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solved into the two BC, CA; and the motion in ba is resolved into the iwo bd, DA ; of which the antecedents Bc. bd, are the velocities with which they directly meet, and the consequents CA, DA, are parallel ; therefore by these the bodies do not impinge on each other, and consequently the motions, according to these directions, will not be changed by the impulse ; so that the velocities with which the bodies meet, are as BC and bd, or their equals EA and Fa. The motions therefore of the bodies B, 6, directly striking each other with the velocities EA, Pa, will be determined by prop. 16 or 14, according as the bodies are elastic or non-elastic ; which being done, let AG be the velocity, so determined, of one of them, as A ; and since there remains also in the body a force of moving in the direction parallel to be, with a velocity as be, make an equal to BE, and complete the rectangle GH : then the two motions in an and Ag, or hi, are compounded into the diagonal ar, which therefore will be the path and velocity of the body B after the stroke. And after the same manner is the motion of the other body b determined after the impact.

If the elasticity of the bodies be imperfect in any given degree, then the quantity of the corresponding lines must be diminished in the same proportion.

THE LAWS OF GRAVITY; THE DESCENT OF HEAVY BODIES; AND THE MOTION OF PRO, JECTILES IN FREE SPACE.

PROPOSITION XVIN.

69. All the properties of Motion delivered in Proposition VI,

its Corollaries and Scholium. for Constant Forces, are true in the Motions of Bodies freely descending by their own Gravity ; namely, that the velocities are as the Times, and the spaces as the Squares of the Times, or as the Squares of the Velocities.

For, since the force of gravity is uniform, and constantly the same, at all places near the earth's surface, or at nearly the same distance from the centre of the earth; and since this is the force by which bodies descend to the surface; they therefore descend by a force which acts constantly and equally; consequently all the motions freely produced by gravity, are as above specified, by that proposition, &c.

SCHOLIUM. 70. Now it has been found, by numberless experiments,

that

that gravity is a force of such a nature, that all bodies, whether light or heavy, fall perpendicularly through equal spaces in the same time, abstracting from the resistance of the air; as lead or gold and a feather, which in an exhausted receiver fall from the top to the bottom in the same time. It is also found that the velocities' acquired by descending, are in the exact proportion of the times of descent: and further, that the spaces descended are proportional to the squares of the timce, and therefore to the squares of the velocities. Hence then it follows, that the weights of gravities, of bodies near the surface of the earth, are proportional to the quantities of matter contained in them ; and that the spaces, times, and velocities, generated by gravity, have the relations contained in the three general proportions before laid down. Further, as it is found, by accurate experiments, that a body in the latitude of London, falls nearly 161'1 feet in the first second of time, and consequently that at the end of that time it has acquired a velocity double, or of 32 feet by corol. I, prop. 6; therefore if g denote 164 feet, the space fallen through in one second of time, or 2g the velocity generated in that time; then, because the velocities are directly proportional to the times, and the spaces to the squares of the times ; therefore it will be,

as 1":":: 28 : 2gc = v the velocity, and 13:13:: g: gta = & the space. So that, for the descents of gravity, we have these general equations, namely, 3 =g12

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48 Hence, because the times are as the velocities, and the spaces as the squares of either, therefore,

if the times be as the numbs. 1, 2, 3, 4, 5, &c. the velocities will also be as 1, 2, 3, 4, 5, &c. and the spaces as their squares 1, 4, 9, 16, 25. &c.

and the space of each time as 1, 3, 5, 7, 9, &c. namely, as the series of the odd numbers, which are the differences of the the squares denoting the whole spaces. So that if the first series of natural numbers be seconds of time, you. II.

$

namely,

3'',

namely, the times in seconds 1, 2';

4"', &c. the velocities in feet will be

32, 64}. 964, 1283, &c. the spaces

in the whole tinies 1614.64, 1441, 257į, &c. and the space for each second 167,481, 80, 4, 1127, &c.

71. These relations, of the times, veloci." ties, and spaces, may be aptly represented

d by certain lines and geometrical figures. Thus, if the line ab denote the time of any 아 ! body's descent, and BC, at right angles to it,

C the velocity gained at the end of that time; by joining ac, and dividing the time AB into

B any number of parts at the points a, b, c; then shall ad, be, cf, parallel to bc, be the velocities at the points of time a, b, c, or at the ends of the times, Aa, ab, Ac ; because these latter lines, by similar triangles are proportional to the former ad, be, cf

, and the times are propore tional to the velocities. Also, the area of the triangle ABC will represent the space decended by the force of gravity in the time AB, in which it generates the velocity BC; because that area is equal to JAB X BC, and the space descended is 8 =

- jtv, or half the product of the time and the last . velocity. And, for the same reason, the less triangles aad,

Abe, Acf, will represent the several spaces described in the corresponding times Aa, ab, ac, and velocities ad, be, cf ; those triangles or spaces being also as the squares of their like sides aa, ab, Ac, which represent the times, or of ad, bez cf, which represent the velocities.

