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be the sum or difference of the momentums before the stroke; namely, the momentum in direction Bc will be
Bv + bv, if the bodies moved the same way, or
Then divide each momentum by the common mass of matter B + b, and the quotient will be the common velocity after the stroke in the direction Bc ; namely, the common velocity will be, in the first case
64. For example, if the bodies, or weights, B and b, be as 5 to 3, and their velocities v and v, as 6 to 4, or as 3 to 2, before the stroke; then 15 and 6 will be as their momentums, and 8 the sum of their weights; consequently, after the stroke, the common velocity will be as
15 + 6 21 - -
8 8 - -
† - - - - - or 14 in the third.
65. If two Perfectly Elastic Bodies impinge on one another; their Relative Velocity will be the same both Before and After the Impulse: that is, they will recede from each other with the Same Velocity with which they approached and met.
For the compressing force is as the intensity of the stroke; which, in given bodies, is as the relative velocity with which they meet or strike. But perfectly elastic bodies restore themselves to their former figure, by the same force by which they were compressed; that is, the restoring force is equal to the compressing force, or to the force with which the bodies approach each other before the impulse. But the bodies are impelled from each other by this restoring force; and therefore this force, acting on the same bodies, will produce a relative velocity equal to that which they had before: or it will make the bodies recede from each other with the
Vol. II. I. sanne
“revelocity with which they before approached, or so as to be equally distant from one another at equal times before and after the impact.
66. Remark. It is not meant by this proposition, that each body will have the same velocity after the impulse as it had before ; for that will be varied according to the relation of the masses of the two bodies; but that the velocity of the one will be, after the stroke, as much increased as that of the other is decreased, in one and the same direction. So, if the elastic body B move with a velocity v, and overtake the elastic body b moving the same way with the velocity v : then their relative velocity, or that with which they strike, is v — v, and it is with this same velocity that they separate from each other aster the stroke. But if they meet each other, or the body b move contrary to the body B; then they meet and strike with the velocity v + v, and it is with the same velocity that they separate and recede from each other after the stroke. But whether they move forward or backward after the impulse, and with what particular velocities, are circumstances that depend on the various masses and velocities of the bodies before the stroke, and which make the subject of the next proposition.
LET the elastic body B move in a the direction Bc, with the velocity B b C
w; and let the velocity of the other body b be v in the same line; which latter velocity v will be positive if b move the same way as B, but negative if b move in the opposite direction to B. Then their relative velocity in the direction Bc is v — v. 5 also the momenta before the stroke are Bv and bw, the sum of which is BV -- bu in the direction BC. Again, put x for the velocity of B, and y for that of b, in the same direction BC, after the stroke; then their relative velocity is y – «, and the sum of their momenta Bz--by in the same direction. But the momenta before and after the collision, estimated in the same direction, are equal, by prop. 10, as also the relative velocities, by the last prop. Whence arise these two
#####". the velocity of b, both in the direction Bc, when v and v are both positive, or the bodies both moved towards c before the collision. But 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 - – 9
* = *H*, the velocity of B, _ (B — b) v + 2B v
y = B + b And if b were at rest before the impact, making its velocity w = 0, the same theorems give B — b B + 6
And, in this case, if the two bodies B and b be equal to each other; then B — b = 0, and #, - # = 1 ; which 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.
, the veloc. of b, in the direction pc,
68. If Bidies strike one another Obliquely, it is proposed to deter
mine their Motions after the Stryke.
LET the two bodies B, 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 concourse; to which draw the perpendiculars B2, bp, and complete the rectanglesce, DF. Then the motion in BA is re- L 2 solved
solved into the two BC, ca; and the motion in ba is resolved into the two bid, DA; of which the antecedents Bc, bo, 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 bp, or their equals EA and FA. The motions therefore of the bodies B, b, directly striking each other with the velocities FA, 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 H1, are compounded into the diagonal A1, 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.
PRO POSITION XVI II. a £9. All the Properties of Motion delivered in Proposition VI, its
Corollaries and Sokolium, 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. 79. Now it has been found, by numberless experiments,
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": to :: 2g : 2gt = v the Velocity, and 1* : * :: g : go = the space. o 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,
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 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. So that if the first series of natural numbers be seconds of time,
- - marãely,