we learn that in order to find the space or height s through which a heavy body falls in a number t of seconds, we have only to multiply the square of this number of seconds byg, that is, by the space described in the first second.

Hence, the height or number of feet through which a heavy body falls during a number t of seconds is so many times 16,1 feet as there are units in the square of this number of seconds.

Thus, when a body has been suffered to fall freely during 7 seconds, we may be assured that it has passed through a space equal to 49 times 16,1 feet, or 788,9 feet. We see, therefore, that when, in the case of falling bodies, the time elapsed is known, nothing is more easy than to determine the velocity acquired, and the space described.

276. If the question were to find the time employed by a body in falling from a known height, the equations=g t2,

that is, we seek how many times the height s contains g, or 16,1 feet, the space described by a body in the first second of its fall, and take the square root of this number.

277. If we would know from what height a heavy body must fall to acquire a given velocity, that is, a velocity by which a certain number of feet is uniformly described in a second; from the equation v = gt, I deduce the value of t, namely, t =

stituting this value in the equation sgt2, I have

g

; sub

by which I learn, that in order to find the height s from which a heavy body must fall to acquire a velocity v, of a certain number of feet in a second, the square of this number of feet is to be divided by double the velocity acquired by a heavy body in one second, that is, by 64,4.

Thus, if I would know, for example, from what height a heavy body must fall, to acquire a velocity of 100 feet in a second, I divide the square of 100, namely, 10000, by 64,4; and the quotient 1155,2 &c., is the height through which a body must fall to acquire a velocity of 100 feet in a second.

We might evidently make use of the same formula in determining to what height a body would rise, when projected vertically upward with a known velocity.

Moreover, from the above equation, s =

v2 = 2gs, or v = √2gs

2

2 g

we obtain

that is, the velocity acquired in falling through any space s, is equal to 2 gs, or equal to eight times the square root of s nearly, v, g, and s, being estimated in feet. Thus the velocity acquired in falling through 1 mile or 5280 feet, is equal to

8,024 、 5200 = 583 feet very nearly.

278. By these examples it will be seen that all the circumstances of the motion of heavy bodies may be easily determined; and it is accordingly to these motions, that we commonly refer all others; so that instead of giving immediately the velocity of a body, we often give the height from which it must fall to acquire this velocity. Occasions will be furnished for examples hereafter.

We will merely observe, therefore, by way of recapitulation, that all the circumstances of accelerated motion, and consequently of the motion of heavy bodies, are comprehended in the two equations gt, sgt2; so that, g being known, and one v=g = of the three things, t, s, v, or the time, space, and velocity, the two others may always be found, either immediately by one or the other of the above equations, or by means of both combined after the manner of article 277.

279. When a body is subjected to the action of a force that is exerted upon it without interruption, but in a different manner at each successive instant, we give to the motion the general denomination of varied. We have examples of varied motion in the unbending of springs; although in this case the velocity goes

on increasing, still the degrees by which it increases go on diminishing. The same may be observed with respect to the degrees by which the motion of a ship arrives at uniformity; the action of the wind upon the sails diminishes according as the ship acquires motion, because it is withdrawn so much the more from this action, according as it has more velocity.

280. The principles necessary for determining the circumstances of this kind of motion are easily deduced from the principles that we have laid down with regard to uniform motion, and motion uniformly accelerated.

(1.) In whatever manner motion is varied, if we consider it with respect to instants infinitely small, we may suppose that the velocity does not change during the lapse of one of these instants. Now, when the motion is uniform, the velocity has for its expression the space described during any time t, divided by this time. Accordingly, when the motion is uniform only for an instant the velocity must have for its expression the infinitely small space described during this instant divided by this instant. Hence, if represents the space described, in the case of a variable motion, during any time t, ds will represent the space uniformly described during the instant dt; we have, therefore,

as the first fundamental equation of varied motion.

281. In the equation v=gt, we have understood by g the velocity which the accelerating force is capable of giving to a body in a determinate time, as one second, by an action that is supposed to continue constantly the same. In the equation

the same thing is to be understood. But we must observe that the accelerating force being supposed to be variable, the quantity g which represents the velocity that the accelerating force is capable of producing, if it were constant for one second, this quantity g, I say, is different for all the different instants of the motion. Indeed, it will be readily conceived, that when the accelerating force becomes less, the velocity that it is capable

of generating in a second by its action repeated equally during each instant of this second, must be less, and vice versâ.

282. From the two equations ds = v dt, dv = g dt, we can obtain a third that may be employed with advantage. Thus from the equation ds = v dt, we deduce d t = ds; substituting

v

this value instead of dt in the equation dog dt, we have,

283. We remark, that in the process by which we have just arrived at the equation d v = g dt, we regarded the velocity as increasing. If it had gone on diminishing, it would have been necessary, instead of dv to put do; so that the two equations dogd t, and g ds = v dv, to become general, must be written

± dv = gdt,

± gds = v dv,

the upper sign being used when the motion is accelerated, and the lower when the motion is retarded.

284. There is a fourth equation that may be deduced from the two fundamental equations, and which should not be omitted. Thus, the equation ds = vd t gives v = d; whence we obtain

ds

t

substituting this value for dv in the equation g d t = ± dv, we have

If we suppose, as we are authorized to do, that d t is constant,

we shall have,

dds

gdt = ±

or gd tdds.

d t

But it must be recollected that, in the equation g dt2 = ±dds, it is supposed that d t is constant. When d t is variable, we make use of the equation

Occasions will occur in which these formulas will be of great use. But we must not forget that the quantity g which they contain, represents, for each instant, the velocity which the accelerating force is capable of giving to the moving body in a known interval of time, as one second, if during the second, it were to act with a uniformly accelerating force; so that as each quantity g measures for each instant, the effect of which the accelerating force is capable, we shall give it, for brevity's sake, the name of accelerating force.

Of the direct Collision of Bodies.

285. We suppose, in what follows, that no account is taken of the gravity of bodies, of friction, or other resistance.

We suppose also that the bodies, whose collision is the subject of consideration, act the one upon the other according to the same straight line, passing through their centres of gravity, and that this straight line, is perpendicular to the plane touching their surfaces at the point where they meet.

We shall consider bodies as divided into two classes, denominated unelastic and elastic; the former are supposed to be such that no force can change their figure; the latter are regarded as capable of having their figure changed, that is, of being compressed, but as endued at the same time with a property by which this figure is resumed after the compressing force is removed.

Although there are not in nature bodies of a sensible mass, that answer perfectly to either of these descriptions, yet it is only by proceeding upon such suppositions, that we are able to determine the action of such bodies as are actually presented to our observation.