speed and the V-n diagram; Lab FMIT 16.01
14Performance of Electric, Hybrid and Sustainable-Fuel AircraftElectric and hybrid aircraft performanceNASA HyTEC
15Performance of Electric, Hybrid and Sustainable-Fuel AircraftSustainable fuel, CO₂ and the full performance picture; final practiceDOE SAF; NASA

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Your progress

Tick units and labs as you finish them, and take the quizzes. Progress and quiz scores are saved only in this browser, never sent to us.

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Unit 1 · Week 1

The Standard Atmosphere, Airspeeds and Density Altitude

Why performance depends on the air, the International Standard Atmosphere (ISA), temperature, pressure and density with altitude, the speed of sound and Mach number, indicated, calibrated, equivalent and true airspeed, and pressure and density altitude.

Learning objectives

By the end of this unit you can:

  • Compute ISA temperature, pressure and density in the troposphere.
  • Find the speed of sound and Mach number.
  • Convert between equivalent and true airspeed.
  • Explain why hot, high airports hurt performance.

Key concepts

  • Lift, drag and engine thrust all depend on air density, so performance is always quoted for a stated atmosphere. The ISA is a reference: 288.15 K, 101,325 Pa and 1.225 kg/m³ at sea level.
  • In the troposphere (up to 11 km) temperature falls 6.5 K per km: T = T₀ − 0.0065h. Hydrostatics plus the ideal-gas law give p/p₀ = (T/T₀)^5.26 and ρ/ρ₀ = (T/T₀)^4.26. Above 11 km temperature is constant at 216.65 K and density falls exponentially.
  • The speed of sound is a = √(γRT); Mach number M = V/a. Because a depends only on temperature, it is lower at cruise altitude.
  • An airspeed indicator senses dynamic pressure ½ρV². Equivalent airspeed EAS = V√σ, with density ratio σ = ρ/ρ₀; true airspeed is higher than EAS at altitude.
  • Pressure altitude is the ISA altitude with the measured pressure; density altitude is the ISA altitude with the actual density. High density altitude (a high, hot or humid airfield) means thinner air, so less lift at a given speed and less engine thrust: takeoff gets longer and climb gets weaker.

Key equations

T=T0−0.0065hT=T_0-0.0065h
Troposphere temperature
ρρ0=(TT0)4.26\frac{\rho}{\rho_0}=\left(\frac{T}{T_0}\right)^{4.26}
Troposphere density
a=γRT, M=Vaa=\sqrt{\gamma RT},\ M=\frac{V}{a}
Speed of sound and Mach
VE=VσV_E=V\sqrt{\sigma}
Equivalent airspeed

Worked examples, step by step

Example 1.1 Find the ISA temperature, pressure and density at 3000 m.

Show the steps
  1. T = 288.15 − 0.0065 × 3000 = 268.65 K.
  2. p = 101,325 × (268.65/288.15)^5.26 = 70.1 kPa.
  3. ρ = 1.225 × (268.65/288.15)^4.26 = 0.909 kg/m³.

Answer: T = 268.65 K, p = 70.1 kPa, ρ = 0.909 kg/m³ (σ = 0.742).

Example 1.2 Find the speed of sound at sea level and at 11 km (216.65 K).

Show the steps
  1. a₀ = √(1.4 × 287 × 288.15).
  2. a = √(1.4 × 287 × 216.65).

Answer: a₀ = 340.3 m/s; a = 295.0 m/s.

Example 1.3 Find the ISA density and pressure at 11 km.

Show the steps
  1. T/T₀ = 216.65/288.15.
  2. ρ = 1.225 × (0.7519)^4.26; p = 101,325 × (0.7519)^5.26.

Answer: ρ = 0.364 kg/m³ and p = 22.6 kPa: less than a third of sea-level density.

Practice problems

Work each on paper first, then check.

  1. Find ISA density at 5000 m.

    Show answer

    0.736 kg/m³.

  2. What Mach number is 250 m/s at sea level?

    Show answer

    250/340.3 = 0.735.

  3. Find the exponent g/(aR) − 1 for density.

    Show answer

    9.81/(0.0065 × 287) − 1 = 4.26.

  4. An airport at 1500 m on a hot day: is its density altitude above or below 1500 m?

    Show answer

    Above: hot air is thinner.

Common misconceptions

  • Myth: True airspeed equals indicated airspeed.

    Actually: Only at sea-level standard conditions.

  • Myth: The speed of sound depends on pressure.

    Actually: For an ideal gas it depends only on temperature.

  • Myth: Density altitude is the airport elevation.

    Actually: It also depends on temperature, pressure and humidity.

Flashcards

  • ISA

    288.15 K, 101,325 Pa, 1.225 kg/m³.

  • Lapse rate

    0.0065 K/m.

  • Density ratio

    σ = ρ/ρ₀.

  • Speed of sound

    √(γRT).

  • EAS

    V√σ.

  • Density altitude

    ISA altitude with the actual density.

Self-check quiz

1. ISA sea-level density is…

Show answer

1.225 kg/m³. Standard value.

2. Troposphere lapse rate is…

Show answer

6.5 K/km. ISA definition.

3. The speed of sound depends on…

Show answer

temperature. a = √(γRT).

4. High density altitude makes takeoff…

Show answer

longer. Less lift and thrust.

How this helps save the planet

EIA reports that about 29% of total U.S. energy use in 2025 went to transporting people and goods, that petroleum products supplied about 89% of transportation energy, and that jet fuel alone was 13% of transportation energy use.

EIA also notes that jet fuel powers jet airplanes and some helicopters, and that biofuels may be blended into jet fuel. Every one of those gallons is burned in the atmosphere modeled in this unit; where an aircraft flies in that atmosphere decides how much it burns.

On Saving Planets: Aerodynamics I · Earth Observation

Sources: U.S. EIA: Use of energy for transportation

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Unit 2 · Week 2

Forces in Flight and the Drag Polar

The four forces, steady level flight, lift and drag coefficients, parasite and induced drag, the parabolic drag polar C_D = C_D0 + KC_L², Oswald efficiency and aspect ratio, maximum lift-to-drag ratio, and drag in cruise.

Learning objectives

By the end of this unit you can:

  • Write the equations of steady level flight.
  • Use the parabolic drag polar.
  • Find (L/D)max and the lift coefficient where it occurs.
  • Compute cruise drag for an airliner.

Key concepts

  • In steady level flight lift equals weight and thrust equals drag: L = W = ½ρV²SC_L and T = D = ½ρV²SC_D.
  • Drag has two parts. Parasite (zero-lift) drag C_D0 comes from skin friction and pressure drag; induced drag KC_L² is the price of making lift, with K = 1/(πeAR) for aspect ratio AR and Oswald efficiency e.
  • The drag polar C_D = C_D0 + KC_L² sets almost every performance number in this course.
  • Lift-to-drag ratio L/D = C_L/C_D is the aircraft’s aerodynamic efficiency. It is largest when induced drag equals parasite drag: C_L* = √(C_D0/K) and (L/D)max = 1/(2√(KC_D0)).
  • At (L/D)max the drag is smallest for a given weight: D_min = W/(L/D)max. Long thin wings (high AR) lower K, which is why gliders and the most efficient airliners have them.

Key equations

L=W=12ρV2SCLL=W=\tfrac12\rho V^2SC_L
Level-flight lift
CD=CD0+KCL2C_D=C_{D0}+KC_L^2
Parabolic drag polar
K=1πeARK=\frac{1}{\pi eAR}
Induced-drag factor
(LD)max=12KCD0\left(\frac LD\right)_{max}=\frac{1}{2\sqrt{KC_{D0}}}
Best lift-to-drag ratio

Worked examples, step by step

Example 2.1 The sample airliner has m = 60,000 kg, S = 120 m², C_D0 = 0.020 and K = 0.045. Find W, (L/D)max, C_L* and the minimum drag.

Show the steps
  1. W = 60,000 × 9.81 = 588.6 kN.
  2. (L/D)max = 1/(2√(0.045 × 0.020)) = 16.67.
  3. C_L* = √(0.020/0.045) = 0.667.
  4. D_min = 588,600/16.67.

Answer: (L/D)max = 16.67 at C_L = 0.667; D_min = 35.3 kN.

Example 2.2 The airliner cruises at 11 km (ρ = 0.364 kg/m³) at 230 m/s. Find q, C_L, C_D and drag.

Show the steps
  1. q = ½ × 0.3636 × 230² = 9618 Pa.
  2. C_L = 588,600/(9618 × 120) = 0.510.
  3. C_D = 0.020 + 0.045 × 0.510² = 0.0317.
  4. D = 9618 × 120 × 0.0317.

Answer: D = 36.6 kN; L/D = 16.1, close to the maximum.

Example 2.3 What is the cruise Mach number?

Show the steps
  1. M = 230/295.0.

Answer: M = 0.780.

Practice problems

Work each on paper first, then check.

  1. Halve C_D0 with K unchanged. How does (L/D)max change?

    Show answer

    Rises by √2: 23.6.

  2. Find K for AR = 9 and e = 0.8.

    Show answer

    1/(π × 0.8 × 9) = 0.0442.

  3. At C_L*, what fraction of drag is induced?

    Show answer

    Half.

  4. Find the cruise L/D above.

    Show answer

    0.510/0.0317 = 16.1.

Common misconceptions

  • Myth: Drag always grows with speed.

    Actually: Induced drag falls with speed; total drag has a minimum.

  • Myth: Bigger wings always mean less drag.

    Actually: They add parasite drag.

  • Myth: L/D is fixed for an aircraft.

    Actually: It depends on C_L, so on speed, weight and altitude.

Flashcards

  • Level flight

    L = W, T = D.

  • Drag polar

    C_D0 + KC_L².

  • K

    1/(πeAR).

  • C_L*

    √(C_D0/K).

  • (L/D)max

    1/(2√(KC_D0)).

  • D_min

    W/(L/D)max.

Self-check quiz

1. In steady level flight…

Show answer

L = W, T = D. Force balance.

2. (L/D)max occurs when…

Show answer

induced = parasite drag. Minimize C_D/C_L.

3. Raising aspect ratio…

Show answer

lowers K. K = 1/(πeAR).

4. D_min equals…

Show answer

W/(L/D)max. D = W/(L/D).

How this helps save the planet

NASA describes the truss-braced wing as a long, thin wing with aerodynamically shaped structural supports that could reduce fuel and operating costs for future airliners. In December 2025, NASA and Boeing tested a semispan model of Boeing’s Subsonic Ultra Green Aircraft Research (SUGAR) concept in a large pressurized wind tunnel in Farnborough, England.

A long, thin wing is a high-aspect-ratio wing: it lowers K in the drag polar and raises (L/D)max. The struts carry the loads that would make such a wing too heavy on its own.

On Saving Planets: Aerodynamics I · Mechanics of Materials

Sources: NASA: NASA, Boeing Advance Truss-Braced Wing Research in Test (Apr. 29, 2026)

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Unit 3 · Week 3

Thrust and Power Required

Thrust required versus speed, the minimum-drag speed, power required and the minimum-power speed, thrust and power available from jets and propellers, maximum and minimum level speeds, and the effect of altitude and weight.

Learning objectives

By the end of this unit you can:

  • Plot thrust and power required against speed.
  • Find the minimum-drag and minimum-power speeds.
  • Explain how altitude changes these speeds.
  • Find maximum level speed from thrust available.

Key concepts

  • Thrust required equals drag: T_R = ½ρV²SC_D0 + 2KW²/(ρV²S). The first term grows with V², the second falls as 1/V², so T_R has a minimum.
  • The minimum-drag speed is V_md = √(2W/(ρS)) (K/C_D0)^¼. It is the speed for (L/D)max.
  • Power required is P_R = T_R V. It is least at V_mp = V_md/3^¼ = 0.760 V_md, where induced drag is three times parasite drag and L/D = 0.866(L/D)max.
  • At altitude, all speeds scale as 1/√σ at the same weight: the true speed for best L/D rises, but the drag is unchanged.
  • Jet thrust is roughly constant with speed and falls with density; propeller power is roughly constant, so propeller thrust falls with speed. Maximum level speed is where available thrust (or power) meets required.

Key equations

TR=12ρV2SCD0+2KW2ρV2ST_R=\tfrac12\rho V^2SC_{D0}+\frac{2KW^2}{\rho V^2S}
Thrust required
Vmd=2WρS(KCD0)1/4V_{md}=\sqrt{\frac{2W}{\rho S}}\left(\frac{K}{C_{D0}}\right)^{1/4}
Minimum-drag speed
Vmp=Vmd31/4V_{mp}=\frac{V_{md}}{3^{1/4}}
Minimum-power speed
PR=TRVP_R=T_RV
Power required

Worked examples, step by step

Example 3.1 Find V_md for the sample airliner at sea level.

Show the steps
  1. √(2 × 588,600/(1.225 × 120)) = 89.5 m/s.
  2. (0.045/0.020)^¼ = 1.225.
  3. V_md = 89.5 × 1.225.

Answer: V_md = 109.6 m/s.

Example 3.2 Find V_md at 11 km and the speed ratio.

Show the steps
  1. Ratio = √(1.225/0.3636) = 1.835.
  2. V_md = 109.6 × 1.835.

Answer: V_md = 201.2 m/s at the same 35.3 kN drag.

Example 3.3 Find V_mp at sea level and the power required at V_md.

Show the steps
  1. V_mp = 109.6/1.316 = 83.3 m/s.
  2. P_R(V_md) = 35,316 × 109.6.

Answer: V_mp = 83.3 m/s; P_R at V_md = 3.87 MW.

Practice problems

Work each on paper first, then check.

  1. Find the drag at V_mp.

    Show answer

    W/(0.866 × 16.67) = 40.8 kN.

  2. Find q at 100 m/s at sea level.

    Show answer

    ½ × 1.225 × 100² = 6125 Pa.

  3. If weight rises 21%, how does V_md change?

    Show answer

    By √1.21 = 1.1 times.

  4. Below V_md, is the aircraft speed-stable?

    Show answer

    No: slowing raises drag (the back side of the drag curve).

Common misconceptions

  • Myth: Minimum drag and minimum power happen at the same speed.

    Actually: V_mp is 24% slower.

  • Myth: Flying higher lowers drag at best L/D.

    Actually: Drag is the same; only the speed is higher.

  • Myth: Thrust required is the same as power required.

    Actually: Power = thrust × speed.

Flashcards

  • Thrust required

    Equals drag.

  • V_md

    Speed for (L/D)max.

  • V_mp

    0.760 V_md.

  • Power required

    T_R V.

  • Altitude

    Speeds ∝ 1/√σ.

  • Back side

    Below V_md, drag rises as speed falls.

Self-check quiz

1. Thrust required is lowest at…

Show answer

V_md. Minimum drag.

2. V_mp/V_md equals…

Show answer

0.760. 3^(−¼).

3. At altitude, V_md (true) is…

Show answer

higher. ∝ 1/√ρ.

4. Propeller thrust with speed…

Show answer

falls. Constant power.

How this helps save the planet

NASA explains that during flight a thin layer of air, the boundary layer, forms very near an aircraft’s surface, and that on most aircraft it becomes turbulent, increasing friction drag and fuel use. NASA’s Crossflow Attenuated Natural Laminar Flow (CATNLF) design aims to keep that flow laminar on large swept wings.

A NASA computational study from 2014 to 2017 estimated that a CATNLF wing on a large, long-range aircraft like the Boeing 777 could save up to 10% of fuel a year. In this unit’s terms, laminar flow lowers C_D0, which lowers thrust and power required at every speed.

