A Theoretical Analysis of the Airborne Path during Take‐off: A Linear Method for Performance Estimation

The solution of the problem of estimating the take‐off distance to a height of 50 feet has to a certain extent been limited by the absence of a theoretical analysis of the airborne part of the take‐off manoeuvre. The three main physical quantities associated with the motion immediately after an aircraft leaves the ground are aircraft speed, the angle the flight path makes with the horizontal and the lift coefficient increment. This latter quantity is the lift coefficient in excess of that required for level flight at the unstick speed, and is produced when the pilot pulls the stick quickly back at take‐off. A linear theoretical analysis is obtained by assuming that variations of the physical quantities already mentioned are small enough for squares and higher powers of such variations to be neglected in comparison with the variations themselves. The results of the analysis depend on the solutions of a pair of ordinary simultaneous linear differential equations with constant coefficients. If the aircraft speed never falls below the unstick speed, the limiting values of the lift coefficient increment which define the safe range of take‐offs can be determined. By considering the mean value of the lift coefficient increment over the safe range of take‐offs it is possible to define a mean safe take‐off, and for such a take‐off, the mean safe airborne distance from the unstick point to a height of 50 feet can be estimated. The application of the theory as a means of estimating the take‐off performance of a bomber aircraft is given as an example at the end of this work.