PROHIBITION 



PHOJKCTILE 



musical composition in harmony, where the key 

 continues unchanged, U called Progression ; where 

 a new key is introduced it is not progression, but 

 Modulation (q.v.). 



Prohibition. See LiguoR LAWS, and TEM- 

 PERANCE. 



Projectile is the name given to any mass 

 thrown so as to describe a path in air near the 

 earth's surface. The path descril>ed is called the 

 trajectory. The importance of the subject springs 

 from its close connection with (Junnery (q.v.). Any 

 moss projected into the air is under the action of 

 two forces: first, its weight, acting downwards and 

 practically constant; second, the resistance of the 

 air to motion through it, which resistance is a 

 function of the speed, and depends also on the 

 form, size, and mass of the projectile. 



If we consider the action of gravity alone, the 

 problem is a very simple one. Since the force 



of gravity is 

 always ver- 

 tical, there 

 can be no 

 change in 

 the value of 

 the horizon- 

 tal compon- 

 ent of the 

 velocity. 

 The projec 



Fig. 1. 



tile, projected from any point O (see fig. 1) at 

 any inclination, will some time or other reach 

 the highest point A. At this point the vertical 

 velocity will be zero ; and, if the horizontal velo- 

 city were here suddenly reversed, the projectile 

 would travel back along the same trajectory to 

 O. As it is, the projectile proceeds along the 

 path AO', which must be exactly similar to AO. 

 In short, the trajectory is symmetrical almut the 

 vertical line drawn through the highest point A. 

 Reckoning from A, let us suppose the projectile 

 to reach P' after t seconds. Then, il the hori- 

 zontal velocity is , the distance of P from the 

 vertical line A~B P'M namely is measured by the 

 product rt. But the projectile in falling through 

 the height AM has acquired a vertical velocity yt, 

 where g is the acceleration due to gravity. Thus 

 the space fallen through, being measured by the 

 product of the average speed and the time, is 



AM = W 



The trajectory is therefore a Parabola (q.v.) with 

 ii - axis vertical. 



If we suppose the projectile to be projected with 

 a velocity whose vertical and horizontal compon- 

 ent* are res|>ectively u and v, then the angle ol 

 projection has its trigonometrical tangent e(|iia! 

 to /B. The time taken to reach the highest 

 point is ii/g ; and the total range on the horizonta 

 plane is 



OO' = 2. OB = 2 rnt/g. 



If we interchange n and i< HO that the tangent ol 

 the angle of projection becomes r/n, instead of it/v 

 we get still the same range, (ienerally, then, a 

 given point, I)', can be reached by two trajectoiie- 

 with the same initial speed of projection. It is 

 easy to show that the two corresponding directions 

 of projection are equally inclined to the line tha 

 makes 45 with the horizontal ; and the rang'- i 

 greater according as the components and < * 

 the given initial velocity are less unequal in 

 magnitude. The greatest range is attained whei 

 = v = V/v'2, V licing the total velocity of pro 

 iection Le. when the angle of projection is 46" 

 In this cane the range is V'/g. Thus, to throw 

 ball to a distance of 100 yards or 300 feet it is 



eceaaary to project it with a velocity of at least 

 00 feet per second (nearly). Practically, however, 

 (fcauseof the atmospheric resistance, it would need 

 distinctly greater speed of projection than that 

 u>t given to attain tlie desired range. 



A very simple observation sulticcs to show that 

 he parabolic trajectory is only approximately 

 ealised in air. A well-driven cncket or golf ball 

 v ill be seen to a spectator suitably placed to de- 

 scribe a trajectory which U distinctly asymmetrical 

 about a vertical line through the highest imint. 

 The path will be found to be less curved during 

 he ascent than during the descent ; while the 

 lighest point is considerably nearer the end than 

 he Iteginning of the trajectory. In fig. 2 the gen- 

 iral character of a real trajectory, A It', ia compared 



Fig. 2. 



with the parabolic trajectory, AB, which would 

 lave been described if the air had offered no resist- 

 ance. AT shows the direction of the initial projec- 

 tion. The same features causing devial ion from the 

 paralxdic form are still more characteristic of the 

 long tlat trajectories of cannon-balls. These, pro- 

 jected with very high speeds, have their approxi- 

 mately horizontal velocities rapidly cut down in 

 the earlier stages by the resistance of the air. 



The first approximately accurate ideas of the 

 resistance presented by tlie air to bodies moving 

 through it at high speeds were obtained by Holiins 

 (see BALLISTIC PENDULUM). In our own times 

 Bashforth, by means of his electric chronograph, 

 has elaborately investigated the subject (see his 

 Motion of Projectiles and The Bashfnrlh C/irono- 

 graph, 1890, the authoritative treatises on this 

 branch of gunnery). Bashforth's results indicate 

 that up to velocities of from 800 to 900 feet per 

 second Newton's theoretic law that the resistance 

 varies as the square of the speed holds practically 

 true. The same law (but with a different en 

 etlicient) holds for all measured velocities abort 

 1300 feet per second ; but between the limits named 

 the resistance depends on higher powers of the 

 speed. Between the velocities of 1000 and 1100 

 feet per second the velocity of sound in air. in 

 fact the resistance grows very rapidly, varying 

 for a certain interval as the sijrlh power of the 

 velocity. The resistance also depend* on the form 

 of the projectile, a spherical M1O< being nearly 

 twice as much resisted as an ogival headed shut of 

 the same diameter ami weight. Kor different si/.cd 

 projectiles of the same form the retardation due to 

 the resistance indirectly n tin- square of the dia- 

 meter and inversely as the weight. It is usual to 

 express the diameter in inches and the weight in 

 pounds; and the following nvmbm are for an 

 ogival-headed projectile, wbOM weight in poumU 

 equals the square of its diameter in inches. 

 (its I line gi\es the NeliH'iiy ami the second the cor- 

 rc-poiiding resistance-acceleration (negative) : 



Velocity . ...1800 

 Acceleration.... SIS 



1*00 

 188 



1100 

 14 



1000 

 TO 



SCO 

 89 



400 

 10 



For a sphere of same weight and size, the resist- 

 ance-acceleration for speeds lower than 850 MM 

 per second is given by the formula 1*181 x 10 "-, 

 where i- is the. velocity. From this it may lie 

 shown that such a sphere falling in air can never 



