Introduction to Aerospace Engineering
How aircraft and spacecraft are designed, built and flown: the atmosphere, lift and drag, performance, stability, propulsion, rockets and orbits, and the path to cleaner flight. Free, self-paced, with four hands-on labs.
Courses › AERO 101
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 first-year aerospace engineering introductions such as MIT 16.00 and the opening of MIT’s Unified Engineering, the introductory aeronautics and astronautics courses at Purdue, Georgia Tech and Embry-Riddle, and the classic textbook Introduction to Flight by John D. Anderson Jr. We link to free materials throughout, especially NASA Glenn’s Beginner’s Guide to Aeronautics. None of these organizations endorse this page.
What we add: every unit connects to cleaner flight and to the spacecraft that watch our planet, every number on the page is re-computed by a checking script before it goes live, and four labs, including a virtual wind tunnel and a glider you build, make the equations real.
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
Aerospace engineering builds the satellites that measure Earth’s climate, forests, ice and oceans, and it also has to shrink its own footprint: every kilogram of jet fuel burned makes about 3.16 kg of CO₂.
Each unit ends with a short, sourced section showing how its ideas help people and the planet.
Before you start
Prerequisites
- High-school algebra and trigonometry; logarithms and exponentials.
- Physics I mechanics (forces, energy, momentum, circular motion) helps; take it alongside if you can.
- Calculus I is useful for a few derivations but not required for the numbers.
How to study
- Keep units in SI throughout and write them on every line.
- Sketch the forces (lift, weight, thrust, drag) before writing any equation.
- Estimate first; aerospace numbers have familiar sizes (sea-level air ≈ 1.2 kg/m³, low orbit ≈ 7.7 km/s).
- Do every practice problem on paper before opening the answer.
- Use the NASA Glenn simulators to check your intuition.
- Retake the quizzes a few days later. Spaced practice beats one long session.
Lab equipment (for the wet labs)
- A computer with a web browser and a spreadsheet (FoilSim runs in the browser).
- For the glider lab: card stock or foam board, a ruler, scissors, modeling clay or coins for nose weight, and a tape measure.
- Optional: a smartphone stopwatch.
Suggested 15-week schedule
About 6 to 9 hours a week: reading, practice problems and, in lab weeks, the lab.
| Week | Unit | Topics | Reading |
|---|---|---|---|
| 1 | History of Flight & the Design Process | History of flight; the design process; four forces | MIT 16.00; Smithsonian |
| 2 | The Atmosphere | Standard atmosphere; speed of sound | NASA Glenn: atmosphere |
| 3 | Basic Aerodynamics: Lift & Drag | Lift and drag equations; Bernoulli | NASA Glenn: lift and drag |
| 4 | Basic Aerodynamics: Lift & Drag | Reynolds number and stall speed; Lab D | FAA PHAK |
| 5 | Airfoils & Wings | Airfoils, lift curve, finite wings; Lab A | NASA FoilSim |
| 6 | Aircraft Performance | Gliding, power required, climb | NASA Glenn: glide angle |
| 7 | Aircraft Performance | Range and endurance; midterm practice | FAA PHAK |
| 8 | Stability & Control | Stability and control; Lab C | NASA Glenn: rotations |
| 9 | Propulsion Basics | Propulsion: jets, fans and propellers | NASA Glenn: propulsion |
| 10 | Rockets & the Rocket Equation | Rocket thrust and the rocket equation | NASA Glenn: rockets |
| 11 | Rockets & the Rocket Equation | Staging and launch vehicles; Lab B | NASA Glenn: rockets |
| 12 | Orbits & Space Missions | Orbits, Kepler’s laws, geostationary orbit | JPL Basics of Space Flight |
| 13 | Orbits & Space Missions | Hohmann transfers and mission design | JPL Basics of Space Flight |
| 14 | Sustainable Aviation & Space | Sustainable aviation and space | NASA; EPA; DOE |
| 15 | Review | Review and final practice set | All units |
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.
Turn on JavaScript to track progress.
Unit 1 · Week 1
History of Flight & the Design Process
From the Wright brothers to Artemis: how aerospace engineers turn a mission into requirements, trade-offs and a design that flies.
Learning objectives
By the end of this unit you can:
- Outline key milestones of aviation and spaceflight.
- Describe the design cycle: requirements, concept, preliminary and detailed design, test.
- Explain trade studies and why weight drives everything.
- Name the four forces of flight.
Key concepts
- On December 17, 1903, the Wright Flyer made a 12-second flight covering 36 m (120 ft); the best flight that day covered 260 m (852 ft) in 59 seconds. The Wrights succeeded by testing: they built gliders and their own wind tunnel before adding an engine.
- Every design starts with a mission: payload, range or orbit, cost, safety. Requirements turn the mission into numbers the design must meet.
- Design is iterative: concept sketch, sizing, analysis, test, redesign. A trade study compares options against weighted criteria.
- Four forces act on an aircraft: lift, weight, thrust and drag. In steady level flight, lift = weight and thrust = drag.
- Weight compounds: extra structure needs more lift, a bigger wing, more drag, more fuel, and more weight again.
Key equations
Worked examples, step by step
Example 1.1 A 1,100 kg light airplane cruises level. How much lift must its wing produce?
Show the steps
- Level flight: L = W = mg.
- L = 1,100 × 9.81.
Answer: 10,791 N ≈ 10.8 kN.
Example 1.2 Two wing concepts are scored out of 10 on cost (weight 0.3), efficiency (0.5) and ease of building (0.2). Concept A scores 8, 6, 9; B scores 5, 9, 6. Which wins?
Show the steps
- A: 0.3(8) + 0.5(6) + 0.2(9) = 7.2.
- B: 0.3(5) + 0.5(9) + 0.2(6) = 7.2.
Answer: A tie at 7.2: the team should test both or revisit the weights it trusts least.
Example 1.3 The Wright Flyer’s first flight covered 36 m in 12 s. What was its average ground speed?
Show the steps
- v = 36/12.
Answer: 3 m/s over the ground. It flew into a strong headwind, so its airspeed was much higher.
Practice problems
Work each on paper first, then check.
An airliner of mass 70,000 kg cruises level. Find its lift.
Show answer
L = 70,000 × 9.81 = 686,700 N ≈ 687 kN.
In level cruise an aircraft’s engines give 30 kN of thrust. What is its drag?
Show answer
30 kN, since thrust balances drag in steady flight.
Why did the Wright brothers build a wind tunnel?
Show answer
Published lift data were unreliable, so they measured many wing shapes themselves: testing before building is the heart of the design process.
Common misconceptions
Myth: Engineers design the aircraft first and then find a mission for it.
Actually: The mission and its requirements come first.
Myth: In level flight thrust must balance weight.
Actually: Thrust balances drag; lift balances weight.
Myth: Shaving a little weight saves only that weight.
Actually: Weight savings compound through wing size, drag and fuel.
Flashcards
Four forces
Lift, weight, thrust, drag.
