The laws of projectiles, or bodies projected by any impulsive force into the atmosphere, are identical with those by which the motions of bodies falling perpendicularly in free space are governed; so that when their relation is understood, a knowledge of the one necessarily leads to an acquaintance with the other. It is well known that if a body be under the influence of a single impulsive force, as a blow with a hammer, or the explosive force of gunpowder, its velocity will be uniform; that is, it will pass over equal space in equal portions of time. It is also well known that a body falling freely in space falls with an accelerated velocity, so that the spaces fallen through in successive equal portions of time, continually increase. Now, if we apply these Tacts to the case of a body projected through the air, we shall find the same laws to be preserved throughout. In the diagram, on the following page, if we suppose a body a projected horizontally, that is, in the direction a b, it would, if not acted upon by the force of gravity, proceed to describe the equal spaces a e, ef, fg, and g b, in equal successive intervals of time.
On the other hand, if we suppose the body simply to fall by its own weight, it will fall through spaces equal to e h, f i, g k, and b d, in exactly the same space of time which it would take to pass over the former spaces. Let us now suppose the two motions to be simultaneous, then the body descending as much as e h while passing from a to e, would be found at h; in passing from efit would descend as far as i, and so on till it reached the point d. In this process it will be seen that neither the horizontal nor the vertical velocity is at all affected by the action of the other. From the spaces e h, fi, g k, and b d. being as the squares of the distances a e, af, a g, ab it is shown that the curve ahikd is a parabola, which, in all cases, is the kind of curve described by bodies under the influence of two forces, such as we have been describing. The altitude to which any projectile would ascend, and the distance it would range in a vacuum are easily ascertained. Let A B be the height through which the projectile would ascend by the force impressed upon it at its outset, and A C the direction in which it is projected.
Describe a semi-circle B D A upon the line A B and where the direction A C cuts the circumference, draw the line ED perpendicular to A B, then will E D be one-fourth of the horizontal range, and E A the altitude to which it will ascend. If the horizontal range and the projectile velocity be given, the direction, so as to hit a given object, may be thus found. Take A G equal to one-fourth of A F, and draw G D perpendicularly to meet the circle, then will A D be the direction in which the projectile must be cast to strike an object at F. If the range A F and the direction AC are known, then the velocity that must be given is found by taking A G, equal to one-fourth of A F, raising the perpendicular G D, and drawing A B perpendicular to A F, till it meets D B, drawn perpendicular to A C; then will A B be the altitude due to the projectile velocity. Since there may be two perpendiculars on the semicircle of equal length, there will be two different elevations that will produce the same range; and since the radius is the longest line that can be drawn in this way, the greatest range will be when the angle of elevation of the projecting machine is 45°, or half a right angle, and in this case it will be just double the altitude due to the initial velocity.
The time which the body would occupy in its flight is always equal to the time a body would take in falling through four times the height of the parabola which it describes.
All the foregoing remarks apply only to the motion of bodies in a vacuum, and would therefore require great correction before they are applied in practice, except in particular cases. When used to regulate the discharge of large shells, or other bodies whose initial velocities do not exceed three or four hundred feet, they may be considered as tolerably accurate. But in cases of great projectile velocities, the theory is quite inadequate without several data drawn from many good experiments; for so great is the effect of the resistance of the air to projectiles of considerable velocity, that some, which, in the air, range only between two and three miles at the most, would, in vacuo, range about ten times as far, or between twenty and thirty miles.