Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/23

Rh the momentum generated in unit time is proportional to the force. This method is best studied in connection with a hypothetical medium suggested by Newton, on which he based several of the problems in the "Principia." This medium is defined as consisting of a large number of material particles, equally distributed in space, having no sensible magnitude, but possessing mass; the particles are not supposed to act upon or be connected to each other in any way. Bodies traversing a region tilled with this medium experience a resistance which is proportional to the momentum communicated per second, and is a quantity that can be calculated mathematically, provided that the velocity of the body and the density of the medium be known, and the surface in presentation of the body be defined.

The employment of this method and its deficiencies in the case of a real fluid are illustrated in the case of the normal plane (Chap. V.), where it is found to give a considerably greater pressure value than actually obtains; the general form of the pressure law is, however, in approximate accord.

§ 3. The Newtonian Method.—Employing absolute units, let $$F =$$ the resistance, let $$m =$$ mass acted upon during time $$t ,$$ and $$v$$ the velocity in the line of motion imparted to the mass $$m ;$$ then the fundamental equation is: $$F = \frac{m\ v}{t} .$$

Now so long as we are dealing with a simple body of mass $$m ,$$ and imparting to it a velocity $$v ,$$ the above equation is merely a statement of the law of motion cited, any constant being eliminated by the fact that we are employing absolute units. The equation, however, holds good whatever the number of parts into which the mass be divided, and however the velocities of the different parts vary amongst themselves. In this case the expression may be written: $$F = \frac{\Sigma\ (m\ v)}{t} .$$ The proof is as follows:—

Let us suppose that the mass acted on per second be divided Rh