On Saving Planets: Fluid Mechanics · Aerodynamics I

Sources: NASA: NASA Tests Technology Offering Potential Fuel Savings for Commercial Aviation (Jan. 21, 2026)

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Unit 4 · Weeks 4–5

Climb Performance

Forces in a steady climb, climb angle and rate of climb, excess thrust and excess power, best-angle and best-rate speeds, specific excess power, time to climb, and absolute and service ceilings.

Learning objectives

By the end of this unit you can:

  • Write the force balance in a steady climb.
  • Compute rate and angle of climb.
  • Distinguish best-angle and best-rate speeds.
  • Estimate time to climb and ceilings.

Key concepts

  • In a steady climb at flight-path angle γ: T − D − W sin γ = 0 and L = W cos γ. So sin γ = (T − D)/W: excess thrust sets the climb angle.
  • Rate of climb RC = V sin γ = (T − D)V/W = (P_avail − P_req)/W: excess power sets the climb rate.
  • The best-angle speed (V_x) maximizes excess thrust and is used to clear obstacles; the best-rate speed (V_y) maximizes excess power and gets to altitude soonest.
  • Specific excess power P_s = (T − D)V/W combines climb and acceleration: it is the rate at which energy height h + V²/(2g) can grow.
  • Excess power falls with altitude. The absolute ceiling is where maximum RC is zero; the service ceiling is where it is 0.5 m/s (100 ft/min).

Key equations

sin⁡γ=T−DW\sin\gamma=\frac{T-D}{W}
Climb angle
RC=(T−D)VWRC=\frac{(T-D)V}{W}
Rate of climb
Ps=(T−D)VWP_s=\frac{(T-D)V}{W}
Specific excess power
hE=h+V22gh_E=h+\frac{V^2}{2g}
Energy height

Worked examples, step by step

Example 4.1 The sample airliner climbs at 130 m/s at sea level with 120 kN of thrust. Find C_L and drag.

Show the steps
  1. q = ½ × 1.225 × 130² = 10,351 Pa.
  2. C_L = 588,600/(10,351 × 120) = 0.474.
  3. D = 10,351 × 120 × (0.020 + 0.045 × 0.474²).

Answer: C_L = 0.474; D = 37.4 kN (taking L ≈ W).

Example 4.2 Find the rate and angle of climb.

Show the steps
  1. RC = (120,000 − 37,394) × 130/588,600.
  2. sin γ = 82,606/588,600.

Answer: RC = 18.2 m/s (3590 ft/min); γ = 8.07°.

Example 4.3 At an average 10 m/s, how long does a climb to 10,000 m take?

Show the steps
  1. t = 10,000/10 = 1000 s.

Answer: t = 16.7 min.

Practice problems

Work each on paper first, then check.

  1. Convert 18.2 m/s to ft/min.

    Show answer

    18.2 × 196.85 = 3590 ft/min.

  2. Which speed clears an obstacle best?

    Show answer

    V_x, best angle.

  3. What is RC at the absolute ceiling?

    Show answer

    Zero.

  4. Why does climb fall with altitude?

    Show answer

    Thrust and excess power fall with density.

Common misconceptions

  • Myth: Climb comes from extra lift.

    Actually: It comes from excess thrust or power; in a climb L < W.

  • Myth: V_x and V_y are the same.

    Actually: V_x is slower.

  • Myth: Climb rate is independent of weight.

    Actually: RC ∝ 1/W.

Flashcards

  • Climb angle

    sin γ = (T − D)/W.

  • Rate of climb

    (T − D)V/W.

  • V_x

    Best angle.

  • V_y

    Best rate.

  • P_s

    Specific excess power.

  • Ceilings

    Absolute: RC = 0; service: 0.5 m/s.

Self-check quiz

1. Climb angle depends on…

Show answer

excess thrust. sin γ = (T − D)/W.

2. Rate of climb depends on…

Show answer

excess power. RC = ΔP/W.

3. Service ceiling RC is…

Show answer

0.5 m/s. 100 ft/min.

4. In a steady climb, lift is…

Show answer

W cos γ. Normal balance.

How this helps save the planet

NASA’s sustainable aviation page notes that needing less energy to fly means using less fuel or, for electric aircraft, less power that must be generated elsewhere, which lowers lifecycle emissions for the whole time an aircraft is in use.

Climb is the most energy-hungry part of a short flight. Specific excess power shows why: every kilogram of weight saved and every newton of drag removed raises the climb rate and shortens the high-thrust phase.

On Saving Planets: Dynamics · Solutions

Sources: NASA: Sustainable Aviation

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Unit 5 · Week 6

Gliding Flight

Forces in an unpowered glide, glide angle and glide ratio, distance from a given height, sink rate and minimum sink, the effect of weight and wind, and engine-out planning.

Learning objectives

By the end of this unit you can:

  • Relate glide angle to L/D.
  • Find glide distance from a height.
  • Compute sink rate.
  • Explain the effects of weight and wind.

Key concepts

  • In a steady glide, thrust is zero: D = W sin γ and L = W cos γ, so tan γ = D/L = 1/(L/D).
  • Glide distance from height h is d = h(L/D). The flattest glide is at (L/D)max, so at V_md.
  • Sink rate V_s = V sin γ ≈ V/(L/D). Minimum sink occurs at the minimum-power speed V_mp, best for staying up longest.
  • Weight does not change the best glide ratio, only the speed at which it happens (heavier gliders fly faster at the same angle). Headwind shortens the distance over the ground; tailwind lengthens it.
  • NASA notes that the Wright brothers measured their gliders’ performance by flying parallel to a dune slope, so the glide angle equalled the slope of the hill.

Key equations

tan⁡γ=1L/D\tan\gamma=\frac{1}{L/D}
Glide angle
d=h (L/D)d=h\,(L/D)
Glide distance
Vs≈VL/DV_s\approx\frac{V}{L/D}
Sink rate
D=Wsin⁡γD=W\sin\gamma
Glide force balance

Worked examples, step by step

Example 5.1 The sample airliner loses all engines at 10 km. How far can it glide at (L/D)max = 16.67 in still air?

Show the steps
  1. d = 10,000 × 16.67.

Answer: d = 166.7 km.

Example 5.2 A sailplane with L/D = 40 is 1000 m up. Find its glide distance and angle.

Show the steps
  1. d = 1000 × 40 = 40 km.
  2. γ = arctan(1/40).

Answer: 40 km at 1.43°.

Example 5.3 Find the sailplane’s sink rate at 30 m/s.

Show the steps
  1. V_s = 30/40.

Answer: 0.75 m/s.

Practice problems

Work each on paper first, then check.

  1. Find the sink rate at 25 m/s with L/D = 35.

    Show answer

    25/35 = 0.714 m/s.

  2. Does adding water ballast change the best glide ratio?

    Show answer

    No, only the best-glide speed.

  3. Glide for longest time or farthest distance: which speed is which?

    Show answer

    V_mp for time, V_md for distance.

  4. Glide angle with L/D = 10?

    Show answer

    arctan(0.1) = 5.71°.

Common misconceptions

  • Myth: Heavier gliders glide less far.

    Actually: Same distance in still air, just faster.

  • Myth: Pull the nose up to stretch a glide.

    Actually: Below best-glide speed the glide gets steeper.

  • Myth: Minimum sink and best glide are the same speed.

    Actually: Minimum sink is slower.

Flashcards

  • Glide angle

    tan γ = 1/(L/D).

  • Glide distance

    h(L/D).

  • Sink rate

    V/(L/D).

  • Best glide

    At V_md.

  • Minimum sink

    At V_mp.

  • Weight

    Changes speed, not ratio.

Self-check quiz

1. tan γ in a glide equals…

Show answer

D/L. Force balance.

2. Best glide distance is at…

Show answer

(L/D)max. Flattest angle.

3. Longest time aloft is at…

Show answer

V_mp. Minimum sink.

4. Extra weight changes…

Show answer

best glide speed. Speed ∝ √W.

