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COLLUSION OF BODIES.
of the momentums before the stroke; namely, the momen-
BV+bu, 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,
in the 2d
64. For example, if the bodies, or weights, в 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
and in the 3d
or 2 in the first case,
or 11 in the second, and
or 17 in the third.
65. If two Perfectly Elastic Bodies impinge on one another:
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 same velo
OF MOTION, FORCES, &c.
city with which they before approached, or so as to be equally
66. Remark. It is not meant by this proposition, that each
67. To determine the Motions of Elastic Bodies after Striking
LET the elastic body в move in
the direction BC, with the velocity
body b be v in the same line; which latter velocity v will be
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-x, and the sum of their momenta вx +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 equations:
COLLISION OF BODIES.
the velocity of B,
viz. BV+bv=Bx+by, and vv y—x; the resolution of which equations gives (Bb)+2bv But-b (B—b) v+2BV · B+b both in the direction BC, when v and are both positive, or the bodies both moved towards c before the collision. But if vbe 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
the velocity of b,
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 2B v, and ข v, the velocities in this case. B+b B+B And in this case, if the two bodies в and b be equal to each 2B 2B other; then в-b — 0, and 1; which give x=0, B+b 2B and yv; that is the body в 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.
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 concourse; to which draw the complete the rectangles CE, DF.
68. If Bodies strike one another Obliquely, it is proposed to determine their Motions after the Stroke.
F perpendiculars BC, bD, and Then the motion in BA is resolved
OF MOTION, FORCES, &c.
solved into the two BC, CA; and the motion in ba is resolved into the two 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 вc and bo, or their equals EA and FA. The motions therefore of the bodies B, 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 B; 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 ah and ag, or HI, are compounded into the diagonal AI, which therefore will be the path and velocity of the body в 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.
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.
70. Now it has been found, by numerous experiments, that
that gravity is a force of such a nature, that all bodies, whether
as 1": "2g: 2gtv the velocity, and 12: 13: g: gt2= s the space.
So that, for the descents of gravity, we have these general equations, namely,
s = gt3
Hence, because the times are as the velocities, and the spaces as the squares of either, therefore,
1, 2, 3,
if the times be as the numbs. the velocities will also be as and the spaces as their squares and the space of each time as namely, as the series of the odd
1, 4, 9, 16, 25, &c.
differences of the squares denoting the whole spaces. So