Minimum drag and minimum power flight conditions

Aerodynamic conditions, speeds and drag at minimum drag (maximum L/D), minimum power and the Carson speed for a parabolic drag polar.

Drafted with Aria, reviewed by the AiCanCode.org team. Spotted an error? Use Give Feedback at the bottom of the page.

Why it matters

Two speeds on the drag curve keep coming back in performance: the speed of minimum drag (maximum L/D) and the speed of minimum power. They decide the best glide, the maximum endurance and range of jets and propeller aircraft, the best climb angle, and where the "back side" of the drag curve begins. Knowing the aerodynamic conditions at each — C_D0 against induced drag — lets you write them down in one line for any aircraft with a parabolic drag polar.

Key ideas

Starting point. Steady level flight, L = W, T = D, parabolic polar C_D = C_D0 + K·C_L². At a given weight, drag is D = W·C_D/C_L, so minimum drag means maximum C_L/C_D. Power required is P = D·V, and since V ∝ 1/√C_L for L = W, minimum power means maximum C_L^(3/2)/C_D.

Minimum drag (maximum L/D). Minimising C_D/C_L = C_D0/C_L + K·C_L with respect to C_L gives

  • induced drag = parasite drag: K·C_L² = C_D0;
  • C_L,md = √(C_D0/K), C_D,md = 2·C_D0;
  • (L/D)_max = 1/(2·√(K·C_D0)) — independent of altitude and weight;
  • D_min = W/(L/D)_max = 2·W·√(K·C_D0). The speed V_md = √(2W/(ρS))·(K/C_D0)^(1/4) grows with √(W/S) and with 1/√σ. In EAS it is independent of altitude. On a graph, the minimum-drag point is where a line from the origin is tangent to the C_L–C_D polar, and on the power-required curve it is where a line from the origin is tangent to the P_R–V curve (because D = P/V is the slope of that line).

Minimum power (maximum C_L^(3/2)/C_D). Minimising C_D/C_L^(3/2) gives

  • induced drag = three times parasite drag: K·C_L² = 3·C_D0;
  • C_L,mp = √(3·C_D0/K) = √3·C_L,md, C_D,mp = 4·C_D0;
  • V_mp = 3^(−1/4)·V_md ≈ 0.760·V_md;
  • (L/D)_mp = (√3/2)·(L/D)_max ≈ 0.866·(L/D)_max, so the drag at V_mp is 1.155 times D_min. The aircraft flies slower than V_md but with more drag; the product D·V is nevertheless smallest there.

Maximum V/D (Carson speed). Minimising D/V, the drag per unit speed, gives K·C_L² = C_D0/3 and V = 3^(1/4)·V_md ≈ 1.316·V_md. It is the best range speed of a constant-TSFC jet and a "most speed for the least fuel" compromise for propeller aircraft. (Its use is covered in the range topics.)

Which matters for what (idealised engines).

  • Maximum L/D: best glide range; best range of a propeller aircraft (constant BSFC and η_p); maximum endurance of a jet (constant TSFC).
  • Minimum power: minimum sink rate in gliding; maximum endurance of a propeller aircraft; maximum rate of climb for a propeller aircraft (at low altitude, roughly).
  • Maximum C_L^(1/2)/C_D (Carson speed): best range of a jet.

Validity. All results assume a parabolic polar with constant C_D0 and K (no Mach or Reynolds effects). V_mp can fall close to or below the stall speed for aircraft with low C_D0; then the practical limit is the stall, not the formula.