How this helps save the planet

NASA’s Beginner’s Guide explains that the smaller the glide angle, the farther an aircraft can fly for a given change in altitude, so the glide angle is a measure of flying efficiency. The Wright brothers used gliders between 1900 and 1902 to learn aerodynamics and piloting before powered flight.

The glide ratio is just L/D, the same number that sets fuel burn in cruise. An aircraft that glides well also cruises on less fuel.

On Saving Planets: Intro to Aerospace Engineering · Aerodynamics I

Sources: NASA Glenn Beginner’s Guide: Glide angle

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Unit 6 · Weeks 7–8

Range and Endurance: the Breguet Equations

Specific fuel consumption for jets and propellers, fuel weight fractions, the Breguet range and endurance equations, best-range and best-endurance speeds, cruise-climb, payload–range trade-offs and reserves.

Learning objectives

By the end of this unit you can:

  • Use thrust- and power-specific fuel consumption.
  • Apply the Breguet range equation for jets and propeller aircraft.
  • Compute endurance.
  • Identify which design changes extend range.

Key concepts

  • A jet’s thrust-specific fuel consumption c is fuel weight burned per unit thrust per unit time (1/s). A piston or turboprop’s power-specific consumption is fuel per unit power per unit time.
  • Integrating dW/dt = −cT with T = W/(L/D) at constant V and L/D gives the Breguet range for jets: R = (V/c)(L/D) ln(W_i/W_f).
  • For propeller aircraft, with propeller efficiency η and power-specific consumption c_p (1/m): R = (η/c_p)(L/D) ln(W_i/W_f). Endurance for jets is E = (1/c)(L/D) ln(W_i/W_f).
  • Range grows with three groups: engine efficiency (V/c or η/c_p), aerodynamics (L/D) and structures (the weight ratio). That is why the equation guides almost every aircraft improvement.
  • Jets fly for best range a little faster than V_md (maximizing V·L/D) and for best endurance at V_md. Propeller aircraft fly for range at V_md and for endurance at V_mp.

Key equations

R=VcLDln⁡WiWfR=\frac{V}{c}\frac{L}{D}\ln\frac{W_i}{W_f}
Breguet range, jet
R=ηcpLDln⁡WiWfR=\frac{\eta}{c_p}\frac{L}{D}\ln\frac{W_i}{W_f}
Breguet range, propeller
E=1cLDln⁡WiWfE=\frac{1}{c}\frac{L}{D}\ln\frac{W_i}{W_f}
Endurance, jet
cp=SFC⋅g3.6×106c_p=\frac{\text{SFC}\cdot g}{3.6\times10^6}
SFC in kg/kWh to 1/m

Worked examples, step by step

Example 6.1 A jet cruises at 230 m/s with c = 0.6 per hour and L/D = 16, burning fuel equal to 25% of its starting weight (W_f/W_i = 0.75). Find the range.

Show the steps
  1. c = 0.6/3600 = 1.667 × 10⁻⁴ s⁻¹.
  2. ln(1/0.75) = 0.2877.
  3. R = (230/1.667 × 10⁻⁴) × 16 × 0.2877.

Answer: R = 6352 km.

Example 6.2 Find the endurance for the same aircraft.

Show the steps
  1. E = (1/1.667 × 10⁻⁴) × 16 × 0.2877 = 27,618 s.

Answer: E = 7.67 h.

Example 6.3 A turboprop has η = 0.8, SFC = 0.3 kg/kWh, L/D = 15 and burns 15% of its weight (W_f/W_i = 0.85). Find the range.

Show the steps
  1. c_p = 0.3 × 9.81/3.6 × 10⁶ = 8.175 × 10⁻⁷ m⁻¹.
  2. ln(1/0.85) = 0.1625.
  3. R = (0.8/8.175 × 10⁻⁷) × 15 × 0.1625.

Answer: R = 2386 km.

Practice problems

Work each on paper first, then check.

  1. Raise L/D from 16 to 18 in the jet example. New range?

    Show answer

    7146 km (12.5% more).

  2. Burn 30% of the weight instead of 25%. Range?

    Show answer

    ln(1/0.7) = 0.3567: 7875 km.

  3. Which term does a laminar-flow wing improve?

    Show answer

    L/D.

  4. Which term does a lighter structure improve?

    Show answer

    The weight ratio.

Common misconceptions

  • Myth: Range is proportional to fuel carried.

    Actually: It grows with the log of the weight ratio.

  • Myth: Best range and best endurance are the same speed.

    Actually: Endurance is flown slower.

  • Myth: Breguet needs constant altitude.

    Actually: It assumes constant V and L/D, which means a slow cruise-climb.

Flashcards

  • TSFC

    Fuel weight per thrust per time.

  • Breguet jet

    (V/c)(L/D) ln(W_i/W_f).

  • Breguet prop

    (η/c_p)(L/D) ln(W_i/W_f).

  • Endurance

    (1/c)(L/D) ln(W_i/W_f).

  • Three levers

    Engine, aerodynamics, structures.

  • Cruise-climb

    Constant V and L/D as fuel burns.

Self-check quiz

1. Jet range is proportional to…

Show answer

V(L/D)/c. Breguet.

2. The weight term in Breguet is…

Show answer

ln(W_i/W_f). Integration of dW/W.

3. Jet endurance is best at…

Show answer

V_md. Minimum drag = minimum fuel flow.

4. Propeller range is best at…

Show answer

V_md. (L/D)max.

How this helps save the planet

NASA and GE Aerospace tested an integrated hybrid-electric jet engine, a modified GE Passport, in December 2025 at GE Aerospace’s Peebles Test Operation site in Ohio. NASA’s Hybrid Thermally Efficient Core (HyTEC) project aims for a hybrid engine that burns up to 10% less fuel than today’s best-in-class engines.

In Breguet terms, a 10% lower fuel burn means a smaller c: the same flight on less fuel, or a longer flight on the same fuel. The engine term, the aerodynamics term and the structures term each give engineers a lever.

On Saving Planets: Thermodynamics · Solutions

Sources: NASA: NASA, GE Aerospace Hybrid Engine System Marks Successful Test (Jan. 26, 2026)

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Unit 7 · Weeks 9–10

Takeoff Performance

Stall speed and C_Lmax, high-lift devices, the takeoff ground roll, liftoff speed, the effect of weight, density altitude, wind and runway slope, the airborne segment over an obstacle, and balanced field length.

Learning objectives

By the end of this unit you can:

  • Find stall and liftoff speeds.
  • Estimate takeoff ground roll.
  • Show how weight and density altitude lengthen takeoff.
  • Explain balanced field length.

Key concepts

  • Stall speed is the slowest steady flight speed: V_s = √(2W/(ρSC_Lmax)). Flaps and slats raise C_Lmax for takeoff and landing.
  • Liftoff speed is typically V_LOF ≈ 1.2V_s, giving a margin above stall.
  • With thrust much larger than drag and friction, the ground roll is about s_g ≈ 1.44W²/(gρSC_LmaxT).
  • Takeoff distance therefore grows roughly with W², and with 1/ρ twice over for jets (through V_s and through lower thrust). A 10 kt headwind shortens the roll; a tailwind lengthens it a lot.
  • Balanced field length is where, if an engine fails at the decision speed V₁, stopping and continuing need the same distance.

Key equations

Vs=2WρSCL,maxV_s=\sqrt{\frac{2W}{\rho SC_{L,max}}}
Stall speed
VLOF≈1.2VsV_{LOF}\approx1.2V_s
Liftoff speed
sg≈1.44W2gρSCL,maxTs_g\approx\frac{1.44W^2}{g\rho SC_{L,max}T}
Ground roll
s2s1=(W2W1)2\frac{s_2}{s_1}=\left(\frac{W_2}{W_1}\right)^2
Weight effect

Worked examples, step by step

Example 7.1 The sample airliner has C_Lmax = 2.4 with takeoff flaps at sea level. Find V_s and V_LOF.

Show the steps
  1. V_s = √(2 × 588,600/(1.225 × 120 × 2.4)).
  2. V_LOF = 1.2 × 57.8.

