PROJECTILES, are such bodies as, being put in a violent motion by any great force, are then cast off or let go from the place where they received their quantity of motion: as a stone thrown from a sling, an arrow from a bow, a bullet from a gun, &c.
It is usually taken for granted, by those who treat of the motion of projectiles, that the force of gravity near the earth's surface is every where the same, and acts in parallel directions; and that the effect of the air's resistance upon very heavy bodies, such as bombs and cannon-balls, is too small to be taken into consideration.
The famous Sir Isaac Newton has shown, that the gravity of bodies which are above the superficies of the earth, is reciprocally as the squares of their distances from its centre; but the theorems concerning the descent of heavy bodies, demonstrated by Galileus, Huygens, and others, are built upon this foundation, that the action of gravity is the same at all distances; and the consequences of this hypothesis are found to be very nearly agreeable to experience. For it is obvious, that the error arising from the supposition of gravity's acting uniformly, and in parallel lines, must
be exceeding small; because even the greatest distance of a projectile above the surface of the earth, is inconsiderable in comparison of the distance from the centre to which the gravitation tends. But then, on the other hand, it is very certain, that the resistance of the air to very swift motions, is much greater than it has been commonly represented. Nevertheless, (in the application of this doctrine to gunnery), if the amplitude of the projection, answering to one given elevation, be first found by experiment, (which we suppose), the amplitudes in all other cases, where the elevations and velocities do not very much differ from the first, may be determined, to a sufficient degree of exactness, from the foregoing hypothesis: because, in all such cases, the effects of the resistance will be nearly as the amplitudes themselves; and were they accurately so, the proportions of the amplitudes, at different elevations, would then be the very same as in vacuo.
Now, in order to form a clear idea of the subject here proposed, the path of every projectile is to be considered as depending on two different forces: that is to say, on the impellent force, whereby the motion is first begun, (and would be continued in a right line); and
Projectiles. and on the force of gravity, by which the projectile, during the whole time of its flight, is continually urged downwards, and made to deviate more and more from its first direction. As whatever relates to the track and flight of a projectile or ball (neglecting the resistance of the air) is to be determined from the action of these two forces, it will be proper, before we proceed to consider their joint effect, to premise something concerning the nature of the motion produced by each, when supposed to act alone, independent of the other; to which end we have premised the two following lemmata.
Lem. I. Every body, after the impressed force whereby it is put in motion ceases to act, continues to move uniformly in a right line; unless it be interrupted by some other force or impediment.
This is a law of nature, and has its demonstration from experience and matter of fact.
Corollary. It follows from hence, that a ball, after leaving the mouth of the piece, would continue to move along the line of its first direction, and describe spaces therein proportional to the times of their description, were it not for the action of gravity, whereby the direction is changed, and the motion interrupted.