VELOCITY

, or Swistness, in Mechanics, is that affection of motion, by which a moving body passes over a certain space in a certain time. It is also called celerity; and it is always proportional to the space moved over in a given time, when the Velocity is uniform, or always the same during that time.

Velocity is either uniform or variable. Uniform, or equal Velocity, is that with which a body passes always over equal spaces in equal times. And it is variable, or unequal, when the spaces passed over in equal times are unequal; in which case it is either accelerated or retarded Velocity; and this acceleration, or retardation, may also be equal or unequal, i. e. uniform or variable, &c. See Acceleration, and Motion.

Velocity is also either absolute or relative. Absolute Velocity is that we have hitherto been considering, in which the Velocity of a body is considered simply in itself, or as passing over a certain space in a certain time. But relative or respective Velocity, is that with which bodies approach to, or recede from one another, whether they both move, or one of them be at rest. Thus, if one body move with the absolute Velocity of 2 feet per second, and another with that of 6 feet per second; then if they move directly towards each other, the relative velocity with which they approach is that of 8 feet per second; but if they move both the same way, so that the latter overtake the former, then the relative Velocity with which that overtakes it, is only that of 4 feet per second, or only half of the former; and consequently it will take double the time of the former before they come in contact together.

Velocity in a Right Line.—When a body moves with a uniform Velocity, the spaces passed over by it, in different times, are proportional to the times; also the spaces described by two different uniform Velocities, in the same time, are proportional to the Velocities; and consequently, when both times and Velocities are unequal, the spaces described are in the compound ratio of the times and Velocities. That is, S <*> TV, and s <*> tv; or S : s :: TV : tv. Hence also, V : v :: S/T : s/t, or the Velocity is as the space directly and the time reciprocally.

But in uniformly accelerated motions; the last degree of Velocity uniformly gained by a body in beginning from rest, is proportional to the time; and the space described from the beginning of the motion, is as the product of the time and Velocity, or as the square of the Velocity, or as the square of the time. That is, in uniformly accelerated motions, v t, and s tv or v² or t². And, in fluxions, s^. = vt^..

Velocity of Bodies moving in Curves.—According to Galileo's system of the fall of heavy bodies, which is now universally admitted among philosophers, the Velocities of a body falling vertically are, at each moment of its fall, as the square roots of the heights from whence it has fallen; reckoning from the beginning of the descent. And hence he inferred, that if a body descend along an inclined plane, the Velocities it has, at the different times, will be in the same ratio: for since its Velocity is all owing to its fall, and it only falls as much as there is perpendicular height in the inclined plane, the Velocity should be still measured by that height, the same as if the fall were vertical.

The same principle led him also to conclude, that if a body fall through several contiguous inclined planes, making any angles with each other, much like a stick when broken, the Velocity would still be regulated after the same manner, by the vertical heights of the different planes taken together, considering the last Velocity as the same that the body would acquire by a fall through the same perpendicular height.

This conclusion it seems continued to be acquiesced in, till the year 1672, when it was demonstrated to be false, by James Gregory, in a small piece of his intitled Tentamina quædam Geometrica de Motu Penduli & Projectorum. This piece has been very little known, because it was only added to the end of an obscure and pseudonymous piece of his, then published, to expose the errors and vanity of Mr. Sinclair, professor of natural philosophy at Glasgow. This little jeu d'esprit of Gregory is intitled, The great and new Art of Weighing Vanity: or a discovery of the Ignorance and Arrogance of the great and new Artist, in his Pseudo-Philosophical writings: by M. Patrick Mathers, Arch-Bedal to the University of S. Andrews. In the Tentamina, Gregory shews what the real Velocity is, which a body acquires by descending down two contiguous inclined planes, forming an obtuse angle, and that it is different from the Velocity a body acquires by descending perpendicularly through the same height; also that the Velocity in quitting the first plane, is to that with which it enters the second, and in this latter direction, as radius to the cosine of the angle of inclination between the two planes.

This conclusion however, Gregory observes, does not apply to the motions of descent down any curve lines, because the contiguous parts of curve lines do not form any angle between them, and consequently no part of the Velocity is lost by passing from one part of the curve to the other; and hence he infers, that the Velocities acquired in descending down a continued curve line, are the same as by falling perpendicularly through the same height. This principle is then applied, by the author, to the motion of pendulums and projectiles.