Answer: V_s = 57.8 m/s; V_LOF = 69.3 m/s.

Example 7.2 Estimate the ground roll with 180 kN thrust.

Show the steps
  1. s_g = 1.44 × 588,600²/(9.81 × 1.225 × 120 × 2.4 × 180,000).

Answer: s_g = 801 m.

Example 7.3 Repeat at a 1500 m airport (ISA ρ = 1.058 kg/m³), same thrust.

Show the steps
  1. s_g = 801 × 1.225/1.058.

Answer: s_g = 927 m, 1.158 times longer (more with the real thrust loss).

Practice problems

Work each on paper first, then check.

  1. The aircraft is loaded to 70,000 kg. Ratio of ground rolls?

    Show answer

    (70/60)² = 1.361.

  2. Stall speed ratio at 70,000 kg?

    Show answer

    √(70/60) = 1.080.

  3. Why do flaps shorten takeoff?

    Show answer

    Higher C_Lmax, lower V_s.

  4. What is V₁?

    Show answer

    The decision speed for a rejected or continued takeoff.

Common misconceptions

  • Myth: Takeoff distance is proportional to weight.

    Actually: Closer to weight squared.

  • Myth: A hot day only affects engines.

    Actually: It also lowers lift at a given speed.

  • Myth: Liftoff happens at stall speed.

    Actually: A margin above stall, about 1.2V_s.

Flashcards

  • Stall speed

    √(2W/(ρSC_Lmax)).

  • V_LOF

    ≈ 1.2V_s.

  • Ground roll

    1.44W²/(gρSC_LmaxT).

  • Weight effect

    ∝ W².

  • Density altitude

    Longer roll.

  • V₁

    Decision speed.

Self-check quiz

1. Stall speed scales with…

Show answer

√W. V_s ∝ √(W/ρ).

2. Ground roll scales roughly with…

Show answer

W². s_g ∝ W²/(ρT).

3. Flaps raise…

Show answer

C_Lmax. High-lift devices.

4. Balanced field length equalizes…

Show answer

stop and go distances. Definition.

How this helps save the planet

NASA and Boeing’s truss-braced wing model included slats, flaps and other moving surfaces so the team could set it to the low-speed, high-lift settings of takeoff and landing. NASA notes that the pressurized QinetiQ tunnel is used to predict airplane behavior in takeoff and landing conditions.

A very efficient cruise wing still has to meet this unit’s stall and runway numbers, which is why high-lift performance is tested as early as cruise drag.

On Saving Planets: Aerodynamics I · Statics

Sources: NASA: NASA, Boeing Advance Truss-Braced Wing Research in Test (Apr. 29, 2026)

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Unit 8 · Week 11

Landing Performance

The approach, flare and ground roll, approach speed, braking, reverse thrust and spoilers, runway condition, landing distance estimates and the energy that must be removed.

Learning objectives

By the end of this unit you can:

  • Find approach speed from stall speed.
  • Estimate landing ground roll from deceleration.
  • Explain how runway condition and spoilers matter.
  • Compute the kinetic energy at touchdown.

Key concepts

  • Landing has three segments: the approach over a 15 m (50 ft) obstacle, the flare and the ground roll.
  • Approach speed is typically V_app ≈ 1.3V_s in the landing configuration.
  • With an average deceleration a, the ground roll is s = V²/(2a). Brakes, spoilers (which dump lift so the wheels can brake) and reverse thrust all raise a.
  • Wet or icy runways lower friction and can greatly lengthen the roll; performance charts give corrections.
  • The kinetic energy at touchdown ½mV² must be absorbed, mostly as heat in the brakes.

Key equations

Vapp≈1.3VsV_{app}\approx1.3V_s
Approach speed
s=V22as=\frac{V^2}{2a}
Ground roll
Ek=12mV2E_k=\tfrac12mV^2
Kinetic energy
a=μga=\mu g
Ideal braking deceleration

Worked examples, step by step

Example 8.1 With landing flaps, C_Lmax = 2.8. Find V_s and V_app for the sample airliner at sea level.

Show the steps
  1. V_s = √(2 × 588,600/(1.225 × 120 × 2.8)) = 53.5 m/s.
  2. V_app = 1.3 × 53.5.

Answer: V_s = 53.5 m/s; V_app = 69.5 m/s.

Example 8.2 Estimate the ground roll from 69.5 m/s at an average 0.4g.

Show the steps
  1. s = 69.5²/(2 × 0.4 × 9.81).

Answer: s = 616 m.

Example 8.3 Find the kinetic energy at 69.5 m/s.

Show the steps
  1. E = ½ × 60,000 × 69.5².

Answer: E = 145 MJ.

Practice problems

Work each on paper first, then check.

  1. Halve the deceleration. New ground roll?

    Show answer

    Doubles: 1232 m.

  2. Why deploy spoilers at touchdown?

    Show answer

    They dump lift so the brakes work.

  3. Approach 10% fast: change in roll?

    Show answer

    1.1² = 1.21, 21% longer.

  4. What is the approach obstacle height?

    Show answer

    15 m (50 ft).

Common misconceptions

  • Myth: Landing distance is just the ground roll.

    Actually: It includes the approach and flare.

  • Myth: A few knots fast doesn’t matter.

    Actually: Roll grows with V².

  • Myth: Brakes work fully at touchdown.

    Actually: Not until lift is dumped.

Flashcards

  • V_app

    ≈ 1.3V_s.

  • Ground roll

    V²/(2a).

  • Spoilers

    Dump lift.

  • Reverse thrust

    Extra deceleration.

  • Wet runway

    Longer roll.

  • Brake energy

    ½mV².

Self-check quiz

1. V_app is about…

Show answer

1.3V_s. Margin above stall.

2. Ground roll with deceleration a is…

Show answer

V²/(2a). Kinematics.

3. Spoilers help braking by…

Show answer

dumping lift. More weight on wheels.

4. 10% faster approach lengthens roll by…

Show answer

21%. V².

How this helps save the planet

NASA lists operations and infrastructure as one of the three main ways to make aviation sustainable: using less energy on the ground or by choice of flight path reduces fuel burned and saves money.

Approach and landing are part of that. A well-planned continuous descent and a landing at the right speed use less fuel and less braking energy than a rushed, fast approach.

On Saving Planets: Dynamics · Solutions

Sources: NASA: Sustainable Aviation

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Unit 9 · Weeks 12–13

Turning Flight and the V-n Diagram

Level coordinated turns, bank angle and load factor, turn radius and rate, stall speed in a turn, maneuvering speed, the V-n diagram with limit load factors and gust loads, and pull-ups.

Learning objectives

By the end of this unit you can:

  • Relate bank angle to load factor.
  • Compute turn radius and rate.
  • Find stall speed in a turn and maneuvering speed.
  • Read a V-n diagram.

Key concepts

  • In a level coordinated turn, lift tilts with the bank angle φ: L cos φ = W and L sin φ = mV²/R. The load factor n = L/W = 1/cos φ.
  • Turn radius R = V²/(g√(n² − 1)) = V²/(g tan φ); turn rate ω = V/R.
  • Stall speed rises with load factor: V_s,turn = V_s√n. At 60° bank, n = 2 and stall speed is 41% higher.
  • Maneuvering speed V_A = V_s√n_limit is the speed at which full control deflection stalls the wing before the limit load is reached.
  • The V-n diagram plots allowed load factor against speed: stall curves bound the left, limit load factors the top and bottom, and dive speed the right. Gust lines add the loads from turbulence.

Key equations

n=1cos⁡ϕn=\frac{1}{\cos\phi}
Load factor in a level turn
R=V2gtan⁡ϕR=\frac{V^2}{g\tan\phi}
Turn radius
ω=VR\omega=\frac{V}{R}
Turn rate
Vs,n=VsnV_{s,n}=V_s\sqrt n
Stall speed under load

Worked examples, step by step

Example 9.1 Find n, turn radius and turn rate at 100 m/s for 30°, 45° and 60° bank.