Formulas

C_L,md = √(C_D0 / K), (L/D)_max = 1 / (2·√(K·C_D0))

V_md = √(2W / (ρS)) · (K / C_D0)^(1/4)

D_min = 2·W·√(K·C_D0)

C_L,mp = √(3·C_D0 / K), C_D,mp = 4·C_D0

(C_L^(3/2) / C_D)_max = (1/4)·(3 / (K·C_D0^(1/3)))^(3/4)

V_mp = 3^(−1/4)·V_md = 0.760·V_md

P_R,min = D_mp·V_mp = (2/√3)·D_min·V_mp

P_R = √(2·W³·C_D² / (ρ·S·C_L³))

Symbols: C_D0 zero-lift drag coefficient; K = 1/(π·e·AR) induced-drag factor; C_L, C_D lift and drag coefficients; W weight (N); S wing area (m²); ρ density (kg/m³); V true airspeed (m/s); D drag (N); P_R power required (W); subscripts md = minimum drag, mp = minimum power. Valid for steady level subsonic flight with a parabolic polar.

Worked examples

Example 1 (standard): minimum-drag and minimum-power points at sea level. Given: W = 50 000 N, S = 30 m², C_D = 0.025 + 0.045·C_L², ρ = 1.225 kg/m³.

  1. C_L,md = √(C_D0/K) = √(0.025/0.045) = 0.745; (L/D)_max = 1/(2√(0.045 × 0.025)) = 14.91.
  2. D_min = W/(L/D)_max = 50 000/14.91 = 3354 N.
  3. V_md = √(2W/(ρS))·(K/C_D0)^(1/4) = 52.16 × 1.158 = 60.4 m/s.
  4. C_L,mp = √3 × 0.745 = 1.291; C_D,mp = 4 × 0.025 = 0.100; L/D = 12.91, so D_mp = 50 000/12.91 = 3873 N.
  5. V_mp = 0.760 × 60.42 = 45.9 m/s.
  6. P_R,min = D_mp·V_mp = 3873 × 45.91 = 177.8 kW, compared with D_min·V_md = 202.7 kW at the minimum-drag speed.

Example 2 (GATE level): from (L/D)_max to minimum power at altitude. Given: an aircraft with (L/D)_max = 16 and C_D0 = 0.02; W = 60 000 N, S = 28 m², flying at 3 km where ρ = 0.9091 kg/m³.

  1. From (L/D)_max = 1/(2√(K·C_D0)): K = 1/(4 × 16² × 0.02) = 0.04883.
  2. C_L,mp = √(3 × 0.02/0.04883) = 1.109; C_D,mp = 0.08; (L/D)_mp = 1.109/0.08 = 13.86 (= 0.866 × 16).
  3. V_mp = √(2W/(ρ·S·C_L)) = √(2 × 60 000/(0.9091 × 28 × 1.109)) = 65.2 m/s.
  4. D_mp = W/(L/D)_mp = 60 000/13.86 = 4330 N.
  5. P_R,min = 4330 × 65.21 = 282.4 kW. Check: V_md = 65.21/0.760 = 85.8 m/s and D_min = 60 000/16 = 3750 N, so P at V_md = 321.8 kW — larger, as expected.

Common mistakes

  • Writing C_D = 2·C_D0 at minimum power. At minimum power C_D = 4·C_D0; at minimum drag it is 2·C_D0.
  • Putting V_mp above V_md. Minimum-power speed is slower (0.76 V_md).
  • Thinking (L/D)_max changes with altitude or weight. Only the speed at which it occurs changes.
  • Using the factor 3^(1/4) for V_mp. It is 3^(−1/4) for minimum power and 3^(+1/4) for the Carson speed.
  • Ignoring the stall: if V_mp < V_stall, the aircraft cannot fly the "theoretical" minimum-power speed.

For GATE AE

This is one of the most examined pieces of performance: expect questions asking for C_L, C_D, L/D or speed at minimum drag or minimum power from a given polar, ratio questions (V_mp/V_md, D_mp/D_min, induced-to-parasite drag ratio), and inverse problems where (L/D)_max is given and K or C_D0 must be found. Practise deriving the conditions by differentiation so you can handle variants such as maximum C_L^(1/2)/C_D.

