B+b both in the direction BC, when v and v are both positive, or the bodies both moved towards ċ before the collision. Eut if v be negative, or the body b moved in the contrary direction before collision, or towards B; then, changing the sign of v, the same theorems become ∞ = y = (Bb) v2bv (Bb) v + 2BV the velocity of B, the veloc. of b, in the direction BC. " B+b And if b were at rest before the impact, making its velocity v = 0, the same theorems give B-b x= v, B+b 2B v, the velocities in this case. And, in this case, if the two bodies в and & be equal to each other; then в — b = 0, and 2B 2B = = 1; which B+b ZB give r = 0, and y=v; that is, the body B will stand still, and the other body b will move on with the whole velocity of the former; a thing which we sometimes see happen in playing at billiards; and which would happen much oftener if the balls were perfectly elastic. PROPOSITION XVII. 68. If Bodies strike one another Obliquely, it is proposed to deter mine their Motions after the Strike. LET the two bodies в, b, move in the oblique directions BA, bA, and strike each other at A, with velocities which are in proportion to the lines BA, ba; to find their motions after the impact. Let CAH represent the plane in which the bodies touch in the point of E G concourse; to which draw the perpendiculars BC, bn, and complete the rectangles CE, DF. Then the motion in BA is re solved into the two BC, CA; and the motion in ba is resolved into the two bD, DA; of which the antecedents вc, 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 в, b, directly striking each other with the velocities EA, FA, 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 AH equal to BE, and complete the rectangle GH: then the two motions in AH and AG, or HI, are compounded into the diagonal AI, 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 PROJECTILES IN FREE SPACE. PROPOSITION XVIII. 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. 770. 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 times, and therefore to the squares of the velocities. Hence then it follows, that the weights or 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 16 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. 1, prop. 6; therefore, if g denote 16 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"t": 2g: 2gtv the velocity, and 12::g : gt2 =s the space. So that, for the descents of gravity, we have these general equations, namely, Hence, because the times are as the velocities, and the spaces as the squares of either, therefore, 1, 2, 3, 4, 5, &c, if the times be as the numbs. the velocities will also be as and the spaces as their squares 1, 4, 9, 16, 25, &c, and the space for each time as 1, 3, 5, 7, 9, &c, namely, as the series of the odd numbers, which are the differences of the squares denoting the whole spaces. that if the first series of natural numbers be seconds of time, namely, So namely, the times in seconds the velocities in feet will be the spaces in the whole times and the space for each second 1", 2", 3", 4", &c, 32, 64, 96, 128, &c, 16,641, 1441, 257, &c, 16, 48, 80, 11272, &c, b 71. These relations, of the times, velocities, and spaces, may be aptly represented by certain lines and geometrical figures. Thus, if the line AB denote the time of any body's descent, and BC, at right angles to it, the velocity gained at the end of that time by joining Ac, and dividing the time AB into 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 proportional to the velocities. Also, the area of the triangle ABC will represent the space descended by the force of gravity in the time AB, in which it generates the velocity BC; because that area is equal to AB X BC, and the space descended is s tv, 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, be, cf, which represent the velocities. h 1 k P 72. But as areas are rather unnatural representations of the spaces passed over by a body in motion, which are lines, the relations may. better be represented by the abscisses and ordinates of a parabola. Thus, if Po be a parabola, PR its axis, and RQ its ordinate; and pa, rb, Pc, &c, parallel to Ro, represent the times from the beginning, or the velocities, then ae, bf, cg, &c, parallel R 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, kg, &c, or pa, rb, Pc, &c, which represent the times or velocities. 73. And because the laws for the destruction of motion, are are the same as those for the generation of it, by equal forces, but acting in a contrary direction; therefore, 1st, A body thrown directly upward, with any velocity, will 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 projected upward, with any velocities, the height accended 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, 1st, To find the space descended by a body in 7 seconds of time, and the velocity acquired. Ans. 788 space; and 2254 velocity, 2d, To find the time of generating a velocity of 100 feet per second, and the whole space descended. 95 Ans. 3" time; 155 space, 193 34, To find the time of descending 400 feet, and the velocity at the end of that time. Ans. 4" time; and 160 velocity, PROPOSITION XIX. 75. If a Body be projected in Free Space, either Parallel to the Horizon, or in an Cblique Direction, by the Force of GunPowder, or any other Impulse; it will, by this Motion, in Conjunction with the Action of Gravity, describe the Curve Line of a Parabola. LET the body be projected from the point A, in the direction AD, with any uniform velocity: then, in any equal portions |