Show the steps
  1. n = 1/cos φ = 1.155, 1.414, 2.00.
  2. R = 100²/(9.81 tan φ) = 1766 m, 1019 m, 589 m.
  3. ω = V/R.

Answer: Turn rates 3.25°/s, 5.62°/s and 9.74°/s.

Example 9.2 How much does stall speed rise at 30°, 45° and 60° bank?

Show the steps
  1. √n = 1.075, 1.189, 1.414.

Answer: 7.5%, 18.9% and 41.4%.

Example 9.3 A light aircraft has V_s = 50 m/s and n_limit = 2.5. Find V_A.

Show the steps
  1. V_A = 50 × √2.5.

Answer: V_A = 79.1 m/s.

Practice problems

Work each on paper first, then check.

  1. Find the turn radius at 60 m/s and 30° bank.

    Show answer

    60²/(9.81 × tan 30°) = 636 m.

  2. Bank angle for n = 2?

    Show answer

    60°.

  3. Why do airliners use about 25–30° bank?

    Show answer

    Comfortable load factor, small stall-speed rise.

  4. What bounds the left of the V-n diagram?

    Show answer

    The stall curve n = (V/V_s)².

Common misconceptions

  • Myth: A steeper bank makes a tighter turn at no cost.

    Actually: It raises load factor and stall speed.

  • Myth: Maneuvering speed is a maximum cruise speed.

    Actually: It protects against full control inputs.

  • Myth: Load factor depends only on speed.

    Actually: It depends on bank angle in a level turn.

Flashcards

  • Load factor

    n = L/W = 1/cos φ.

  • Turn radius

    V²/(g tan φ).

  • Turn rate

    V/R.

  • Stall in turn

    V_s√n.

  • V_A

    V_s√n_limit.

  • V-n diagram

    Flight envelope.

Self-check quiz

1. Load factor at 60° bank is…

Show answer

2. 1/cos 60°.

2. Turn radius is…

Show answer

V²/(g tan φ). Centripetal balance.

3. Stall speed in a turn is…

Show answer

V_s√n. More lift needed.

4. The V-n diagram shows…

Show answer

allowed loads vs speed. Flight envelope.

How this helps save the planet

NASA’s 2026 CATNLF flight tests mounted a roughly 3-foot (40-inch) scale wing model vertically under an F-15B research jet. The first flight, on January 29, 2026, lasted 75 minutes and confirmed the aircraft could maneuver safely with the test article attached.

NASA says the first flight focused on envelope expansion, with turns, steady holds and gentle pitch changes between about 20,000 and nearly 34,000 feet, and that it was the first of up to 15 planned flights. Expanding an envelope step by step is the V-n diagram of this unit put into practice.

On Saving Planets: Dynamics · Mechanics of Materials

Sources: NASA: NASA Completes First Flight of Laminar Flow Scaled Wing Design (Feb. 11, 2026)

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Unit 10 · Weeks 14–15

Performance of Electric, Hybrid and Sustainable-Fuel Aircraft

Range of battery-electric aircraft, why weight does not fall as batteries discharge, specific energy of batteries versus jet fuel, hybrid-electric engines, sustainable aviation fuel, CO₂ from fuel burn, and putting all the performance levers together.

Learning objectives

By the end of this unit you can:

  • Derive the range equation for a battery-electric aircraft.
  • Compare the specific energy of batteries and jet fuel.
  • Compute CO₂ from fuel burned.
  • Rank technologies by which Breguet term they improve.

Key concepts

  • A battery-electric aircraft does not get lighter as it flies, so there is no logarithm: R = (e_b η/g)(L/D)(m_b/m), with battery specific energy e_b (J/kg), total efficiency η and battery mass fraction m_b/m.
  • Jet fuel holds about 43 MJ/kg (about 11,900 Wh/kg, a textbook value). Batteries of about 250 Wh/kg (a sample value) hold roughly 48 times less per kilogram, which is why electric aircraft today are small and short-range.
  • Hybrid-electric engines use electric motors to help a jet engine run more efficiently, lowering the specific fuel consumption c.
  • Burning a kilogram of jet fuel makes about 3.16 kg of CO₂ (a textbook value). Fuel saved by better L/D, lighter structures or better engines cuts CO₂ in direct proportion.
  • Sustainable aviation fuel (SAF) has properties similar to conventional jet fuel, so it does not change the performance equations; it changes where the carbon comes from.

Key equations

R=ebηgLDmbmR=\frac{e_b\eta}{g}\frac{L}{D}\frac{m_b}{m}
Battery-electric range
mCO2≈3.16 mfuelm_{CO_2}\approx3.16\,m_{fuel}
CO₂ from jet fuel
efueleb=43 MJ/kgeb\frac{e_{fuel}}{e_b}=\frac{43\ \text{MJ/kg}}{e_b}
Specific-energy ratio
R=VcLDln⁡WiWfR=\frac{V}{c}\frac{L}{D}\ln\frac{W_i}{W_f}
Breguet, for comparison

Worked examples, step by step

Example 10.1 An electric aircraft has 250 Wh/kg batteries (sample value), η = 0.8, L/D = 15 and a 30% battery fraction. Find its range.

Show the steps
  1. e_b = 250 × 3600 = 900,000 J/kg.
  2. R = (900,000 × 0.8/9.81) × 15 × 0.3.

Answer: R = 330 km.

Example 10.2 How much more energy per kilogram does jet fuel hold?

Show the steps
  1. 43 × 10⁶/3600 = 11,944 Wh/kg.
  2. 11,944/250.

Answer: About 47.8 times.

Example 10.3 How much CO₂ comes from burning 1000 kg of jet fuel?

Show the steps
  1. 3.16 × 1000.

Answer: 3160 kg.

Practice problems

Work each on paper first, then check.

  1. Repeat the electric range with 400 Wh/kg cells.

    Show answer

    528 km.

  2. An aircraft burns 2500 kg of fuel per hour. CO₂ per hour?

    Show answer

    7900 kg.

  3. Which Breguet term does a hybrid engine improve?

    Show answer

    c, the fuel consumption.

  4. Why is there no logarithm in the battery range equation?

    Show answer

    Mass stays constant.

Common misconceptions

  • Myth: Electric aircraft get lighter as they fly.

    Actually: Battery mass stays the same.

  • Myth: SAF needs new engines.

    Actually: It is designed to work in existing aircraft.

  • Myth: Only engines matter for emissions.

    Actually: Aerodynamics, weight and operations all count.

Flashcards

  • Electric range

    (e_bη/g)(L/D)(m_b/m).

  • Jet fuel

    ≈ 43 MJ/kg.

  • Battery

    ≈ 250 Wh/kg (sample).

  • CO₂

    ≈ 3.16 kg per kg fuel.

  • Hybrid engine

    Lowers c.

  • SAF

    Drop-in fuel from biomass and wastes.

Self-check quiz

1. Battery-electric range is proportional to…

Show answer

e_b(L/D)(m_b/m). Constant mass.

2. Jet fuel vs 250 Wh/kg battery holds about…

Show answer

48×. 11,944/250.

3. CO₂ per kg of jet fuel is about…

Show answer

3.16 kg. Carbon plus oxygen.

4. SAF changes…

Show answer

the carbon source. Similar fuel properties.

How this helps save the planet

DOE says sustainable aviation fuel made from biomass and waste resources can deliver performance similar to petroleum jet fuel. DOE’s 2023 Billion-Ton Report concluded that the United States could triple its biomass production to more than 1 billion tons a year, producing an estimated 60 billion gallons of liquid biofuels, enough feedstock to meet the projected fuel demand of U.S. aviation.

NASA notes that SAF does not release new CO₂ into the environment and has been shown to reduce contrails that can trap heat in Earth’s atmosphere. NASA also estimates that with today’s fleet and efficiency, worldwide air traffic in 2050 would need more than 620 megatons of fuel and produce close to 2,000 megatons of CO₂; the United States has committed to net-zero greenhouse gas emissions from U.S. aviation by 2050.

