PROJECTILES, are such bodies as, be ing 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 grant ed, 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 up on very heavy bodies, such as bombs and cannon-balls, is too small to be taken into consideration.
Sir Isaac Newton has shown, that the gravity of bodies which are above the su perficies 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 Gallileo, and 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 gra vity's acting uniformly, and in parallel lines, must be exceedingly small ; be cause even the greatest distance of a pro jectile above the surface of the earth, is inconsiderable, in comparison of its dis. tance from the centre, to which the gra vitation tends. But then, on the other hand, it is very certain, that the resist ance of the air to very swift motions, is much greater than it has been commonly represented. Nevertheless, (in the appli cation of this doctrine to gunnery) if the amplitude of the projection, answering to one given elevation, be first found by ex periment (which we suppose) the ampli tudes in all other cases, where the eleva tions and velocities do not very much dif fer 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 eleva tions, 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 i every projectile is to be considered as depending on two different forces ; that is to say; on the impellant force, whereby the motion is first begun, (and would be continued in a right line) and on the force of gravity, by which the projectile, during the whole time of its flight, is con tinually urged downwards, and made to deviate more and more from its first di rection. As whatever relates to the track and flight of a projectile, or ball, (neg lecting the resistance of the air) is to be determined from the action of these two forces, it will be proper, before we ceed to consider their joint effects, to premise something concerning the nature of the motion produced by each, when supposed to act alone, independently of the other ; to which end we have premis• ed the two following lemmata.
Lemma I. Every body, after the im pressed force whereby it is put in mo tion ceases to act, continues to move uniformly in a right line ; unless it be interrupted by some other force or im pediment.
This is a law of nature, and has its de monstration 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 direc, tion is changed, and the motion inter, rupteci, Lemma II. The motion, or velocity, acquired by a ball, in freely descending from rest, by the force of an uniform gra vity., is as the time of the ; and the space fallen through, as the square of that time.
The first part of this lemma is ex tremely obvious : for since every motion is proportional to the force whereby it is generated, that generated by the force of an uniform gravity must be as the time of the descent ; because the whole effort of such a force is proportional to the time P —, A of its action ; that is, as the time of the descent.