Requirements
Measurable targets a design must meet.
Trade study
Weighted comparison of design options.
Design cycle
Concept → preliminary → detailed design → test → iterate.
Wright Flyer
First powered, controlled airplane flights, Dec. 17, 1903.
Weight spiral
Extra weight grows the whole design.
Self-check quiz
How this helps save the planet
Design priorities change with the times. Requirements for new aircraft now include fuel burn and emissions, so the same trade-study tools weigh climate alongside cost and performance.
NASA lists three main ways to make aviation sustainable: efficient aircraft technology, sustainable aviation fuel, and better operations and infrastructure.
On Saving Planets: Artemis II
Sources: Smithsonian: 1903 Wright Flyer · NASA: Sustainable aviation
Free learning resources
- Smithsonian: 1903 Wright Flyer: The Wright Flyer’s story and specifications.
- MIT OpenCourseWare 16.00: Introduction to Aerospace Engineering and Design: MIT’s design-centered introductory course.
- NASA Glenn: Beginner’s Guide to Aeronautics: Start here for every topic.
Unit 2 · Week 2
The Atmosphere
The air an aircraft flies through: how temperature, pressure, density and the speed of sound change with altitude.
Learning objectives
By the end of this unit you can:
- Use a standard atmosphere model to find T, p and ρ at altitude.
- Apply the ideal-gas law to air.
- Compute the speed of sound and Mach number.
- Explain why airliners cruise near 11 km.
Key concepts
- A standard atmosphere is an average model fitted to measurements. In NASA’s model the troposphere runs from the ground to 11,000 m, and temperature falls about 6.5 °C per kilometer.
- From 11 to 25 km (lower stratosphere) the model’s temperature is constant at −56.46 °C, while pressure keeps falling exponentially.
- Density follows from the ideal-gas law, ρ = p/(RT). At 11 km air is about 30% as dense as at sea level.
- The speed of sound is a = √(γRT), so it depends only on temperature. Mach number M = V/a.
- Thin air means less drag at a given speed, which is why jets cruise high, but wings and engines still need enough air to work.
Key equations
Worked examples, step by step
Example 2.1 Find T, p and ρ at 5,000 m with NASA’s model.
Show the steps
- T = 15.04 − 0.00649 × 5,000 = −17.41 °C.
- p = 101.29 × (255.69/288.08)^5.256 = 54.11 kPa.
- ρ = 54.11/(0.2869 × 255.69).
Answer: T ≈ −17.4 °C, p ≈ 54.1 kPa, ρ ≈ 0.738 kg/m³ (60% of sea level).
Example 2.2 Find the speed of sound at sea level (15 °C) and at 11 km (−56.5 °C).
Show the steps
- a = √(1.4 × 287 × 288.15) = 340.3 m/s.
- a = √(1.4 × 287 × 216.65) = 295.0 m/s.
Answer: 340 m/s and 295 m/s.
Example 2.3 An airliner flies at 230 m/s true airspeed at 11 km. What is its Mach number?
Show the steps
- M = 230/295.0.
Answer: M ≈ 0.78.
Practice problems
Work each on paper first, then check.
Find the air density at 11,000 m with the NASA model (T = −56.46 °C, p = 22.72 kPa).
Show answer
ρ = 22.72/(0.2869 × 216.64) ≈ 0.366 kg/m³, about 30% of sea level.
Find the temperature at 1,500 m in the NASA model.
Show answer
T = 15.04 − 9.74 = 5.3 °C.
What is Mach 2 at sea level in m/s?
Show answer
2 × 340 = 680 m/s.
Find the sea-level density from p = 101.40 kPa and T = 15.04 °C.
Show answer
ρ = 101.40/(0.2869 × 288.14) ≈ 1.227 kg/m³.
Common misconceptions
Myth: The speed of sound depends on pressure.
Actually: For an ideal gas it depends only on temperature.
Myth: It gets colder all the way up.
Actually: Temperature levels off above 11 km and rises again higher in the stratosphere.
Myth: Thinner air always helps.
Actually: It cuts drag but also lift and engine thrust, so wings must fly faster or at higher lift coefficients.
Flashcards
Standard atmosphere
Average model of T, p, ρ versus altitude.
Troposphere
Ground to ~11 km; temperature falls with height.
Lapse rate
≈ 6.5 °C per km in the troposphere.
Ideal gas law
p = ρRT.
Speed of sound
a = √(γRT).
Mach number
M = V/a.
Self-check quiz
How this helps save the planet
Aircraft fly where weather and climate happen. NASA studies aircraft emissions through computer modeling, combustion laboratory testing and flight testing, because what engines release at altitude matters for the atmosphere.
The same atmosphere physics underlies how satellites turn the radiation they measure into profiles of the air’s temperature.
On Saving Planets: Atmospheric drag
Sources: NASA Glenn: Earth atmosphere model (metric) · NASA: Sustainable aviation
Free learning resources
- NASA Glenn: Earth atmosphere model (metric): The model and equations used in this unit.
- FAA Pilot’s Handbook of Aeronautical Knowledge: Free FAA handbook chapters on the atmosphere and aerodynamics.
Unit 3 · Weeks 3–4
Basic Aerodynamics: Lift & Drag
Where lift and drag come from, and the two equations that size every wing: L = ½ρV²SC_L and D = ½ρV²SC_D.
Learning objectives
By the end of this unit you can:
- Explain lift as the result of turning a flow, with pressure differences described by Bernoulli’s equation.
- Use the lift and drag equations.
- Compute dynamic pressure, stall speed and Reynolds number.
- Distinguish skin-friction, pressure and induced drag.
Key concepts
- A wing turns the airflow downward; by Newton’s third law the air pushes the wing up. Equivalently, faster flow over the top means lower pressure there (Bernoulli). Both describe one flow.
- Dynamic pressure q = ½ρV² sets the scale of every aerodynamic force. Lift and drag coefficients C_L and C_D capture shape and angle of attack.
- Lift grows with V², so halving speed needs four times the lift coefficient or wing area.
- Drag comes from skin friction, pressure (form) drag and induced drag, the cost of making lift with a finite wing.
- Reynolds number Re = ρVc/μ compares inertial and viscous effects; wind-tunnel models must account for it to predict full-size behavior.
Key equations
Worked examples, step by step
Example 3.1 A 1,100 kg airplane with a 16.2 m² wing flies level at 55 m/s at sea level (ρ = 1.227 kg/m³). Find C_L.
Show the steps
- L = W = 10,791 N.
- q = ½ × 1.227 × 55² = 1,855 Pa.
- C_L = 10,791/(1,855 × 16.2).
Answer: C_L ≈ 0.36.
Example 3.2 With C_L,max = 1.6, find this airplane’s stall speed at sea level.
Show the steps
- V_s = √(2W/(ρSC_L,max)).
- = √(2 × 10,791/(1.227 × 16.2 × 1.6)).
Answer: V_s ≈ 26.1 m/s (about 51 knots).
Example 3.3 Find the Reynolds number of a 1.5 m chord at 55 m/s at sea level (μ = 1.789 × 10⁻⁵ kg/(m·s)).
Show the steps
- Re = 1.227 × 55 × 1.5/(1.789 × 10⁻⁵).
Answer: Re ≈ 5.7 million.
Practice problems
Work each on paper first, then check.
If speed doubles at the same C_L, how does lift change?
Show answer
Lift ∝ V², so it quadruples.
Find the dynamic pressure at 230 m/s at 11 km (ρ = 0.366 kg/m³).
Show answer
q = ½ × 0.366 × 230² ≈ 9,680 Pa.
Find the drag of the Example 3.1 airplane if C_D = 0.032.
Show answer
D = 1,855 × 16.2 × 0.032 ≈ 962 N.
How much faster is the stall speed at 1,500 m (ρ = 1.060 kg/m³) than at sea level?
Show answer
V_s ∝ 1/√ρ: √(1.227/1.060) = 1.076, so about 7.6% faster (≈ 28.0 m/s true airspeed).
Common misconceptions
Myth: Air over the top must meet air under the bottom at the trailing edge (equal transit time).
Actually: It does not; the upper flow actually arrives first. NASA Glenn lists equal transit as an incorrect theory of lift.
Myth: Lift is only Bernoulli, or only Newton.
Actually: Both describe the same flow; pressure differences and flow turning go together.
Myth: Drag is just air friction.
Actually: Pressure drag and induced drag are often larger.
Flashcards
Dynamic pressure
q = ½ρV².
Lift equation
L = qSC_L.
Drag equation
D = qSC_D.
Stall speed
V_s = √(2W/(ρSC_L,max)).
Reynolds number
ρVc/μ.
Induced drag
Drag due to lift on a finite wing.
Self-check quiz
How this helps save the planet
Drag is fuel. In cruise, engine thrust equals drag, so every newton of drag removed is fuel not burned and CO₂ not emitted. Needing less energy to fly is the first of NASA’s three paths to sustainable aviation.
The same lift physics drives wind-turbine blades, which turn the wind’s aerodynamic force into clean electricity.
On Saving Planets: Physics I
Sources: NASA Glenn: the drag equation · NASA: Sustainable aviation
Free learning resources
- NASA Glenn: the lift equation: Lift equation and coefficients.
- NASA Glenn: the drag equation: Drag equation.
- NASA Glenn: Bernoulli’s equation: Pressure and velocity.
- NASA Glenn: Bernoulli and Newton: Why the equal-transit explanation is wrong.
Unit 4 · Week 5
Airfoils & Wings
Airfoil geometry, the lift curve, stall, and why long, slender wings make less drag.
Learning objectives
By the end of this unit you can:
- Name airfoil geometry: chord, camber, thickness, angle of attack.
- Use thin-airfoil theory: c_l ≈ 2πα.
- Correct the lift slope for a finite wing of aspect ratio AR.
- Use the drag polar C_D = C_D0 + C_L²/(πeAR) to find the best lift-to-drag ratio.
Key concepts
- An airfoil is a wing’s cross-section. Camber lets it lift at zero angle of attack; thickness adds strength and room for fuel.
- Thin-airfoil theory predicts a lift slope of 2π per radian (about 0.11 per degree) until the flow separates and the wing stalls.
- A finite wing loses lift slope to its tip vortices: a = a₀/(1 + a₀/(πeAR)), with aspect ratio AR = b²/S.
- Total drag C_D = C_D0 + KC_L² with K = 1/(πeAR). Maximum L/D occurs where induced drag equals parasite drag.
- High aspect ratio (sailplanes, albatrosses, modern airliner wings) means less induced drag and higher L/D.
Key equations
Worked examples, step by step
Example 4.1 Thin-airfoil theory: find c_l at α = 5°.
Show the steps
- α = 5π/180 = 0.0873 rad.
- c_l = 2π × 0.0873.
Answer: c_l ≈ 0.548.
Example 4.2 Find the lift slope of a wing with AR = 8 and e = 0.9.
Show the steps
- a = 2π/(1 + 2π/(π × 0.9 × 8)) = 6.283/1.278.
Answer: 4.92 per rad ≈ 0.086 per degree, 22% below the 2D value.
Example 4.3 With C_D0 = 0.025, e = 0.8 and AR = 7.5, find (L/D)max and the C_L where it occurs.
Show the steps
- K = 1/(π × 0.8 × 7.5) = 0.0531.
- (L/D)max = 1/(2√(0.025 × 0.0531)).
- C_L* = √(0.025/0.0531).
Answer: (L/D)max ≈ 13.7 at C_L ≈ 0.69.
Practice problems
Work each on paper first, then check.
Find the aspect ratio of a wing with span 11 m and area 16.2 m².
Show answer
AR = 121/16.2 ≈ 7.5.
Using c_l = 2πα, what angle gives c_l = 0.5?
Show answer
α = 0.5/(2π) = 0.0796 rad ≈ 4.6°.
For the Example 4.3 airplane, find C_D at C_L = 0.36.
Show answer
C_D = 0.025 + 0.0531 × 0.1296 ≈ 0.0319.
If AR rises from 7.5 to 15 (same e and C_D0), how does (L/D)max change?
Show answer
K halves, so (L/D)max grows by √2 to ≈ 19.4.
Common misconceptions
Myth: More angle of attack always means more lift.
Actually: Past the stall angle the flow separates and lift drops.
Myth: Symmetric airfoils cannot lift.
Actually: They lift at a positive angle of attack.
Myth: Wingtip devices make induced drag zero.
Actually: They reduce it; only an infinitely long wing would have none.
Flashcards
Chord
Leading edge to trailing edge.
Camber
Curvature of the mean line.
Angle of attack
Angle between chord and oncoming flow.
Stall
Flow separation; lift drops.
Aspect ratio
b²/S.
Drag polar
C_D = C_D0 + KC_L².
Self-check quiz
How this helps save the planet
Long, slender wings are one of the most direct ways to cut fuel burn, because induced drag falls as aspect ratio rises. That is exactly the “efficient aircraft technology” path in NASA’s sustainable-aviation work.
The albatross, which glides for hours over the ocean, uses the same physics: a long, thin wing with a high lift-to-drag ratio.
On Saving Planets: Calculus II
Sources: NASA Glenn: induced drag coefficient · NASA: Sustainable aviation
Free learning resources
- NASA Glenn: FoilSim: Explore camber, thickness and angle of attack.
- NASA Glenn: induced drag coefficient: Aspect ratio and efficiency factor.
- MIT OpenCourseWare 16.01–16.04: Unified Engineering: MIT’s fluids lectures.
Unit 5 · Weeks 6–7
Aircraft Performance
How far, how long, how fast and how steep: gliding, climbing, range and endurance.
Learning objectives
By the end of this unit you can:
- Relate glide ratio to L/D.