On Saving Planets: Solutions · Thermodynamics

Sources: U.S. DOE: Sustainable Aviation Fuels · NASA: Sustainable Aviation

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The lab (Flight performance lab)

The lab is the hands-on part of the course. Here it is free and for practice only, with no credit.

Every lab can be done in a free simulation, a spreadsheet, or with paper and a tape measure. Fly paper gliders only in a clear space away from people’s faces, stairs and traffic, and keep walkways free of trip hazards.

Safety reference: OSHA: Walking-working surfaces (slip, trip and fall hazards)

Lab A: Build a standard-atmosphere calculator

Week 1

Goal: Build a spreadsheet of ISA temperature, density and speed of sound from sea level to 11 km.

T=T0−0.0065h,  ρ=ρ0(T/T0)4.26,  a=γRTT=T_0-0.0065h,\ \ \rho=\rho_0(T/T_0)^{4.26},\ \ a=\sqrt{\gamma RT}
Troposphere model

Procedure

  1. Make a column of altitudes every 500 m from 0 to 11,000 m.
  2. Compute T, then ρ with the exponent 9.81/(0.0065 × 287) − 1.
  3. Compute a = √(1.4 × 287 × T).
  4. Plot ρ and a against altitude and compare with the values in Unit 1.

Data table

Altitude (m)T (K)ρ (kg/m³)a (m/s)
0288.151.225340.3
3000268.650.909328.5
5000255.650.736320.5
11,000216.650.364295.0

Computed ISA values.

Sample calculation

  1. ρ(3000) = 1.225 × (268.65/288.15)^4.26 = 0.909 kg/m³.
  2. a(5000) = √(1.4 × 287 × 255.65) = 320.5 m/s.

Safety

  • Computer only.

Home or virtual alternative

  • Any spreadsheet works.

NASA Glenn: Dynamic pressure · MIT OpenCourseWare 16.01–16.04 Unified Engineering (2005–2006)

Planet link: EIA reports jet fuel was 13% of U.S. transportation energy use in 2025. U.S. EIA: Use of energy for transportation

Lab B: Paper glider glide ratio

Week 6

Goal: Measure the glide ratio of a paper glider from its launch height and flight distance, and find its glide angle and sink rate.

LD=dh,  γ=arctan⁡hd\frac LD=\frac dh,\ \ \gamma=\arctan\frac hd
Glide ratio and angle

Procedure

  1. Fold a simple glider from printer paper; add a paper clip at the nose to trim it.
  2. In a clear space, release it gently (no throw) from a measured height, about 1.5 m.
  3. Measure where it lands and time the flight; repeat five times and average.
  4. Move the paper clip and see how the glide ratio changes.

Data table

QuantityValue
Release height1.5 m
Distance6.0 m
Glide ratio4.0
Glide angle14.0°
Flight time2.0 s
Average speed3.09 m/s

Sample values; real paper gliders have far lower L/D than sailplanes.

Sample calculation

  1. L/D = 6.0/1.5 = 4.0; γ = arctan(1.5/6.0) = 14.0°.
  2. Speed along the path = √(6² + 1.5²)/2.0 = 3.09 m/s; sink rate = 1.5/2.0 = 0.75 m/s.

Safety

  • Fly only in a clear space, never toward people’s faces; keep walkways clear.
  • Do not stand on chairs or stairs to launch; use your standing height.

Home or virtual alternative

  • This is a home lab.

NASA Glenn Beginner’s Guide: Glide angle · PhET: Projectile Motion

Planet link: NASA notes the Wright brothers studied gliders from 1900 to 1902 and used glide angle to measure performance. NASA Glenn Beginner’s Guide: Glide angle

Lab C: Measuring a drag polar in FoilSim

Week 2

Goal: Collect lift and drag coefficients at several angles of attack, fit a parabolic drag polar and find (L/D)max.

CD=CD0+KCL2C_D=C_{D0}+KC_L^2
Parabolic drag polar

Procedure

  1. Open NASA FoilSim and choose a wing shape and speed.
  2. Record C_L and C_D at five angles of attack.
  3. Plot C_D against C_L² and fit a straight line: the intercept is C_D0 and the slope is K.
  4. Compute (L/D)max = 1/(2√(KC_D0)).

Data table

C_LC_DL/D
0.20.02189.17
0.60.036216.6
1.00.065015.4

Sample values generated from the course polar (C_D0 = 0.020, K = 0.045); your FoilSim numbers will differ.

Sample calculation

  1. C_D(0.6) = 0.020 + 0.045 × 0.36 = 0.0362.
  2. (L/D)max = 1/(2√(0.045 × 0.020)) = 16.67 at C_L = 0.667.

Safety

  • Computer only.

Home or virtual alternative

  • Any browser works.

NASA Glenn: FoilSim · NASA Glenn: Induced drag coefficient

Planet link: NASA’s CATNLF wing aims to keep laminar flow on swept wings, lowering C_D0. NASA: NASA Tests Technology Offering Potential Fuel Savings for Commercial Aviation (Jan. 21, 2026)

Lab D: Breguet range trade study

Week 8

Goal: Use a spreadsheet to see how L/D, fuel consumption and fuel fraction change a jet’s range.

R=VcLDln⁡WiWfR=\frac{V}{c}\frac{L}{D}\ln\frac{W_i}{W_f}
Breguet range

Procedure

  1. Enter V = 230 m/s, c = 0.6 per hour and W_f/W_i = 0.75.
  2. Compute range for L/D = 14, 16 and 18.
  3. Then cut c by 10% and compare.
  4. Rank the changes by kilometres gained.

Data table

L/DRange (km)
145558
166352
187146

Computed values for a sample jet.

Sample calculation

  1. R(14) = (230/1.667 × 10⁻⁴) × 14 × 0.2877 = 5558 km.
  2. Cutting c by 10% multiplies range by 1/0.9 = 1.111.

Safety

  • Computer only.

Home or virtual alternative

  • Any spreadsheet works.

MIT OpenCourseWare 16.01–16.04 Unified Engineering (2005–2006)

Planet link: NASA’s HyTEC project targets a hybrid engine burning up to 10% less fuel than today’s best engines. NASA: NASA, GE Aerospace Hybrid Engine System Marks Successful Test (Jan. 26, 2026)

Lab E: Takeoff distance and density altitude

Week 10

Goal: Compute how takeoff ground roll grows with airport elevation and aircraft weight.

sg≈1.44W2gρSCL,maxTs_g\approx\frac{1.44W^2}{g\rho SC_{L,max}T}
Ground roll

Procedure

  1. Use the sample airliner: W = 588.6 kN, S = 120 m², C_Lmax = 2.4, T = 180 kN.
  2. Compute s_g at 0, 1500 and 3000 m ISA density, holding thrust constant.
  3. Repeat at 70,000 kg.
  4. Discuss why real jets lose even more because thrust also falls.

Data table

Elevation (m)ρ (kg/m³)Ground roll (m)
01.225801
15001.058927
30000.9091079

Computed values with constant thrust (a simplification).

Sample calculation

  1. s_g(0) = 1.44 × 588,600²/(9.81 × 1.225 × 120 × 2.4 × 180,000) = 801 m.
  2. s_g(3000) = 801 × 1.225/0.909 = 1079 m.

Safety

  • Computer only.

Home or virtual alternative

  • Any spreadsheet works.

FAA PHAK Chapter 11: Aircraft Performance (PDF) · FAA Airplane Flying Handbook

Planet link: NASA notes truss-braced wing models are tested in takeoff and landing high-lift settings. NASA: NASA, Boeing Advance Truss-Braced Wing Research in Test (Apr. 29, 2026)

Lab F: Turn performance table

Week 13

Goal: Build a table of load factor, turn radius and stall-speed increase against bank angle.

n=1cos⁡ϕ,  R=V2gtan⁡ϕn=\frac{1}{\cos\phi},\ \ R=\frac{V^2}{g\tan\phi}
Level-turn relations

Procedure

  1. Set V = 100 m/s.
  2. Compute n and R for bank angles from 15° to 60°.
  3. Add √n for the stall-speed increase.
  4. Mark the bank angle where stall speed rises 10%.

