Exercise set

Aircraft Stall Speed, Load Factor, and Margin Exercises

Worked aircraft exercises for stall speed, load factor, bank angle, equivalent airspeed, flaps, gusts, icing, AoA bias and release margins.

Branch: Aerospace Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on aircraft stall speed and operating margins: lift coefficient limits, load factor, bank angle, equivalent airspeed, flap configuration, gusts, contamination, angle-of-attack bias and release gates. Aerodynamic coefficient and drag-polar practice is handled in a separate specialist exercise set.

Use the calculations as release screens. Real flight decisions need configuration control, air-data validation, calibrated weight and CG, flight-test evidence, icing/contamination rules, pilot procedures and guarded uncertainty margins.

How to use these exercises

Use the set as an aircraft operating-margin review. Exercises 1 to 4 establish baseline stall speed, load-factor effects and banked-turn stall speed. Exercises 5 to 9 check equivalent airspeed, approach margin, flap effect, hot-and-high true speed and contamination penalty. Exercises 10 to 17 add gust loads, manoeuvring speed, V-n stall boundary, AoA bias, climb margin, wing loading and guarded uncertainty. Exercise 18 combines the gates into a release decision.

Before calculating, state weight, CG, configuration, airspeed type, density condition, load-factor case, contamination state, AoA sensor source and procedure limit. Stall margin is not a single speed: it changes with load factor, flap setting, density, contamination, bias and uncertainty. The engineering comment below each exercise identifies which operating boundary must be verified before release.

Release Evidence Notes

Stall-margin evidence must state weight, configuration, load factor, reference speed type, protection threshold, sensor source and uncertainty allowance. A safe margin in one configuration can disappear after icing, flap changes, bank angle or sensor bias.

The evidence package should separate aerodynamic capability, air-data validity and operational procedure. Aerodynamic capability covers $C_{L,max}$ , flap state, contamination and load factor. Air-data validity covers indicated, calibrated, equivalent and true airspeed plus AoA calibration. Operational procedure covers approach speed, bank limits, turbulence, go-around climb margin and restrictions.

Release evidence should also identify the limiting case. The lowest stall speed may not be the release driver if a contaminated wing, hot-and-high operation, banked turn, gust case or negative climb margin controls the operation.

Engineering Boundary Notes

These exercises use simplified steady-flight relationships. Dynamic stall, buffet onset, high-lift hysteresis, unsteady gust response, propeller slipstream and flight-control blending require dedicated validation. Treat pass results as screening evidence for the stated configuration, not as approval for every loading or weather condition.

The main boundary is speed definition. Stall, loads and runway performance may require different speed references, so the record should state whether a value is indicated, calibrated, equivalent or true. The second boundary is uncertainty: small margins over guarded stall speed should be treated as operating restrictions or retest triggers.

Common Release Mistakes

mixing true airspeed and equivalent airspeed in stall checks;
applying clean-wing $C_{L,max}$ after contamination;
checking level-flight stall but not banked-turn load factor;
ignoring AoA sensor bias in envelope-protection thresholds;
treating approach speed margin as valid after flap or weight changes;
accepting a stall gate without uncertainty and procedure evidence.

Another common mistake is treating stall margin separately from climb margin. Contamination, density altitude and configuration can simultaneously raise stall speed and reduce climb or go-around capability. Both gates should pass for an unrestricted release.

Do not use a nominal AoA warning threshold without bias and failure-mode review. A sensor can trigger early, late or inconsistently depending on calibration, local flow, ice, sideslip and voting logic.

Scenario Map

Scenario	Main calculation	Release decision
Baseline stall	weight, wing area and $C_{L,max}$	Establish reference speed.
Load factor	bank, maneuver and gust	Guard accelerated stall.
Configuration	flap, contamination and weight	Approve or restrict operation.
Air-data margin	equivalent speed and AoA bias	Validate protection threshold.
Release gate	combined margins and uncertainty	Fly, restrict or hold release.

Validation Package Checklist

weight, wing area, configuration and $C_{L,max}$ source;
airspeed type: indicated, calibrated, equivalent or true;
load-factor case, bank angle and gust assumption;
contamination, icing and high-lift configuration evidence;
AoA sensor calibration, bias and envelope-protection threshold;
guarded release margin and operational restriction.
climb or go-around margin where contamination or density altitude applies;
flight-test, simulation or handbook source for each protected condition;
release status states fly, restrict, retest, derate or hold.

A complete validation package should make the operating decision reproducible. Another engineer should be able to see which configuration was protected, which speed reference was used, which uncertainty was guarded and which restriction applies when a margin fails.

Exercise 1: Baseline Stall Speed

An aircraft weighs $W=24000\ \text{N}$ , wing area is $S=16\ \text{m}^2$ , density is $1.225\ \text{kg/m}^3$ and $C_{L,max}=1.55$ . Compute stall speed.

Solution

V_s=\sqrt{\dfrac{2W}{\rho S C_{L,max}}}

V_s=\sqrt{\dfrac{2(24000)}{(1.225)(16)(1.55)}}=39.8\ \text{m/s}

Engineering Comment

This is a clean calculation only if weight, area, density and configuration match the release condition.

