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time; but as it continues to act with the same force which it exerted in the first second, the space passed over must be four times 16.15 feet at the end of this second. It has then acquired an uniform velocity, that would carry it over eight times the space of 16.15 feet in the third second, if gravity ceased to act; but during the third second gravity continues to act as formerly and will carry the body over another 16.15 feet; so that at the end of the third second, it has descended through nine times 16.15 feet. But in the two first of these seconds, it had passed over four of these spaces, and in the third five of them. Hence the spaces descended through in the separate second of time, will be as the odd numbers, 126.96.36.199.9, &c. And hence as the square of 1 is 1. and the sum of 1 and 3, viz. the sum of the spaces passed over at the end of two seconds is 4, the square of two seconds; and the sum of 4 and 5, the spaces passed over at the end of three seconds, is 9, the square of three seconds; and the sum of 9 and 7, the spaces passed over at the end of four seconds, is 16, the square of four seconds, &c. It is evident that the spaces descended through at the end of any time must be as the square of the time, as we have seen before.
Now as these odd numbers grow large, they approach nearer to an equality, and consequently as they represent the spaces passed over in each separate portion of time, the motion approaches nearer to an equable motion, the longer it is continued.
Bodies projected perpendicularly upwards are retar ded by gravity in the same manner as they are accelerated by gravity in their descent. Hence they spend the same time in rising to a given height, as in falling from it. The heights to which they will rise, when projected with any given velocities will be as the squares
of the velocities, or the squares of the times of ascent. The times of ascent and descent will be equal; and so will the velocities at any particular point of the line of ascent and descent; and consequently at the place of projection and the end of the fall.*
If the descent be in a resisting medium, the motion will sooner become uniform, the denser the medium is; and as the quantity of matter in the falling body is lessened. But to understand this matter more clearly, we must next consider
THE MOTION OF BODIES IN A RESISTING MEDIUM.
THE whole doctrine of resistance in any medium may be comprised in the following articles, which may be easily understood, when considered separately.
1. If a globe move in two mediums of different den
As the spaces descended are as the squares of the times, if a heavy body fall through 16.45 feet in a second of time; in 60′′ or one minute it would fall through the square of 60 such spaces, or 3600 times 16.15 feet = 58140 feet, almost ten miles, at the surface of the earth.
But if the falling body were at the distance of the moon, or 60 semidiameters of the earth from its center, then the force of gravity there would be diminished as the square of the distance was increased, and therefore it would be 3600 times less at the distance of the moon. Consequently it would bring down a heavy body towards the earth but a 3600th part of 58140 feet in a minute, which is 16.15 feet. So that a heavy body at the distance of the moon would require a minute of time to fall as far, as it would fall in one second, at the surface of the earth. Now if the projectile force of the moon was destroyed and it was at liberty to fall towards the earth, it would fall through 16.15 feet in the first minute; and this being exactly equal to the force by which the moon is retained in her orbit, it proves that the moon is retained in her orbit by the force of gravity: as we shall see here. after.
sity, with the same velocity; it is plain, that the number of particles which it strikes in a given time must be proportional to the density of the medium, and consequently the resistance will be in the same proportion. Let R= the resistance and D= the density of the medium; then R=D.
2. Let the globe move in any resisting medium with different velocities; the quicker it moves, it strikes the greater number of particles in a given time, and it strikes every one of them with a greater force; and as action and reaction are equal and contrary, the resistance will be proportional to the velocity, on a double account; viz. on account of the number of particles and the force with which each particle is struck; that is the resistance will be proportional to the square of the velocity. R=VV.
3. Suppose two globes of different diameters, but containing the same quantity of matter, move in the resisting medium with the same velocity, they will strike every particle with the same momentum, but the number of particles struck in a given time will be proportional to the surfaces or half surfaces of the globes, and these surfaces are as the squares of the diameters: therefore the resistance in this case will be proportional to the squares of the diameters. R=dd. (d=diam.)
4. But if we consider the globes as containing different quantities of matter under equal surfaces, one being, for instance, of lead, and the other of cork, but of the same diameter; the resistance they meet with, will be inversely as their quantities of matter; because the greater that the quantity of matter is in a body moving with a given velocity, the more easily will it overcome the resistance by its greater momentum, and the less will it be retarded in the resisting medium.
Now the quantity of matter in globes of different diameters will be as the cubes of their diameters. Therefore the resistance in this case will be inversely as the
cubes of their diameters. 1 R=
5. Now if we combine these four cases together, and suppose that two globes of different diameters and densities or quantities of matter, move through resisting mediums of different densities, as air and water, and with different velocities; the whole resistance will be in the direct ratio of the density of the medium, the square of the velocity, and the square of the diameters; and in the inverse ratio of the cubes of the diameters. But, because the direct ratio of the square of the diameters combined with the inverse ratio of the cubes of the same makes only the inverse ratio of the diameters simply, the resistance will be proportional to the density of the medium and square of the velocity directly, and to the Dvvdd Dvv d
diameters of the globes inversely. R= ddd
In all that I have said on this subject, I have neglected the small resistance arising from the asperity of the moving body, or the tenacity of the fluid medium, which is subject to no certain rules of computation, and is very small in comparison with what arises from the vis inertia of the particles of the medium, which we have only considered in the foregoing articles.
From these propositions we are able to account for many phenomena of bodies in motion in a resisting medium; as why the accelerated motion of a body falling by the force of gravity through air or water becomes uniform, when the resistance of the medium, which increases with the squares of the velocity, becomes equal to the constant increment of velocity generated by gra
vity, why a bullet will go farther and faster through air, than the same weight of small shot, with a given charge of powder, as the resistance increases with the surfaces of the moving bodies, and as the surface of the small shot is much greater in proportion to its weight than that of the bullet, the shot is most resisted,-why a ball of lead will go farther, than a ball of cork, of the same diameter, with a given force of powder, through the air or in water: the resistance being universally as the quantity of matter in each, or the cubes of their diameters; whereas, if they moved in an unresisting medium, the cork ball would fly quickest and farthest,— why the motion of a body projected downwards through a resisting medium continues uniform throughout; when the velocity with which it is projected is equal to the velocity of a uniform descent in that medium; but if the projectile velocity were greater than that of a uniform descent, the motion would be retarded, and if less, it would be accelerated. Hence an arrow spends less time in rising than in falling, the latter part of its descent being either uniform or nearly so, according to the weight of the arrow. Hence also the slow descent of dust, feathers, and light bodies through the air. And lastly we see the reason why a ball shot with considerably velocity, passes through a board slightly supported, without throwing it down; because of the resistance of the air behind it: and why it might be flattened on the surface of water, without entering far into it, when projected perpendicularly to the surface; or reflected from it, when the direction is oblique to the surface.
If a body be projected upwards in a resisting medium, such as air or water, with a given velocity, it will not rise to the same height it would do in vacuo,