- Find thrust and power required in level flight.
- Compute climb rate from excess power.
- Use the Breguet range equation.
Key concepts
- A glider with lift-to-drag ratio L/D travels L/D meters forward for every meter of height lost: tan γ = D/L.
- Thrust required in level flight equals drag; power required is drag × speed. Excess power becomes climb: rate of climb = (P_avail − P_req)/W.
- For a jet, range follows the Breguet equation: R = (V/c)(L/D) ln(W_i/W_f). Three levers: engine efficiency (V/c), aerodynamics (L/D) and structures (weight fraction).
- Endurance (time aloft) is maximized at a different speed than range.
Key equations
Worked examples, step by step
Example 5.1 A sailplane with L/D = 20 is 1,000 m up. How far can it glide in still air, and at what angle?
Show the steps
- Distance = 1,000 × 20.
- γ = arctan(1/20).
Answer: 20 km, at a glide angle of about 2.9°.
Example 5.2 A jet cruises at 230 m/s with L/D = 17 and thrust-specific fuel consumption c = 0.6 per hour. Burning fuel lowers its weight from W_i to W_i/1.25. Find its range.
Show the steps
- c = 0.6/3,600 = 1.667 × 10⁻⁴ s⁻¹.
- R = (230/1.667 × 10⁻⁴) × 17 × ln 1.25.
Answer: ≈ 5,235 km.
Example 5.3 The Example 3.1 airplane, with the Example 4.3 drag polar, has about 957 N of drag at 55 m/s. What power does level flight require?
Show the steps
- P = DV = 957 × 55.
Answer: ≈ 52.6 kW (about 71 hp).
Practice problems
Work each on paper first, then check.
A paper glider has L/D = 5. Launched level from 2 m, about how far does it glide?
Show answer
10 m.
An aircraft weighing 10,800 N has 30 kW of excess power. Find its rate of climb.
Show answer
30,000/10,800 ≈ 2.8 m/s.
In Example 5.2, how does range change if L/D rises from 17 to 20?
Show answer
Range ∝ L/D: 5,235 × 20/17 ≈ 6,159 km.
How far can the Example 4.3 airplane glide from 1,500 m at its best L/D of 13.7?
Show answer
1,500 × 13.7 ≈ 20.6 km.
Common misconceptions
Myth: A heavier glider glides farther.
Actually: Glide ratio depends on L/D, not weight; weight changes the best speed.
Myth: Range depends only on the fuel load.
Actually: It depends on engine efficiency, L/D and the weight fraction (Breguet).
Myth: Flying faster always gives more range.
Actually: Best range comes at a moderate speed tied to the best L/D.
Flashcards
Glide ratio
Horizontal distance per unit height = L/D.
Power required
P_R = DV.
Rate of climb
(P_A − P_R)/W.
Breguet range
R = (V/c)(L/D) ln(W_i/W_f).
TSFC
Fuel flow per unit thrust.
Endurance
Time aloft.
Self-check quiz
How this helps save the planet
The Breguet equation is aviation’s climate equation: for a given trip, fuel burned falls when engines get more efficient, wings get higher L/D, or structures get lighter. With the other terms fixed, range is proportional to L/D.
NASA estimates that with today’s fleet and operations, worldwide air traffic in 2050 would need more than 620 million tonnes of fuel and emit close to 2,000 million tonnes of CO₂, which is why every term in the equation matters.
On Saving Planets: CO2M: measuring CO₂ from space
Sources: NASA: Sustainable aviation · FAA Pilot’s Handbook of Aeronautical Knowledge
Free learning resources
- NASA Glenn: glide angle: Gliding flight.
- FAA Pilot’s Handbook of Aeronautical Knowledge: Aircraft performance chapter.
- NASA Glenn: thrust-to-weight ratio: Thrust, weight and climb.
Unit 6 · Week 8
Stability & Control
Why a well-designed aircraft returns to steady flight by itself, and how pilots and computers control it.
Learning objectives
By the end of this unit you can:
- Name the three axes, pitch, roll and yaw, and their control surfaces.
- Explain static and dynamic stability.
- Locate the center of gravity and the neutral point; compute static margin.
- Explain why a tail stabilizes an aircraft.
Key concepts
- An aircraft rotates about its center of gravity: pitch (elevator), roll (ailerons), yaw (rudder).
- Static stability: after a disturbance, the aircraft initially tends to return. Dynamic stability: the motion actually dies out over time.
- Pitch stability needs the center of gravity ahead of the neutral point. Static margin = (x_np − x_cg)/c̄ must be positive.
- A tail far behind the CG works like the feathers on an arrow: a nose-up disturbance raises the tail’s lift and pushes the nose back down.
- Too much stability makes an aircraft sluggish and costs trim drag; some modern fighters are designed unstable and flown by computer.
Key equations
Worked examples, step by step
Example 6.1 The neutral point is at 40% of the mean chord and the CG at 25%. Find the static margin.
Show the steps
- SM = 0.40 − 0.25.
Answer: 0.15, or 15%: stable.
Example 6.2 A glider has a 50 g wing at x = 10 cm, a 10 g tail at x = 40 cm and 20 g of nose weight at x = 0. Find the CG.
Show the steps
- Σmx = 50(10) + 10(40) + 20(0) = 900 g·cm.
- Σm = 80 g.
Answer: x_cg = 11.25 cm from the nose.
Example 6.3 For the Example 6.2 glider (mean chord 8 cm), the neutral point is at 13 cm. Is it stable?
Show the steps
- SM = (13 − 11.25)/8.
Answer: SM ≈ 0.22: stable. Remove nose weight and the CG moves aft toward instability.
Practice problems
Work each on paper first, then check.
Which surface controls pitch?
Show answer
The elevator, on the horizontal tail.
CG at 30% chord, neutral point at 28%. Stable?
Show answer
SM = −0.02 < 0: statically unstable.
In Example 6.2, how much nose weight m puts the CG at 12 cm?
Show answer
900/(60 + m) = 12 gives m = 15 g.
Why does a paper airplane with no nose weight pitch up and stall?
Show answer
Its CG is too far aft, near or behind the neutral point.
Common misconceptions
Myth: Statically stable means the oscillation dies out.
Actually: That is dynamic stability; static stability is the initial tendency.
Myth: More stability is always better.
Actually: Too much makes control heavy and wastes trim drag.
Myth: The CG can be anywhere as long as the wing lifts.
Actually: An aft CG makes the aircraft unstable.
Flashcards
Pitch / roll / yaw
Elevator / ailerons / rudder.
Static stability
Initial tendency to return.
Dynamic stability
Disturbance dies out over time.
Neutral point
CG location for neutral pitch stability.
Static margin
(x_np − x_cg)/c̄.
Trim
Balanced, zero net moment.
Self-check quiz
How this helps save the planet
Stability margins cost fuel: a conventional tail usually pushes down to balance the aircraft, and the wing must make extra lift, with extra drag, to carry it. Trimming that penalty is part of the efficient-aircraft path NASA describes.
The same balance and control ideas keep Earth-observing satellites pointed steadily so their instruments can map the planet.
On Saving Planets: Orbits guide
Sources: NASA Glenn: aircraft rotations · NASA: Sustainable aviation
Free learning resources
- NASA Glenn: aircraft rotations: Pitch, roll and yaw.
- FAA Pilot’s Handbook of Aeronautical Knowledge: Stability and flight controls chapters.
Unit 7 · Week 9
Propulsion Basics
How propellers, jets and rockets make thrust by throwing mass backward, and why bigger, slower jets are more efficient.
Learning objectives
By the end of this unit you can:
- Use the momentum thrust equation.
- Compute propulsive efficiency.
- Compare turbojets, turbofans and propellers.
- Define specific impulse and thrust-specific fuel consumption.
Key concepts
- Every engine makes thrust by accelerating a mass flow: F = ṁ(v_e − v₀) for an air-breathing engine (ignoring fuel mass and pressure terms).
- Propulsive efficiency η_p = 2/(1 + v_e/v₀): moving a lot of air a little is more efficient than moving a little air a lot. That is why turbofans have huge fans.
- Turbojet: all air passes through the core. Turbofan: most air bypasses the core through the fan. Turboprop: a propeller driven by a gas turbine.
- Specific impulse I_sp = F/(ṁ_fuel g₀) measures thrust per fuel weight flow, in seconds.
Key equations
Worked examples, step by step
Example 7.1 A turbofan moves 500 kg/s of air from 250 m/s to 320 m/s. Find thrust and propulsive efficiency.
Show the steps
- F = 500 × 70 = 35,000 N.
- η_p = 2/(1 + 320/250) = 2/2.28.
Answer: F = 35 kN, η_p ≈ 0.88.
Example 7.2 A turbojet makes the same 35 kN with 100 kg/s of air at v₀ = 250 m/s. What exhaust speed does it need, and what is η_p?
Show the steps
- v_e = 250 + 35,000/100 = 600 m/s.
- η_p = 2/(1 + 2.4).
Answer: v_e = 600 m/s, η_p ≈ 0.59: much less efficient.
Example 7.3 An engine makes 100 kN while burning 2.5 kg/s of fuel. Find its I_sp.
Show the steps
- I_sp = 100,000/(2.5 × 9.80665).
Answer: ≈ 4,079 s. Air-breathers beat rockets on this measure because they do not carry their oxidizer.
Practice problems
Work each on paper first, then check.
Find the thrust for ṁ = 200 kg/s, v₀ = 200 m/s, v_e = 300 m/s.
Show answer
20 kN.
Find η_p when v_e = v₀.
Show answer
1.0, but the thrust is zero: efficiency and thrust trade off.
Why do airliners use high-bypass turbofans?
Show answer
They move a large mass of air slowly, which raises propulsive efficiency and cuts fuel burn and noise.
Common misconceptions
Myth: Jets push against the air behind them.
Actually: They accelerate air through the engine; the reaction to that is thrust.
Myth: Faster exhaust always means a better engine.
Actually: For efficiency, exhaust speed should be close to flight speed.
Myth: Specific impulse is a speed.
Actually: It is measured in seconds; I_sp × g₀ is an effective exhaust speed.
Flashcards
Momentum thrust
F = ṁ(v_e − v₀).
Propulsive efficiency
2/(1 + v_e/v₀).
Turbofan
Fan plus core; most air bypasses.
Bypass ratio
Bypass air ÷ core air.
Specific impulse
F/(ṁ g₀), in seconds.
TSFC
Fuel flow per unit thrust.
Self-check quiz
How this helps save the planet
Higher propulsive efficiency is a direct fuel and CO₂ saving. Every kilogram of jet fuel burned produces about 3.16 kg of CO₂ (the fuel’s carbon combined with oxygen from the air; see Unit 10), so efficiency gains add up over an engine’s decades of service.
Sustainable aviation fuel, made from wastes and biomass, is blended to match conventional jet fuel’s properties for safe operation. NASA notes that SAF does not release new CO₂ into the environment.
On Saving Planets: General Chemistry I
Sources: NASA: Sustainable aviation · U.S. DOE: Sustainable aviation fuels
Free learning resources
- NASA Glenn: thrust-to-weight ratio: Engines and thrust.
- NASA Glenn: specific impulse: Rocket and jet efficiency.
- MIT OpenCourseWare 16.01–16.04: Unified Engineering: MIT propulsion lectures.
Unit 8 · Weeks 10–11
Rockets & the Rocket Equation
Why reaching orbit is so hard: Tsiolkovsky’s rocket equation, mass ratios and staging.
Learning objectives
By the end of this unit you can:
- Use Δv = I_sp g₀ ln(m₀/m_f).
- Compute required mass ratios.
- Explain why staging works.
- Compute rocket thrust-to-weight ratio.
Key concepts
- A rocket carries all its propellant, fuel and oxidizer. Its thrust is F = ṁv_e + (p_e − p_a)A_e.
- The ideal rocket equation: Δv = v_e ln(m₀/m_f), with v_e = I_sp g₀. Speed grows only with the logarithm of the mass ratio.
- Reaching low Earth orbit takes roughly 9.4 km/s of Δv once gravity and drag losses are included, so most of a launch vehicle’s mass is propellant.
- Staging drops empty tanks and engines so later stages do not carry dead weight.
- Thrust-to-weight ratio must exceed 1 to lift off.
Key equations
Worked examples, step by step
Example 8.1 An engine with I_sp = 450 s fires until m₀/m_f = 5. Find Δv.
Show the steps
- v_e = 450 × 9.80665 = 4,413 m/s.
- Δv = 4,413 × ln 5.
Answer: ≈ 7,102 m/s.
Example 8.2 What mass ratio does a single stage need for 9.4 km/s with I_sp = 300 s? With 450 s?
Show the steps
- 300 s: e^(9,400/2,942) = e^3.195.
- 450 s: e^(9,400/4,413) = e^2.130.
Answer: ≈ 24.4 versus ≈ 8.4: about 96% versus 88% of the rocket must be propellant.
Example 8.3 A two-stage rocket: each stage has I_sp = 300 s and mass ratio 4. Find the total Δv.
Show the steps
- Each stage: 2,942 × ln 4 = 4,078 m/s.
- Add the stages.
Answer: ≈ 8,157 m/s, twice what one stage with mass ratio 4 gives.
Practice problems
Work each on paper first, then check.
Find Δv for I_sp = 311 s and m₀/m_f = 3.
Show answer
311 × 9.80665 × ln 3 ≈ 3,351 m/s.
A 500,000 kg rocket has 7.5 MN of thrust. Find T/W at liftoff.
Show answer
7.5 × 10⁶/(500,000 × 9.81) ≈ 1.53.
What I_sp gives Δv = 3,000 m/s with mass ratio 2?
Show answer
3,000/(9.80665 × ln 2) ≈ 441 s.