Plausibility Check

A light aircraft stall speed around $40\ \text{m/s}$ is reasonable.

Exercise 2: Load-Factor Stall Speed

Using $V_s=39.8\ \text{m/s}$ , compute accelerated stall speed at load factor $n=2.0$ .

Solution

V_{s,n}=V_s\sqrt{n}=39.8\sqrt{2.0}=56.3\ \text{m/s}

Engineering Comment

Maneuvering and banked flight can move stall speed into the operating envelope.

Plausibility Check

Doubling load factor increases stall speed by $\sqrt{2}$ , not by two.

Exercise 3: Bank Angle Load Factor

For a coordinated level turn at bank angle $\phi=45^\circ$ , compute load factor:

n=\dfrac{1}{\cos\phi}

Solution

n=\dfrac{1}{\cos45^\circ}=1.414

Engineering Comment

Bank-angle stall margin belongs in procedures, displays and envelope-protection logic.

Plausibility Check

A $45^\circ$ turn increases load factor by about $41\%$ .

Exercise 4: Banked-Turn Stall Speed

Using $V_s=39.8\ \text{m/s}$ and $n=1.414$ , compute banked-turn stall speed.

Solution

V_{s,\phi}=39.8\sqrt{1.414}=47.3\ \text{m/s}

Engineering Comment

The stall warning threshold should account for banked turns, not only wings-level flight.

Plausibility Check

The result is higher than baseline but below the $2g$ stall speed.

Exercise 5: Equivalent Airspeed at Altitude

At altitude, true airspeed is $82\ \text{m/s}$ and density ratio is $\sigma=0.62$ . Compute equivalent airspeed:

V_E=V_T\sqrt{\sigma}

Exercise 10: Gust Load Factor

Cruise load factor is $1.0$ . A vertical gust increment is estimated as $\Delta n=0.42$ . Compute total positive load factor.

Solution

n=1.0+0.42=1.42

Engineering Comment

Gust load can combine with maneuver load; release checks should define the combined case.

Plausibility Check

The gust adds less than half a g.

Exercise 11: Gust-Adjusted Stall Speed

Solution

Effective true warning angle:

\alpha_{warn,true}=13.5-0.9=12.6^\circ

Margin to stall:

m=15.0-12.6=2.4^\circ

Since $2.4^\circ>1.0^\circ$ , it passes.

Engineering Comment

Bias sign matters. A bias can make protection trigger too early or too late.

Plausibility Check

Positive indicated bias means true angle is lower than indicated at warning.

Exercise 15: Drag Penalty and Climb Margin

Contamination adds $12\%$ drag. Available climb excess thrust margin is $9\%$ before contamination. Does climb margin remain positive?

Solution

Net margin:

m=9\%-12\%=-3\%

The climb margin becomes negative.

Engineering Comment

Stall margin and climb margin must both pass for dispatch or test release.

Plausibility Check

The drag penalty is larger than the original excess margin.

Exercise 16: Wing-Loading Effect

Aircraft A has wing loading $W/S=1500\ \text{N/m}^2$ and Aircraft B has $1800\ \text{N/m}^2$ . With the same density and $C_{L,max}$ , compute ratio $V_{s,B}/V_{s,A}$ .

Solution

\dfrac{V_{s,B}}{V_{s,A}}=\sqrt{\dfrac{1800}{1500}}=\sqrt{1.2}=1.095

Aircraft B has about $9.5\%$ higher stall speed.

Engineering Comment

Wing loading is a compact way to compare stall and takeoff/landing tendency across configurations.

Plausibility Check

The speed ratio is the square root of the loading ratio.

Exercise 17: Guarded Stall-Speed Uncertainty

Computed stall speed is $47.4\ \text{m/s}$ . Airspeed uncertainty is $1.2\ \text{m/s}$ and model uncertainty allowance is $1.8\ \text{m/s}$ . Compute guarded stall speed by direct addition.

Solution

V_{guard}=47.4+1.2+1.8=50.4\ \text{m/s}

Engineering Comment

Use a conservative guard for release gates unless the uncertainty combination method is justified.

Plausibility Check

The guarded value is $3.0\ \text{m/s}$ above the computed value.

Exercise 18: Aircraft Stall-Margin Release Gate

A test card has approach speed $52\ \text{m/s}$ , guarded stall speed $50.4\ \text{m/s}$ , AoA warning margin $2.4^\circ$ against $1.0^\circ$ required, contamination climb margin $-3\%$ and banked-turn stall speed $47.3\ \text{m/s}$ . Decide release status.

Solution

Speed margin over guarded stall is:

m_V=52-50.4=1.6\ \text{m/s}

AoA warning passes and banked-turn stall is below approach speed, but climb margin fails:

-3\%<0

Release should be held or restricted until contamination drag and climb margin are resolved.

Engineering Comment

Stall margin alone is not enough if the same contamination invalidates climb or go-around performance.

Plausibility Check

One hard operational gate is negative, so the release cannot be unconditional.

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