Data table

BanknR (m)
15°1.0353804
30°1.1551766
45°1.4141019
60°2.000589

Computed values.

Sample calculation

  1. R(15°) = 100²/(9.81 × tan 15°) = 3804 m.
  2. n(60°) = 1/cos 60° = 2.000.

Safety

  • Computer only.

Home or virtual alternative

  • Any spreadsheet works.

FAA Airplane Flying Handbook

Planet link: NASA’s first CATNLF flight included turns and holds from about 20,000 to nearly 34,000 feet. NASA: NASA Completes First Flight of Laminar Flow Scaled Wing Design (Feb. 11, 2026)

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Exam prep practice sets

Time yourself: about 10 minutes per problem. Then compare with the full solutions.

Midterm practice set (Units 1–5)

  1. Find the ISA temperature at 2000 m.

    Show full solution
    1. 288.15 − 0.0065 × 2000.

    Answer: 275.15 K.

  2. Find the ISA density at 2000 m.

    Show full solution
    1. 1.225 × (275.15/288.15)^4.26.

    Answer: 1.006 kg/m³.

  3. Find (L/D)max for C_D0 = 0.025 and K = 0.040.

    Show full solution
    1. 1/(2√(0.04 × 0.025)).

    Answer: 15.81.

  4. An aircraft weighing 500 kN has 50 kN excess thrust at 120 m/s. Find the rate of climb.

    Show full solution
    1. 50,000 × 120/500,000.

    Answer: 12.0 m/s.

  5. How far can an aircraft with L/D = 12 glide from 2000 m?

    Show full solution
    1. 2000 × 12.

    Answer: 24 km.

Final practice set (all units)

  1. Find the range of a jet with V = 240 m/s, c = 0.55 per hour, L/D = 17 and W_f/W_i = 0.8.

    Show full solution
    1. (240/(0.55/3600)) × 17 × ln 1.25.

    Answer: 5959 km.

  2. Find the sea-level stall speed for W = 100 kN, S = 30 m², C_Lmax = 1.8.

    Show full solution
    1. √(2 × 100,000/(1.225 × 30 × 1.8)).

    Answer: 55.0 m/s.

  3. Takeoff weight rises 20%. Ratio of ground rolls?

    Show full solution
    1. 1.2².

    Answer: 1.44.

  4. Find the landing roll from 60 m/s at 3 m/s².

    Show full solution
    1. 60²/(2 × 3).

    Answer: 600 m.

  5. Find the turn radius at 80 m/s and 45° bank.

    Show full solution
    1. 80²/(9.81 × tan 45°).

    Answer: 652 m.

  6. Find the range of an electric aircraft with 300 Wh/kg, η = 0.85, L/D = 18 and a 35% battery fraction.

    Show full solution
    1. (300 × 3600 × 0.85/9.81) × 18 × 0.35.

    Answer: 590 km.

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Formula sheet

T=T0−0.0065h, ρ/ρ0=(T/T0)4.26, a=γRTT=T_0-0.0065h,\ \rho/\rho_0=(T/T_0)^{4.26},\ a=\sqrt{\gamma RT}
L=W=12ρV2SCL, CD=CD0+KCL2, K=1/(πeAR)L=W=\tfrac12\rho V^2SC_L,\ C_D=C_{D0}+KC_L^2,\ K=1/(\pi eAR)
(L/D)max=1/(2KCD0), CL∗=CD0/K(L/D)_{max}=1/(2\sqrt{KC_{D0}}),\ C_L^*=\sqrt{C_{D0}/K}
Vmd=2W/(ρS)(K/CD0)1/4, Vmp=Vmd/31/4V_{md}=\sqrt{2W/(\rho S)}(K/C_{D0})^{1/4},\ V_{mp}=V_{md}/3^{1/4}
sin⁡γ=(T−D)/W, RC=(T−D)V/W\sin\gamma=(T-D)/W,\ RC=(T-D)V/W
tan⁡γglide=1/(L/D), d=h(L/D)\tan\gamma_{glide}=1/(L/D),\ d=h(L/D)
R=(V/c)(L/D)ln⁡(Wi/Wf), E=(1/c)(L/D)ln⁡(Wi/Wf)R=(V/c)(L/D)\ln(W_i/W_f),\ E=(1/c)(L/D)\ln(W_i/W_f)
Vs=2W/(ρSCL,max), sg≈1.44W2/(gρSCL,maxT)V_s=\sqrt{2W/(\rho SC_{L,max})},\ s_g\approx1.44W^2/(g\rho SC_{L,max}T)
s=V2/(2a), Vapp≈1.3Vss=V^2/(2a),\ V_{app}\approx1.3V_s
n=1/cos⁡ϕ, R=V2/(gtan⁡ϕ), VA=Vsnlimn=1/\cos\phi,\ R=V^2/(g\tan\phi),\ V_A=V_s\sqrt{n_{lim}}
Relec=(ebη/g)(L/D)(mb/m)R_{elec}=(e_b\eta/g)(L/D)(m_b/m)

Constants

ConstantValue
International Standard Atmosphere, sea level (textbook values)ρ₀ = 1.225 kg/m³, T₀ = 288.15 K, p₀ = 101,325 Pa, lapse rate 0.0065 K/m to 11 km
AirR = 287 J/(kg·K), γ = 1.4
Gravityg = 9.81 m/s²
Sample airliner used throughout (sample values)m = 60,000 kg, W = 588.6 kN, S = 120 m², C_D0 = 0.020, K = 0.045
Jet fuel (textbook values)about 43 MJ/kg; about 3.16 kg CO₂ per kg burned
1 m/s196.85 ft/min

Source: NIST Guide to the SI, Appendix B.9: conversion factors

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Free texts and simulations

Use these free lectures, handbooks and simulations alongside the course.

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Glossary

Absolute ceiling
Altitude where the maximum rate of climb is zero.
Balanced field length
Runway length where stopping and continuing after an engine failure need the same distance.
Breguet equation
Range or endurance from engine efficiency, L/D and weight ratio.
Density altitude
ISA altitude with the same air density as the actual air.
Drag polar
Relation between drag and lift coefficients.
Equivalent airspeed
True airspeed times √σ.
Glide ratio
Horizontal distance per unit of height lost; equal to L/D.
Induced drag
Drag caused by producing lift.
Load factor
Lift divided by weight.
Maneuvering speed
Speed at which full control input stalls the wing before limit load.
Parasite drag
Drag not due to lift.
Rate of climb
Vertical speed in a climb.
Service ceiling
Altitude where maximum climb rate is 0.5 m/s.
Specific excess power
Excess power per unit weight.
Specific fuel consumption
Fuel used per unit thrust or power per unit time.
Stall speed
Slowest steady flight speed, at C_Lmax.
Standard atmosphere
Reference model of temperature, pressure and density with altitude.
Sustainable aviation fuel
Jet fuel made from biomass and wastes.
V-n diagram
Allowed load factor against airspeed.
Wing loading
Weight divided by wing area.

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Self-assessment rubric

No grade or credit is given. Use this to judge honestly where you are in each unit.

LevelWhat it looks like
4 · MasteryStarts from the atmosphere and the force balance, picks the right speed (V_md, V_mp, V_x, V_y, V_s), keeps weight and mass straight, and explains which design lever changes the answer.
3 · ProficientSolves the practice problems with occasional unit slips.
2 · DevelopingFollows worked examples but mixes up range and endurance speeds or forgets density.
1 · BeginningPlugs numbers into formulas without a diagram; needs to revisit the unit.

Suggested weighting if you want a score

PartWeight
Practice problems20%
Labs25%
Midterm practice set25%
Final practice set30%

Aim for level 3 or better in every unit before moving to the next course.

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