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2,400 lb light airplane from its drag polar: (L/D)max, top speed, best climb, ceilings, Breguet range and endurance, take-off and landing over 50 ft
| Tool | Aircraft Performance |
| Source | Anderson, J. D. (1999). Aircraft Performance and Design. McGraw-Hill, New York. Chapters 5 and 6 (polar closed forms, best climb, Breguet, ground roll with the thrust at 0.7 V_LO); Raymer, D. P. (2018). Aircraft Design: A Conceptual Approach, 6th ed. AIAA, Reston, VA. Chapter 17 (take-off and landing segments); Gasich, W. E. (1947). Experimental Verification of Two Methods for Computing the Take-Off Ground Run of Propeller-Driven Aircraft. NACA TN 1258 (the ground-run integral, eq. 9); Federal Aviation Administration (2023). Pilot's Handbook of Aeronautical Knowledge, FAA-H-8083-25C, Chapter 11 (weight, wind and altitude effects reproduced). |
| Level | Undergraduate |
Problem
A light airplane of 2,400 lb with 160 ft² of wing has the drag polar C_D = 0.025 + K C_L² with e = 0.8 and AR = 7.4, C_Lmax 1.6 clean, 1.8 with take-off flaps and 2.0 with landing flaps, a 180 hp normally aspirated engine driving a propeller at a constant efficiency of 0.80 with a BSFC of 0.45 lb/hp-h, and 240 lb of fuel. At sea level find (L/D)max and the characteristic speeds, the top speed, the best rate and angle of climb, the absolute and service ceilings, the Breguet range and endurance, and the take-off and landing distances over a 50 ft obstacle on dry concrete (mu_r 0.02, mu_b 0.40, C_L 0.3 in the ground roll, Delta C_D0 0.02 take-off and 0.04 landing, k_LO 1.1, k_TD 1.15, 1 s rotation, 2 s free roll, 3 deg approach, no wind).
Inputs to Enter
- W 2,400 lb, S 160 ft², C_D0 0.025, e 0.8, AR 7.4; C_Lmax 1.6 / 1.8 / 2.0 (clean / take-off / landing)
- Propeller: P_sh 180 hp, eta 0.80 constant, Gagg-Ferrar power lapse, BSFC 0.45 lb/hp-h, W_f 240 lb; altitude 0 ft
- Runway: mu_r 0.02, mu_b 0.40, h_obs 50 ft, C_L ground roll 0.3, Delta C_D0 take-off 0.02, C_L braking 0, Delta C_D0 landing 0.04, no reverse thrust, k_LO 1.1, k_TD 1.15, t_R 1 s, t_fr 2 s, gamma_a 3 deg, V_w 0
Answers
- K = 1/(pi e AR) = 0.05377; C_L at (L/D)max = sqrt(C_D0/K) = 0.682, (L/D)max = 1/(2 sqrt(C_D0 K)) = 13.64; V_s 52.6 kt, V_md 80.6 kt (D_min 176 lb), V_mp 61.3 kt (P_R,min 38.2 hp)
- Power available 144 hp: top speed 147.0 kt (248 ft/s) where P_R = P_A; minimum level speed = V_s
- Best rate of climb 1,455 ft/min at 61.3 kt (= V_mp, closed form eta P/W − P_R,min/W; Anderson's 0.8776 form 1,454.7); best angle 15.7 deg at the stall speed
- Absolute ceiling 24,180 ft, service ceiling (100 ft/min) 22,200 ft by the altitude sweep with the Gagg-Ferrar lapse
- Breguet range (eta/c)(L/D)max ln(W0/W1) = 832.4 nmi at (L/D)max; endurance 12.09 h at minimum power (c = 2.2727e-7 /ft)
- Take-off: V_LO 54.6 kt, thrust at 0.7 V_LO 1,228 lb, ground roll 276.5 ft (Anderson's 0.7 V_LO estimate 276.2 ft), rotation 92 ft, transition 341 ft (R 1,441 ft, gamma 13.7 deg), climb 38 ft: 746.9 ft over 50 ft
- Landing: approach 915 ft, flare 78 ft, free roll 183 ft, braking roll 308 ft (V_TD 54.1 kt): 1,484 ft from 50 ft
- SI: V_max 75.6 m/s, (R/C)max 7.39 m/s, ceilings 7,371 / 6,766 m, range 1,542 km, take-off 227.6 m, landing 452.2 m
Notes
The tool's default inputs are this airplane; press Submit. The case is the tool's own default (the brief's light airplane), not an example printed in a book: every number is checked in test_aircraft_performance.php (192 checks) against the closed forms of Anderson's chapters and against an independent scipy construction (brentq, minimize_scalar, solve_ivp of the ground-roll equation of motion, quad over the fuel burn; References/Aircraft-Performance/checks/perf_check.py), which agrees to 1e-8. With a constant propeller efficiency the take-off thrust eta P/(0.7 V_LO) = 1,228 lb is optimistic; the efficiency table option gives a realistic static thrust.