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Aerospace
Thrust required versus true airspeed
Drag, and so thrust required, in steady level unaccelerated flight with lift equal to weight W.
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- AER-25
- Rev
- v1
- Field
- Aerospace
- Result
- D [N]
With lift equal to weight, the lift coefficient is CL = 2W/(ρSV²), and the drag is D = ½ρSV²(CD0 + kCL²) = ½ρSCD0V² + 2kW²/(ρSV²). The first term, parasite drag, rises with V². The second, induced drag, falls with V². Their sum is least at the speed where the two are equal, and the curve rises steeply on both sides, with the left side leading to stall.
Drag, and so thrust required, in steady level unaccelerated flight with lift equal to weight W. The first term is zero-lift drag and rises with V²; the second is lift-dependent drag and falls with V². Constant density and the parabolic polar; restrict V to the valid flight envelope.
Calculator
- Parasite drag, ½ρSCD0V²480 N
- Induced drag, 2kW²/(ρSV²)270 N
- Lift coefficient, 2W/(ρSV²)0.5
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 50 m/s the aircraft needs 750 N of thrust, with 64 % of the drag being parasite and the rest induced.
- The minimum drag of 720 N occurs at 43.301 m/s, where both terms are equal.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| W Weight | 2,000 to 100,000 N | 487.5 | 19,230 | 18,740 | +10% | +7.56 % |
| V True airspeed | 15 to 150 m/s | 721.3 | 4,350 | 3,629 | +10% | +7.19 % |
| S Wing area | 5 to 100 m² | 721.2 | 3,043 | 2,322 | +10% | +3.13 % |
| CD0 Zero-lift drag coefficient | 0.01 to 0.08 | 510 | 2,190 | 1,680 | +10% | +6.4 % |
| ρ Air density | 0.3 to 1.3 kg/m³ | 720 | 1,200 | 480 | +10% | +3.13 % |
| k Induced drag factor | 0.02 to 0.1 | 600 | 1,080 | 480 | +10% | +3.6 % |
Sweep of True airspeed
| True airspeed (m/s) | Drag (thrust required) (N) |
|---|---|
| 15 | 3,043 |
| 31.88 | 859.4 |
| 48.75 | 740.3 |
| 65.63 | 983.6 |
| 82.5 | 1,406 |
| 99.38 | 1,964 |
| 116.3 | 2,645 |
| 133.1 | 3,441 |
| 150 | 4,350 |
Sweep one variable
Variables and how each one contributes
Drag (thrust required)
What the equation gives you. Everything else feeds it.
Air density
The air density, in kg/m³. Raises parasite drag and lowers induced drag. Raising it by 10 % (to 1.32) moves the result by +3.13 %, from 750 to 773.5 N.
Wing area
The wing area, in m². Raises parasite drag and lowers induced drag in direct and inverse proportion. Raising it by 10 % (to 17.6) moves the result by +3.13 %, from 750 to 773.5 N.
Zero-lift drag coefficient
The zero-lift drag coefficient. Raises the parasite drag term in direct proportion. Raising it by 10 % (to 0.022) moves the result by +6.4 %, from 750 to 798 N.
True airspeed
The true airspeed, in m/s. Raises parasite drag with V² and lowers induced drag with 1/V². Raising it by 10 % (to 55) moves the result by +7.19 %, from 750 to 803.9 N.
Induced drag factor
The induced drag factor. Raises the induced drag term in direct proportion. Raising it by 10 % (to 0.0495) moves the result by +3.6 %, from 750 to 777 N.
Weight
The aircraft weight, in N. Raises induced drag with the square of the weight. Raising it by 10 % (to 13,200) moves the result by +7.56 %, from 750 to 806.7 N.
Worked example with the values above
- Parasite: ½ρSCD0V² = 0.5 × 1.2 × 16 × 0.02 × 2,500 = 480 N.
- Induced: 2kW²/(ρSV²) = 1.296×107 / 48,000 = 270 N.
- D = 480 + 270 = 750 N.
Assumptions and limits
- A parabolic polar CD = CD0 + kCL², constant over the speed range and with no compressibility rise.
- Steady level, unaccelerated flight with lift equal to weight, at constant density.
Common mistakes
- Using a polar from a different configuration, such as flaps or gear down.
- Forgetting stall and Mach limits on the mathematical solution.
- Mixing up the minimum-drag speed and the minimum-power speed.
Statistics and uncertainty
The parabolic drag polar fits real aircraft well only over the middle of the lift range. Speeds from it are usually accurate to 5 % when CD0 and k come from flight tests; with handbook estimates they can be off by 10 % to 20 %, and the result moves with the one-quarter power of the drag ratio.
Where it is used
- Plotting thrust and power curves and finding best speeds for range and endurance.
- Estimating how stall speed changes in turns.
Maths you need
- Drag polar: CD = CD0 + kCL²
- Level flight: L = W and T = D
- Load factor: n = L/W