Varignon too, in the year 1693, followed in the same track, shewing that the Velocity lost in passing from one right lined direction to another, becomes indefinitely small in the course of a curve line; and that therefore the doctrine of Galileo holds good for the descent of bodies down a curve line, viz, that the Velocity | acquired at any point of the curve, is equal to that which would be acquired by a fall through the same perpendicular altitude.

The nature of every curve is abundantly determined by the ratio of the ordinates to the corresponding abscisses; and the essence of curves in general may be conceived as consisting in this ratio, which may be varied in a thousand different ways. But this same ratio will be also that of two simple Velocities, by whose joint effect a body may describe the curve in question; and consequently the essence of all curves, in general, is the same thing as the concourse or combination of all the forces which, taken two by two, may move the same body. Thus we have a most simple and general equation of all possible curves, and of all possible Velocities. By means of this equation, as soon as the two simple Velocities of a body are known, the curve resulting from them is immediately determined.

It may be observed, in particular, according to this equation, that an uniform Velocity, combined with a Velocity that always varies as the square roots of the heights, the two produce the particular curve of a parabola, independent of the angle made by the directions of the two forces that give the Velocities; and consequently a cannon ball, shot either horizontally or obliquely to the horizon, must always describe a parabola, were it not for the resistance of the air.

Circular Velocity. See Circular.

Initial Velocity, in Gunnery, denotes the Velocity with which military projectiles issue from the mouth of the piece by which they are discharged. This, it is now known, is much more considerable than was formerly apprehended. For the method of estimating it; and the result of a variety of experiments, by Mr. Robins, and myself, &c, see the articles Gun, GUNNERY, Projectile, and Resistance.

Mr. Robins had hinted in his New Principles of of Gunnery, at another method of measuring the Initial Velocities of military projectiles, viz, from the arc of vibration of the gun itself, in the act of expulsion, when it is suspended by an axis like a pendulum. And Mr. Thompson, in his experiments (Philos. Trans. vol. 71, p. 229) has pursued the same idea at considerable length, in a number of experiments, from whence he deduces a rule for computing the Velocity, which is somewhat different from that of Mr. Robins, but which agrees very well with his own experiments.

This rule however being drawn only from the experiments with a musket barrel, and with a small charge of powder, and besides being different from that in the theory as proposed by Robins; it was suspected that it would not hold good when applied to cannon, or other large pieces of ordnance, of different and various lengths, and to larger charges of powder. For this reason, a great multitude of experiments, as related in my Tracts, vol. 1, were instituted with cannon of various lengths and charged with many different quantities of powder; and the Initial Velocities of the shot were computed both from the vibration of a ballistic pendulum, and from the vibration of the gun itself; but the consequence was, that these two hardly ever agreed together, and in many cases they differed by almost 400 feet per second in the Velocity. A brief abstract for a comparison between these two methods, is contained in the following tablet, viz.

In this table, the first column shews the number of the gun, as they were of different lengths; viz, the length of number 1 was 30 1/3 inches, number 2 was 40 1/3 inches, number 3 was 60 inches, and number 4 was 83 inches, nearly. After the first column, the rest of the table is divided into four spaces, for the four charges, 2, 4, 8, 16 ounces of powder: and each of these is divided into three columns: in the first of the three is the Velocity of the ball as determined from the vibration of the gun; in the second is the Velocity as determined from the vibration of the pendulum; and in the third is the difference between the two, being so many feet per second, which is marked with the nega- tive sign, or —, when the former Velocity is too little, otherwise it is positive.

From the comparison contained in this table, it appears, in general, that the Velocities, determined by the two different ways, do not agree together; and that therefore the method of determining the Velocity of the ball from the recoil of the gun, is not generally true, although Mr. Robins and Mr. Thompson had suspected it to be so: and consequently that the effect of the inflamed powder on the recoil of the gun, is not exactly the same when it is fired without a ball, as when it is fired with one. It also appears, that this difference is no ways regular, neither in the different | guns with the same charge of powder, nor in the same gun with different charges: That with very small charges, the Velocity by the gun is greater than that by the pendulum; but that the latter always gains upon the former, as the charge is increased, and soon becomes equal to it; and afterwards goes on to exceed it more and more: That the particular charge, at which the two Velocities become equal, is different in the different guns; and that this charge is less, or the equality sooner takes place, as the gun is longer. And all this, whether we use the actual Velocity with which the ball strikes the pendulum, or the same increased by the Velocity lost by the resistance of the air, in its flight from the gun to the pendulum.