Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/24

§ 3 into $$n$$ parts, and that each part be acted on by the force $$F$$ for $$1 / n$$ th of a second. Then the momentum communicated to each part $$= \frac{F}{n} ,$$ and the total momentum per second $$= n\ \frac{F}{n} = F ,$$ which holds good when the number of parts becomes infinite and the communication of momentum continuous. And since the communication of momentum for each of the periods of $$1 / n$$ th second is independent of the masses of the individual parts, it is in nowise essential that the $$n$$ parts are of equal mass; consequently the velocities acquired by the different parts may vary amongst themselves to any extent, without thereby affecting the total quantity of momentum communicated.

This principle in its application to fluid dynamics has sometimes been termed the Doctrine of the Continuous Communication of Momentum.

§ 4. Application of the Newtonian Method in the Case of the Normal Plane.—To illustrate the method in the case of the normal plane in motion in a region supposed filled with the medium of Newton, we must first define the mode in which the surface of the plane imparts velocity to the constituent particles. If, on the one hand, the body and the particles be supposed perfectly elastic, then the particles on colliding with the surface will bounce off with a velocity equal and opposite (relatively) to that with which they strike; that is to say, if $$V$$ be the velocity of the plane, and $$v$$ be the velocity given to the projected particles, $$v$$ will be double of $$V .$$ If, on the other hand, we suppose that the plane is inelastic, and that it eats up or absorbs the particles on impact, then the velocity imparted to them will be equal to that of the plane, or, $$v = V .$$ It is thus of little consequence which hypothesis we take, the one will give a result exactly twice as great as the other. We will select the second hypothesis, which will give the lesser value of the two.

Let us assume the medium as of the same density as air at