PROJECTILES, Mcariox or. By this is under stood the path followed by a particle of matter projected either obliquely upward, or horizon tally from a height above the earth's surface. The problem of predicting this path was solved by Galileo, the solution depending upon the assumption that the horizontal velocity of pro jection of the particle is unaffected by the 1.er tieal force of gravity wide!' produces a constant vertical acceleration y (approximately 9S0 on the C'.13.S. system). If the particle is projected in an oblique direction upward, which makes the angle B with the horizon, with a velocity V. it will have a horizontal component Vera: 0, and a vertical one Vsiti0. The former remains unaltered; the latter is subject to a negatirc acceleration g. The particle will continue to rise until the initial vertical velocity is de creased to zero. If t is the time of ascent gt =Vsin0 or t = Vsiney. In this time the particle will have gone horizontally a dis tance Vcos0 X t or 2 - 9 After the particle reaches its highest point, it, will fall and will take the same time to reach the horizontal plane through its point of pro jection as it did to rise to the summit of its path. ]n the entire time, therefore. of rising and falling, the particle will more horizontally Vzsin20 a distance Since the time taken to Vsin0 rise to its highest point was against an acceleration g, the height of this point is ;gtz, or For a given value of V, the great Vzsin20 est distance of horizontal motion, is when sin20 has its greatest value, viz. 1 ; for this 20 = 90°, and hence = 45°. (This con clusion is seriously (modified in practice by the resisting action of the air.) The path of the particle may be deduced: if horizontal distances are called 4.e, and vertical ones y, then at a time t after projection x= tVcos0 tVsin0 If t is eliminated from these equations, = which is the equation of a parabola.
In the simplest case, when the point of projec tion is at a height above the surface of the earth, and the particle is projected horizontally with a velocity V, x =Vt, where y is vertically down.
Eliminating t, these equations 2 give x-' As a solid moves through the air, it meets opposition of various kinds due to the air. There is an opposing force which diminishes the linear speed. For speeds less than 100 feet per second the resistance of the air varies directly as the square of the velocity, as stated by New ton. According to Duchemin (1842), this resist ance = orz for speeds below 1370 feet per second, and = ea' for higher speeds. In these expressions is the speed of the projectile and a, b, c, are factors of proportionality. The first formula has been verified by the recent work of Dr. A. F. Zahm.
If the projectile is rotating on an axis, the angular speed is decreased, owing to friction; and owing to the unequal friction on the various sides, there is a sidewise force producing the 'curves' of a baseball and the 'drift' of a bullet. If the projectile is elongated or broad, the centre of pressure of the air against it and its centre of inertia are not in general in the line of motion; so there is a moment tending to make the projectile turn around an axis at right angles to the plane including the line of motion, the centre of pressure, and the centre of inertia. if the projectile is not rotating. it will turn so as to move with its broadest face front ; e.g.
penny falling in water falls face down, not edge ' down; a sheet of cardboard falling through the air tries to fall face down. lf, however, the projectile is rotating around an axis: e.g. an I elongated bullet, the effect is to change the .
direction of the axis.