72. But as arcas are rather unnatural

Pa bod representations of the spaces passed over hF by a body in motion, which are lines, the

il relations may better be represented by the abscisses and ordinates of a parabola. h 제 Thus, if pa be a parabola, PR its axis, and Rq its ordinate; and pa, rb, pc, &c. parallel to RQ, represent the times from R

Q the beginning, or the velocities, then aė, bf, cg, &c. parallel to the axis pr, will represent the spaces described by a fall. ing body in those times ; for, in a parabola, the abscisses ph, pi, pk, &c. or ae, bf, cg, &c. which are the spaces described, are as the squares of the ordinates he, if, ky, &c. or pa, pbo PC, &c. which represent the times or velocities.

73. And because the laws for the destruction of motion,

are

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are the same as those for the generation of it, by equal forces,
but acting in a contrary direction; therefore,
ist

, A body thrown directly upward, with any velocity vill lose equal velocities in equal times.

2d, If a body be projected upward, with the velocity it acquired in any time by descending freely, it will lose all its velocity in an equal time, and will ascend just to the same height from which it fell, and will describe equal spaces in equal times, in rising and falling, but in an inverse order ; and it will have equal velocities at any one and the same point of the line described, both in ascending and descending.

3d, If bodies be projocted upward, with any velocities, the height ascended to, will be as the squares of those velocities, or as the squares of the times of ascending, till they lose all their velocities.

74. To illustrate now the rules for the natural descent of bodies by a few examples, let it be required,

lat, To find the space descended by a body in 7 seconds of time, and the volocity acquired.

Ans. 788, space ; and 225} velocity, 2d, To find the time of generating a velocity of 100 feet per second, and the whole space descended.

Ans. 311 's time ; 155,155 space. 3d, To find the time of descending 400 feet, and the velocity at the end of that time.

Ans. 4"7 time ; and 1604velocity.

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PROPOSITION XIX.

*75. Ifa Body be projected in Free Space either Parallel to the

Horizon, or in an Oblique Direction, by the Force of Gun.
Powder, or any other Impulse ; it will by this Motion, in
Conjunction with the Action of Gravity describe the Curve
Line of a Parabola.

2 I

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A I

1 Let the body be projected from the point a, in the direction sd, with any uniform velocity: then, in any equal portions of time, it would by prop. 4, describe the equal spaces AB, BC, CD, &c. in the line ad, if it were not drawn continually down below that line by the action of gravity. Draw BE, CF, DG &c. in the direction of gravity, or perpendicular to the horizon, and equal to the spaces through which the body would descend by its gravity in the same time in which it would uniformly pass over the corresponding spaces AB, AC, AD, &c. by the projectile motion. Then, since by these two motions the body is carried over the space AB, in the same time as over the space BE, and the space ac in the same time as the space cf, and the space ad in the same time as the space Dc, &c; therefore, by the composi. tion of motions, at the end of those times, the body will be found respectively in the points E, F, G, &c; and consequently the real path of the projectile will be the curve line ABFG &c. But the spaces AB, AC, AD, &c. described by uniform motion, are as the times of description; and the spaces BE, CF, DG, &c. described in the same times by the accelerating force of gravity, are as the squares of the times ; consequently the perpendicular descents are as the squares of the spaces in Ad, that is BE, C8, DG, &c, are respectively proportional to AB”, AC3, AD', &c; which is the property of the parabola by theor. 8, Con. Sect. Therefore the path of the projectile is the parabolic line ALFG &c, to which ad is a tangent at the point A.

portions

on, are

76. Corol. 1. The horizontal velocity of a projectile, is always the same constant quantity, in every point of the curve ; because the horizontal motion is in a constant ratio to the motion in AD, which is the uniform projectile motion. And the projectile velocity is in proportion to the constant horizontal velocity, as radius to the cosine of the angle DAM, or angle of elevation or depression of the piece above or below the horizontal line an.

77. Corol. 2. The velocity of the projectile in the direction of the curve, or of its tangent at any point A is as the secant of its angle Bal of direction above the horizon. For the motion in the horizontal direction as is constant, and as is to AB, as radius to the secant of the angle A ; therefore the motion at a, in AB, is every vhere as the secant of the angle A.

78. Corol. 3. The velocity in the direction dg of gravity, or perpendicular to the horizon , at any point G of the curve, is to the first uniform projectile velocity at A, or point of contact of a tangent, as 260 is to AD. For, the times in AD and og being equal, and the velocity acquired by freely de

scending

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