Quick check

  1. At minimum drag, what is the ratio of induced to parasite drag?
  2. At minimum power, what is C_D in terms of C_D0?
  3. What is V_mp/V_md for a parabolic polar?
  4. If C_D0 = 0.02 and K = 0.05, what is (L/D)_max?

Answers: 1. 1. 2. C_D = 4·C_D0. 3. 3^(−1/4) ≈ 0.760. 4. 1/(2√0.001) = 15.8.

Interview questions

All Aircraft Performance interview questions

Try answering each one aloud before you open it.

  1. 1.What is the aerodynamic condition for minimum drag in steady level flight?Concept

    With a parabolic polar C_D = C_D0 + K·C_L², drag at fixed weight is D = W·C_D/C_L, so minimum drag means maximum L/D. Differentiating gives K·C_L² = C_D0: induced drag equals parasite drag. Then C_L = √(C_D0/K), C_D = 2C_D0 and (L/D)_max = 1/(2√(K·C_D0)). The minimum drag itself is W/(L/D)_max, independent of altitude.

  2. 2.Why is the minimum-power speed lower than the minimum-drag speed?Concept

    Power is drag times speed. Just below V_md the drag rises only slowly (the curve is flat at its minimum) while the speed falls in proportion, so the product D·V keeps falling. It reaches a minimum where induced drag is three times parasite drag, at V_mp = 3^(−1/4)·V_md ≈ 0.76·V_md. There the drag is about 15 % above D_min, but the lower speed more than compensates.

  3. 3.Which flight conditions are optimised at maximum L/D and which at minimum power?Concept

    Maximum L/D (minimum drag) gives the flattest glide and best glide range, the maximum endurance of a jet with constant TSFC, and the best range of a propeller aircraft with constant BSFC and propeller efficiency. Minimum power gives the minimum sink rate in a glide and the maximum endurance of a propeller aircraft. The Carson speed, 3^(1/4)·V_md, maximises V/D and gives the best range of an idealised jet.

  4. 4.Does (L/D)_max change with altitude or weight?Concept

    No. (L/D)_max = 1/(2√(K·C_D0)) depends only on the drag polar, i.e. on the aircraft's shape and configuration (and, at high speed, on Mach number). Altitude and weight change only the speed at which it is reached: V_md ∝ √(W/(ρS)). In EAS the minimum-drag speed is the same at every altitude for a given weight.

  5. 5.An aircraft has C_D0 = 0.02 and K = 0.05. Find (L/D)_max and the lift coefficient at which it occurs.Concept

    (L/D)_max = 1/(2√(K·C_D0)) = 1/(2√(0.05 × 0.02)) = 1/(2 × 0.03162) = 15.8. It occurs at C_L = √(C_D0/K) = √0.4 = 0.632, where C_D = 2 × 0.02 = 0.04. A quick check: 0.632/0.04 = 15.8.

  6. 6.How would you find the minimum-drag point graphically from a power-required curve?Concept

    Drag is D = P/V, which is the slope of a straight line from the origin to a point on the P_R–V curve. The minimum drag is therefore at the point where a line from the origin is tangent to the P_R curve. The lowest point of the curve itself is the minimum-power condition. On a C_L–C_D polar, the tangent from the origin likewise marks (L/D)_max.

  7. 7.What practical limit can prevent an aircraft from flying at its theoretical minimum-power speed?Concept

    V_mp = 0.76·V_md can lie close to or below the stall speed, especially for clean aircraft with low C_D0 or at high weight, so the real minimum is set by the stall margin. It is also on the back side of the drag curve, where speed is unstable: a small speed loss increases drag and the aircraft slows further unless the pilot adds power. In practice endurance speeds are flown slightly faster than the theoretical value.

Finished this topic? Mark it so your progress, study plan and readiness keep up.

Stuck on something here?

PreviousThrust and power available for jet and propeller aircraftNext Steady climb and descent: rate of climb and ceilings