Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/254



§ 163. Energy Expended in Flight.—There are certain general propositions relating to the Economics of Flight that may now be demonstrated, and which are essential to the further development of our subject.

The energy expended in flight is utilised in two directions: firstly, in the renewal of the aerodynamic disturbance, or wave necessary to the support of the weight, that is, the energy expended aerodynamically; secondly, the energy expended in overcoming the direct resistance, i.e., that due to skin friction and eddy making, which varies approximately as the square of the velocity. Let the latter be denoted by the symbol $$x,$$ and let $$y$$ be the aerodynamic resistance.

Now we have seen that the aerodynamic resistance varies approximately in the inverse ratio of the velocity squared (§§ 159 and 160), for any given weight sustained, so that if we take the case of an aerodrome supporting a given load (inclusive of its own weight) we have the relation, $$y \propto \frac{1}{V^2} ,$$ and if we further assume the factors which give rise to direct resistance to undergo no change, we have, $$x \propto V^2 .$$ And the total resistance $$= x + y ,$$ the energy expended in flight per unit distance $$= x + y ,$$ and energy per unit time, or power $$= V \times (x + y).$$

'''§ 164. Minimum Energy. Two Propositions.'''—Taking the achievement of flight for granted, the problem of least energy presents itself in two forms: