PROJECTILES, THEORY OF, is the investigation of the path or trajectory, as it is called, of a body which is projected into space in a direction inclined to that of gravita tion. A body thus projected is acted upon by two forces, the force of projection, which, if acting alone, would carry the body onward forever in the same direction and at the same rate; and the force of gravity, which tends to draw the body downward toward the earth. The force of projection acts only'at the commencement of the body's motion; the force of gravity, on the contrary, continues to act effectively during the whole time of the body's motion, drawing it further and further from its original direction, and causing it to describe a curved path, which, if the body moved in a vacuum, would be accurately a parabola. This is readily seen by considering Fig. 1, in which A represents the point from which the body is projected (suppose the embrasure of a fort); AB the direction of projection (horizontal in this instance); Al the distance which would be passed over by the projectile in unit of time if gravity did not act; '1-2, the distance which would similarly be described in second unit of time; 2-3, 3-4. etc., the distances corresponding to the third, fourth, etc., units of time—all these distances being necessarily equal, from the impulsive nature of the force of projection; Al', again, represents the dis tance which the projectile would fall 'under the action of gravity alone in the first unit of time; 1'-2' the distance due to gravity in the second unit of time; 2'-3' the distance due to the third unit, etc., the distances Al', A2', A3', etc., being in the proportion of 1, 4, 9, ttc. (see FALLING BODIES); hence, by the well-known principle of the composition of forces end velocities (q.v.), we find at once, by completing• the series of parallelograms, that at the end of the first unit of time the body is at c, at the end of the second at b, at the end of the third at e, etc. Now, as the lines l'c, 2'b, 3'e, etc., increase as the numbers 1, 2, etc., and the lines Al', A2', A3', etc., as the numbers it follows that the curve. Acts is a parabola (q.v.). As, by the second law of motion, each force produces its full undisturbed by the other, it follows that the projectile reaches f in the same time as it would, without being projected, have taken to fall to 4'. A greater velocity of projection would make it take a wider flight; but at the end of four seconds, it must still be at some point in the same horizontal line—at g, for example.
In order to determine exactly the motion of a projectile, and to find its range, greatest altitude, and time of flight, it will be necessary to examine its nature more technically— for which some slight knowledge of algebrtrand trigonometry is requisite. Let the body in this instance be projected obliquely to the direction of gravity, from the point A (Fig.
2) in the direction AT, and let the velocity of projection v be sufficient, if gravity were not to act, to carry it to T in t units of time, and let the force of gravity, if allowed to act upon it. at rest, carry it to G in the same time; then, as before, the body, under the action of both forces, will be found at P (which is found by completing a parallelogram of which AT and AG are the sides) at the end of t units of time, having fallen through a distance equal to TP (not at once, but in a constant succession of minute deflections, as indicated in Fig. 1) in that time! Let t represent
the time of flight, v the velocity due to projection,. g the accelerating force of gravity, and let A be. the angle of elevation TAB; then AT .= TP = '10, TM = at sin. A; and consequently PM (or y) = vt sin. A— (I.), and AM (or x) = vt cos. A. Now, if we find from the last. two equations the values of t, and equate these vahtes, we obtain, by an easy alge braic process, the equation y = x tan.
A and if the height through which the body must fall to acquire a velocity equal to the velocity of projection be called Is, then = 2g.h, is = 4/4 = , and 1 g a' = p, substituting which in the equation, we obtain y = a tan. (II.), as the equation to the path of a projectile, where x is the horizontal distance, and y the corresponding height above the level of the point of projection. Suppose, now, that we wish to find the time of flight on the horizontal plane, it is evident that at the end of its flight the projectile will be at B, and y will be equal to zero; hence, putting y = o in equation I., we obtain t = 2v sin. The range or distance AB is similarly found by g putting y = o in equation II., when a is found to be equal to 414 sin. A cos. A, or 214 sin.
2A. The greatest altitude is evidently the point which the projectile has attained at the end of half the time of flight, or after it has traversed half its horizontal range; hence, by putting x = 2h sin. A cos. A in equation II., or t = ro sin. A in equation I., we obtain y = is A slight examination of the expression for the range will show that it is greatest when the body is projected upward at an angle of 45° to the horizon, and that a. body projected at a greater angle than 45° has the same range as one projected at an angle correspondingly less (Fig. 3).
These results, however., do not correspond to the actual circumstances of the case, except when the projectile possesses considerable density and its motion is slow, for in all other cases, the resistance of the air, which increases in a rapid ratio with the velocity of the projectile, causes it to deviate very considerably from a parabolic orbit, especially