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Aerospace
Level-flight speed limits for constant available thrust
The two speeds at which a constant thrust T equals the drag D = AV² + B/V², with A = ρSCD0/2 and B = 2kW²/(ρS).
- Ref
- AER-26
- Rev
- v1
- Field
- Aerospace
- Result
- Vlow [m/s]
For a constant thrust T, level flight needs D = T, and with D = AV² + B/V² this gives a quadratic in V²: AV⁴ − TV² + B = 0, with roots V² = [T ± √(T² − 4AB)]/(2A). The lower root is the slowest speed the thrust can hold, the higher root the fastest. Between them thrust exceeds drag. If T² < 4AB the thrust is below the minimum drag and there is no steady speed. Stall or a Mach limit can remove either root.
The two speeds at which a constant thrust T equals the drag D = AV² + B/V², with A = ρSCD0/2 and B = 2kW²/(ρS). Setting T = D gives a quadratic in V², with roots for the lowest and highest steady speeds. The page returns the lower speed; the upper one is in the derived rows. It needs T² ≥ 4AB; stall or a Mach limit can remove either root.
Calculator
- Upper level-flight speed63.25 m/s
- Minimum drag, 2√(AB)800 N
- Minimum-drag speed, (B/A)^¼44.72 m/s
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 this thrust the aircraft can fly level between 31.623 m/s and 63.246 m/s.
- The minimum drag of 800 N is 80 % of the available thrust; the margin T − Dmin is what supports climbing and acceleration.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| B Induced drag factor | 100,000 to 1,200,000 N·m²/s² | 10.1 | 44.72 | 34.62 | +10% | +6.76 % |
| T Constant available thrust | 800 to 3,000 N | 16.48 | 44.72 | 28.24 | +10% | −7.13 % |
| A Parasite drag factor | 0.05 to 0.3 N·s²/m² | 28.89 | 36.51 | 7.621 | +10% | +1.8 % |
Sweep of Constant available thrust
| Constant available thrust (N) | Lower level-flight speed (m/s) |
|---|---|
| 800 | 44.72 |
| 1,075 | 29.87 |
| 1,350 | 25.62 |
| 1,625 | 22.94 |
| 1,900 | 21.01 |
| 2,175 | 19.52 |
| 2,450 | 18.32 |
| 2,725 | 17.33 |
| 3,000 | 16.48 |
Sweep one variable
Variables and how each one contributes
Lower level-flight speed
What the equation gives you. Everything else feeds it.
Constant available thrust
The constant available thrust, in N. A higher thrust widens the speed range, lowering the slow root and raising the fast one. Raising it by 10 % (to 1,100) moves the result by −7.13 %, from 31.62 to 29.37 m/s.
Parasite drag factor
The parasite-drag factor ρSCD0/2, in N·s²/m². A higher value lowers the fast root, as drag at high speed is higher. Raising it by 10 % (to 0.22) moves the result by +1.8 %, from 31.62 to 32.19 m/s.
Induced drag factor
The induced-drag factor 2kW²/(ρS), in N·m²/s². A higher value raises the slow root, as drag at low speed is higher. Raising it by 10 % (to 880,000) moves the result by +6.76 %, from 31.62 to 33.76 m/s.
Worked example with the values above
- Discriminant: T² − 4AB = 1,000,000 − 640,000 = 360,000; root 600.
- Squared speeds: V² = (T ± root)/(2A) = 1,000 and 4,000 m²/s².
- Lower speed: V = 31.6228 m/s (upper 63.2456 m/s).
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