Common misconceptions
Myth: Rockets push against the air.
Actually: They work best in vacuum; they push on their own exhaust.
Myth: Double the propellant, double the Δv.
Actually: Δv grows with the logarithm of the mass ratio.
Myth: Orbit needs only 7.7 km/s of Δv.
Actually: Gravity and drag losses push the total to roughly 9.4 km/s.
Flashcards
Rocket equation
Δv = I_sp g₀ ln(m₀/m_f).
Mass ratio
m₀/m_f.
Staging
Dropping spent stages.
Thrust-to-weight
Must exceed 1 at liftoff.
Δv to LEO
≈ 9.4 km/s including losses.
I_sp
Seconds of thrust per weight of propellant.
Self-check quiz
How this helps save the planet
Every Earth-observing satellite, from Landsat to the missions that track sea level and greenhouse gases, rides a rocket sized with this equation. It is why each kilogram of instrument is precious and why satellites are built to be light.
Our Delta-v calculator lets you run the same numbers for real missions.
On Saving Planets: Delta-v calculator · Landsat 9
Sources: NASA Glenn: ideal rocket equation · NASA Glenn: rocket thrust equation
Free learning resources
- NASA Glenn: ideal rocket equation: Derivation and use.
- NASA Glenn: rocket thrust equation: Pressure and momentum thrust.
- NASA Glenn: specific impulse: Comparing propellants.
Unit 9 · Weeks 12–13
Orbits & Space Missions
Circular and elliptical orbits, periods, geostationary orbit and the Hohmann transfer, plus how Earth-observing missions choose their orbits.
Learning objectives
By the end of this unit you can:
- Compute circular orbital speed and period.
- Find the geostationary radius.
- Compute a Hohmann transfer Δv and transfer time.
- Describe common Earth orbits and their uses.
Key concepts
- In a circular orbit gravity supplies the centripetal force: v = √(μ/r), with μ = GM.
- Kepler’s third law: T = 2π√(a³/μ). Higher orbits are slower and take longer.
- A geostationary orbit has a period of one sidereal day (86,164 s) and sits over the equator at a radius of about 42,164 km.
- A Hohmann transfer uses two burns along an ellipse that touches both circular orbits; it is the most efficient two-burn transfer between coplanar circular orbits.
- Low Earth orbit (about 400 km for the ISS), sun-synchronous polar orbits for imaging, and geostationary orbit for weather each serve different missions.
Key equations
Worked examples, step by step
Example 9.1 Find the speed and period of a circular orbit 400 km up.
Show the steps
- r = 6,771 km.
- v = √(3.986 × 10¹⁴/6.771 × 10⁶) = 7,673 m/s.
- T = 2π√(r³/μ) = 5,545 s.
Answer: 7.67 km/s and ≈ 92.4 minutes.
Example 9.2 Find the geostationary radius.
Show the steps
- r = (μT²/4π²)^(1/3) with T = 86,164 s.
Answer: ≈ 42,164 km from Earth’s center (about 35,786 km above the equator).
Example 9.3 Find the Hohmann Δv from a 300 km circular orbit to geostationary radius.
Show the steps
- r₁ = 6,671 km, r₂ = 42,164 km.
- Δv₁ = 2,428 m/s; Δv₂ = 1,468 m/s.
- Transfer time = half the transfer ellipse’s period.
Answer: Total ≈ 3.90 km/s; transfer ≈ 5.27 hours (ignoring any plane change).
Practice problems
Work each on paper first, then check.
Find v for a circular orbit at r = 7,000 km.
Show answer
√(3.986 × 10¹⁴/7 × 10⁶) ≈ 7,546 m/s.
If the orbit radius quadruples, how does the period change?
Show answer
T ∝ r^(3/2): 8 times longer.
Why are many weather satellites geostationary?
Show answer
They stay over one spot and can image the same hemisphere continuously.
Common misconceptions
Myth: Astronauts float because there is no gravity in orbit.
Actually: At 400 km gravity is about 89% of its surface value; they are in free fall.
Myth: To catch a spacecraft ahead of you, just speed up.
Actually: Speeding up raises your orbit and lowers your angular rate.
Myth: Higher orbits need more speed.
Actually: Circular speed falls with height, though reaching higher orbits takes more total Δv.
Flashcards
Circular speed
√(μ/r).
Kepler’s third law
T = 2π√(a³/μ).
Vis-viva
v² = μ(2/r − 1/a).
Hohmann transfer
Two-burn elliptical transfer.
GEO
Period = 1 sidereal day; r ≈ 42,164 km.
Sun-synchronous orbit
Near-polar orbit that passes over each place at about the same local time.
Self-check quiz
How this helps save the planet
Climate monitoring depends on orbit choice. Sun-synchronous orbits let Landsat image the same place under similar lighting, so changes in forests, crops and ice stand out, while geostationary weather satellites watch storms develop continuously.
Orbits also need care: debris threatens these missions, so spacecraft are increasingly designed to be disposed of at the end of their lives.
On Saving Planets: Orbits guide · Orbital velocity calculator · Orbital period calculator
Sources: NASA JPL: Basics of Space Flight · NASA Earth fact sheet
Free learning resources
- NASA JPL: Basics of Space Flight: NASA JPL’s free primer on orbits and missions.
- Physics I, Unit 7: gravitation and orbits: Our orbits unit.
Unit 10 · Week 14
Sustainable Aviation & Space
Putting it together: the physics of cutting aviation’s climate impact, from efficient aircraft to sustainable fuels and new energy sources, and keeping space usable.
Learning objectives
By the end of this unit you can:
- Compute CO₂ from fuel burned.
- Use the Breguet equation to compare efficiency improvements.
- Explain the roles and limits of sustainable aviation fuel, batteries and hydrogen.
- Describe space sustainability: debris and disposal.
Key concepts
- Burning one kilogram of kerosene-type jet fuel produces about 3.16 kg of CO₂, fixed by chemistry. The only ways to cut it are to burn less fuel or to use fuel whose carbon came recently from the air.
- EPA reports that the aircraft covered by its greenhouse gas rule account for 10% of U.S. transportation greenhouse gas emissions and 3% of the U.S. total.
- NASA’s three paths: efficient aircraft technology, sustainable aviation fuel, and better operations and infrastructure.
- Today’s batteries store far less energy per kilogram than jet fuel, so electric aircraft suit short, small flights; hydrogen stores a lot of energy per kilogram but needs large tanks.
- Space sustainability means planning disposal so useful orbits stay open for the satellites that monitor Earth.
Key equations
Worked examples, step by step
Example 10.1 Where does 3.16 come from? Model jet fuel as C₁₂H₂₃ and find kg of CO₂ per kg of fuel.
Show the steps
- Molar mass: 12(12.011) + 23(1.008) = 167.3 g/mol.
- 12 CO₂ per molecule: 12 × 44.01 = 528.1 g.
- 528.1/167.3.
Answer: ≈ 3.16 kg of CO₂ per kg of fuel.
Example 10.2 NASA estimates 620 million tonnes of fuel in 2050 at today’s efficiency. How much CO₂ is that?
Show the steps
- 620 × 3.16.
Answer: ≈ 1,959 million tonnes, matching NASA’s “close to 2,000 megatons.”
Example 10.3 A flight burns 20 t of fuel. A new wing raises L/D from 17 to 19. Estimate the fuel and CO₂ saved.
Show the steps
- Fuel ≈ 20 × 17/19 = 17.9 t.
- Fuel saved ≈ 2.1 t; × 3.16.
Answer: ≈ 2.1 t of fuel and ≈ 6.7 t of CO₂ per flight (a simplified estimate).
Practice problems
Work each on paper first, then check.
A flight burns 5,000 kg of fuel. How much CO₂?
Show answer
≈ 15,800 kg.
If engines become 15% more fuel-efficient, by about how much do CO₂ emissions per trip fall?
Show answer
About 15%, since CO₂ is proportional to fuel burned.
Why can’t today’s batteries power long-haul airliners?
Show answer
Their energy per kilogram is far below jet fuel, and the Breguet trade makes extra weight very costly.
Common misconceptions
Myth: Sustainable aviation fuel emits no CO₂ when burned.
Actually: It emits CO₂ like jet fuel; the benefit is that its carbon came recently from the air, so lifecycle emissions are lower.
Myth: Electric airplanes will soon replace all jets.
Actually: Battery energy density limits them to short, small flights for now.
Myth: Space is too big to pollute.
Actually: Debris in popular orbits is a real hazard.
Flashcards
CO₂ per kg fuel
≈ 3.16 kg.
SAF
Sustainable aviation fuel from wastes and biomass.
Three paths
Efficient aircraft, SAF, operations.
Energy density
Energy per kg; batteries ≪ jet fuel.
Space debris
Defunct objects that threaten useful orbits.
Lifecycle emissions
All emissions from fuel production to use.
Self-check quiz
How this helps save the planet
This unit is the course’s planet tie-in in full: each equation from Units 3–7 is a lever on fuel burn, and fuel burn is CO₂. Engineers who can run these numbers design the cleaner aircraft that NASA and industry are working toward.
EPA reports that the aircraft covered by its rule produce 10% of U.S. transportation greenhouse gas emissions, and NASA notes the U.S. committed in 2021 to net-zero aviation greenhouse gas emissions by 2050.
On Saving Planets: CO2M: measuring CO₂ from space · Delta-v calculator
Sources: NASA: Sustainable aviation · EPA: greenhouse gas emissions from aircraft
Free learning resources
- NASA: Sustainable aviation: NASA’s overview and goals.
- EPA: greenhouse gas emissions from aircraft: U.S. emissions standards and data.
- U.S. DOE: Sustainable aviation fuels: How SAF is made.
Labs
Four labs put the course to work: a virtual wind tunnel with NASA’s FoilSim, a rocket-equation calculator, a glider you design and fly, and an atmosphere and stall-speed spreadsheet.
Sample data is labeled as such. Your own measurements will differ, and explaining why is part of the lab.
Safety reference: FAA Pilot’s Handbook of Aeronautical Knowledge
Lab A: Virtual wind tunnel with NASA FoilSim
Week 5
Goal: Measure an airfoil’s lift curve in FoilSim and compare it with thin-airfoil theory.
Procedure
- Open FoilSim and choose a symmetric airfoil (zero camber) about 12% thick.
- Record the lift coefficient at α = 0°, 2°, 4°, 6° and 8°.
- Add a column with 2πα and compare. Then add camber and repeat.
- Increase α until FoilSim warns of stall; note the angle.
Data table
| α (deg) | Theory c_l = 2πα |
|---|---|
| 0 | 0.000 |
| 2 | 0.219 |
| 4 | 0.439 |
| 6 | 0.658 |
| 8 | 0.877 |
Computed theory values. Your FoilSim readings will differ somewhat; explain why (thickness, model assumptions).
Sample calculation
- 2π × (4π/180) = 0.439.
- Theory slope: 0.110 per degree. A finite wing with AR = 8 and e = 0.9 would show only about 0.086 per degree (Unit 4).
Safety
- None beyond normal computer use.
Home or virtual alternative
- Runs in a web browser.
Planet link: Wind-tunnel testing, real and virtual, is how designers find low-drag wings that burn less fuel. NASA: Sustainable aviation
Lab B: Rocket-equation calculator
Week 11
Goal: Build a spreadsheet that gives Δv from I_sp and mass ratio, and use it to size a two-stage rocket.
Procedure
- Columns: I_sp, m₀, m_f, mass ratio, Δv.
- Fill rows for I_sp = 300 s and 450 s at mass ratios 2, 4 and 8.
- Add a two-stage sheet: total Δv = sum of the stages. Find stage mass ratios that reach 9.4 km/s.
- Check one result against our Delta-v calculator.
Data table
| I_sp (s) | Mass ratio | Δv (m/s) |
|---|---|---|
| 300 | 2 | 2,039 |
| 300 | 4 | 4,078 |
| 300 | 8 | 6,118 |
| 450 | 4 | 6,118 |
| 450 | 8 | 9,177 |
Computed values (ideal: no gravity or drag losses).
Sample calculation
- 300 × 9.80665 × ln 2 = 2,039 m/s.
- Each doubling of the mass ratio adds the same 2,039 m/s at 300 s: the logarithm at work.
- Two stages at 300 s, each with mass ratio 5, give 2 × 4,735 = 9,470 m/s, just above 9.4 km/s.
Safety
- None beyond normal computer use. Do not build or fly rocket motors without trained supervision and local permission.
Home or virtual alternative
- A spreadsheet is enough.
Saving Planets Delta-v calculator · NASA Glenn: ideal rocket equation
Planet link: Every kilogram saved on a climate satellite shrinks the rocket needed to launch it. NASA Glenn: ideal rocket equation
Lab C: Design, build and fly a glider
Week 8
Goal: Design a card-stock or foam glider, set its CG for a positive static margin, then measure its glide ratio.
Procedure
- Sketch a design: pick a wing span and chord for an aspect ratio between 5 and 8, a tail about 20% of the wing area, and a fuselage stick.
- Build it. Find the CG by balancing it on a finger; add nose weight until the CG is about 25% of the wing chord behind the wing’s leading edge.
- Launch gently and level from a measured height in still air, at least five times; record each distance.
- Compute L/D. Then move the CG aft in small steps and record what happens.
Data table
| Trial | Height (m) | Distance (m) |
|---|---|---|
| 1 | 2.0 | 9.6 |
| 2 | 2.0 | 10.4 |
| 3 | 2.0 | 10.0 |
| Mean | 2.0 | 10.0 |
Sample data for practice, not a measured result.
Sample calculation
- Mean L/D ≈ 10.0/2.0 = 5.0.
- Glide angle = arctan(1/5) ≈ 11.3°.
- Typical finding: with the CG too far aft the glider pitches up and stalls (negative static margin); too far forward, it dives.
Safety
- Cut away from your body; an adult should handle any craft knife.
- Fly away from people’s faces, in an open space.
Home or virtual alternative
- Card stock, tape and a coin for nose weight work well.
Planet link: Gliders show how far good aerodynamics alone can take you, with no fuel at all.
Lab D: Atmosphere and stall-speed spreadsheet
Week 4
Goal: Implement NASA’s atmosphere model and see how altitude changes density, the speed of sound and stall speed.
Procedure
- Implement NASA’s troposphere and lower-stratosphere fits from 0 to 12 km in 1 km steps.
- Compute ρ, a and V_s for W = 10,791 N, S = 16.2 m², C_L,max = 1.6.
- Plot ρ/ρ₀ and V_s against altitude.
Data table
| Altitude (m) | ρ (kg/m³) | V_s (m/s) |
|---|---|---|
| 0 | 1.227 | 26.1 |
| 5,000 | 0.738 | 33.6 |
| 10,000 | 0.414 | 44.8 |
Computed from NASA’s model.
Sample calculation
- At 5,000 m: √(2 × 10,791/(0.738 × 16.2 × 1.6)) = 33.6 m/s.
- V_s grows as 1/√ρ: at 10 km it is 1.72 times its sea-level value.
Safety
- None beyond normal computer use.
Home or virtual alternative
- A spreadsheet is enough.
NASA Glenn: Earth atmosphere model (metric)
Planet link: Standard-atmosphere physics also underlies how satellites measure the air.
Exam prep practice sets
Time yourself: about 10 minutes per problem. Then compare with the full solutions.
Midterm practice set (Units 1–5)
Find ρ at 5,000 m with NASA’s model.
Show full solution
- T = −17.41 °C; p = 54.11 kPa.
- ρ = 54.11/(0.2869 × 255.69).
Answer: ≈ 0.738 kg/m³.
A 1,100 kg airplane (S = 16.2 m², C_L,max = 1.6): find its stall speed at 5,000 m.
Show full solution
- V_s = √(2 × 10,791/(0.738 × 16.2 × 1.6)).
Answer: ≈ 33.6 m/s.
Find (L/D)max for C_D0 = 0.025, e = 0.8, AR = 7.5.
Show full solution
- K = 0.0531; (L/D)max = 1/(2√(C_D0 K)).
Answer: ≈ 13.7.
A glider with L/D = 13.7 starts 1,500 m up. How far can it glide?
Show full solution
- 1,500 × 13.7.
Answer: ≈ 20.6 km.
Final practice set (all units)
Find the static margin for x_np = 0.40c̄ and x_cg = 0.25c̄.
Show full solution
- 0.40 − 0.25.
Answer: 0.15 (stable).
An engine moves 500 kg/s of air from 250 to 320 m/s. Find thrust and η_p.
Show full solution
- F = 500 × 70; η_p = 2/(1 + 1.28).
Answer: 35 kN; ≈ 0.88.
Find Δv for I_sp = 450 s and mass ratio 5.
Show full solution
- 4,413 × ln 5.
Answer: ≈ 7,102 m/s.
Find the period of a circular orbit 400 km up.
Show full solution
- T = 2π√(r³/μ) with r = 6.771 × 10⁶ m.
Answer: ≈ 92.4 min.
Find the Hohmann Δv from a 300 km orbit to geostationary radius.
Show full solution
- Δv₁ = 2,428 m/s; Δv₂ = 1,468 m/s.
Answer: ≈ 3.90 km/s.
A flight burns 20 t of fuel. How much CO₂?
Show full solution
- 20 × 3.16.
Answer: ≈ 63 t.
Formula sheet
Constants
| Constant | Value |
|---|---|
| Standard gravity g₀ | 9.80665 m/s² |
| Sea-level air density (NASA model) | 1.23 kg/m³ |
| Sea-level pressure (NASA model) | 101.3 kPa |
| Speed of sound, sea level (15 °C) | 340 m/s |
| Gas constant of air R | 287 J/(kg·K) |
| Ratio of specific heats γ (air) | 1.4 |
| Earth’s GM | 3.986 × 10¹⁴ m³/s² |
| Earth’s mean radius | 6,371 km |
Simulators and references
Free tools and references to use alongside the course.
- NASA Glenn: Beginner’s Guide to Aeronautics: NASA’s free reference for every equation in this course, with simulators.
- NASA Glenn: FoilSim: Virtual wind tunnel for airfoils (Lab A).
- MIT OpenCourseWare 16.00: Introduction to Aerospace Engineering and Design: MIT’s first-year design course.
- FAA Pilot’s Handbook of Aeronautical Knowledge: Free FAA handbook on aerodynamics, performance and stability.
- NASA JPL: Basics of Space Flight: Free primer on orbits and missions.
Glossary
- Angle of attack
- Angle between the chord line and the oncoming air.
- Aspect ratio
- Span squared divided by wing area.
- Breguet range equation
- Range of a jet from speed, fuel use, L/D and weight fraction.
- Camber
- Curvature of an airfoil’s mean line.
- Center of gravity
- Point where the weight acts.
- Drag polar
- C_D as a function of C_L.
- Dynamic pressure
- ½ρV².
- Geostationary orbit
- Equatorial orbit with a one-sidereal-day period.
- Hohmann transfer
- Two-burn elliptical transfer between circular orbits.
- Induced drag
- Drag caused by producing lift with a finite wing.
- Mach number
- Speed divided by the local speed of sound.
- Neutral point
- CG position giving neutral pitch stability.
- Propulsive efficiency
- Fraction of jet power that becomes useful thrust power.
- Reynolds number
- Ratio of inertial to viscous effects, ρVc/μ.
- Rocket equation
- Δv = I_sp g₀ ln(m₀/m_f).
- Specific impulse
- Thrust per weight flow of propellant, in seconds.
- Stall
- Loss of lift when flow separates from the wing.
- Standard atmosphere
- Average model of air properties versus altitude.
- Static margin
- Distance from CG to neutral point in mean chords.
- Sustainable aviation fuel
- Jet fuel made from wastes or biomass.
Self-assessment rubric
No grade or credit is given. Use this to judge honestly where you are in each unit.
| Level | What it looks like |
|---|---|
| 4 · Mastery | Sets up problems from forces and requirements, checks units and orders of magnitude, and explains trade-offs. |
| 3 · Proficient | Solves the practice problems with occasional slips. |
| 2 · Developing | Follows worked examples but struggles to choose the right equation. |
| 1 · Beginning | Plugs in numbers without checking units or assumptions; revisit the unit. |
Suggested weighting if you want a score
| Part | Weight |
|---|---|
| Practice problems | 25% |
| Labs | 20% |
| Midterm practice set | 25% |
| Final practice set | 30% |
Aim for level 3 or better in every unit before moving to the next course.