before the stroke, and which make the subject of the next proposition. It may further be remarked, that the sums of the two velocities, of each body, before and after the stroke, are equal to each other. Thus, v, v being the velocities before the impact, if x and y be the corresponding ones after it; since v v=y x, therefore v + x = v + y.

149. PROP. To determine the motions of elastic bodies after striking each other directly.

Let the elastic body в move in the direction BC, with the velocity v; and let the velocity of the other

B

C

body b be v in the same line; which latter velocity v will be positive if b move the same way as в, but negative if b move in the opposite direction to B. Then their relative velocity in the direction BC is v v; also the momenta before the stroke are BV and bv, the sum of which is BV + bv 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 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 art. 136, as also the relative velocities, by the last prop. Whence arise these two equations :

viz. BV + bv = BX + by,
and v- ༧=་

the resolution of which equations gives

the velocity of B,

y

, the velocity of b.

which two velocities are in the ratio of b to B, or reciprocally as the two bodies themselves.

Corol. 1. The velocity lost by в drawn into B, and the velocity gained by b drawn into b, give each of them

COLLISION OF BODIES.

2Bb B+b

(vv), for the momentum gained by the one and lost by the other, by the stroke; which increment and decrement being equal, they cancel one another, and leave the same mo. mentum BV+bv after the impact, as it was before it.

=

Bx2+by, or the sum Corol. 2. Hence also, BV + by3 of the vires vivarum is always preserved the same, both before and after the impact. For, since

BV + bv=Bx + by,

or BV

by bv,

and v + x = y + v, these two equas. multiplied, give BV2 Bx2= by2— bv3,

or BV2 + bv2 = Bx2+ by3,

the equation of the so called living forces.

Corol. 3. 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

x=

y ===

(B-b)v-2bv
B+b

(B-b)v+2BV
B+b

,

the velocity of B,

the veloc. of b, in the direction BC.

And if b were at rest before the impact, making its velocity v0, the same theorems give

x=

B-b

B+,

v, and y =

And, in this case, if the two bodies в and b be equal to

0, 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.

Scholium.

150. If the bodies be elastic only in a partial degree, the sum of the momenta will still be the same, both before and after collision, but the velocities after, will be less than in the case of perfect elasticity, in the ratio of the imperfection. Hence, with the same notation as before, the two equations =Bx + by, will now be BV + bv

where m to n denotes the ratio of perfect to imperfect elasticity. And the resolution of these two equations, gives the following values of x and y, viz.

for the velocities of the two bodies after impact in the case of imperfect elasticity: and these would become the same as the former if n were = m.

Hence, if the two bodies в and b be equal, then

where the velocity lost by в is just equal to that gained by b. And if in this case 6 was at rest before the impact, or v = 0, then the resulting motions would be

which are in the ratio of m n to m+n.

Also, if m=n, or the bodies perfectly elastic, then x = 0, and y = v; or B would be at rest, and b go on with the first motion of B.

Further, in this case also, the velocity of в before the im

pact, is to that of b after it, as v to

m+n

2m

v, or as 2m to

mn. But, if the bodies be now supposed to vibrate in circles, as pendulums, in which case the chords (c and c) of the arcs described are known to be proportional to the velocities; then it will be 2m:m+n::c:c; hence m:n:: c: 2c -C. So that, by measuring these chords, of the arcs thus experimentally described, the ratio of m to n, or the degree of elasticity in the bodies, may be determined.

151. PROP. The greatest velocity which can be generated by the propagation of motion through a row of contiguous perfectly elastic bodies, will be when those bodies are in geometrical progression.

First, take three bodies, A, x, and c: then (art. 149) the

velocity communicated from a to x =

2ла

A+x'

a being the

velocity of A: and when the body x impinges upon c at rest with this velocity, the vel. communicated to c will

(A÷x+1)(x+c) A+X+ (AC÷x)+c

This fraction is evidently a max. when its denominator is a min. that is, since a and c are given, when x2 = AC, or when x is a mean proportional between a and c.

For the same reason the velocity communicated from the second body through c the third, to a fourth, D, will be greatest when c is a mean proportional between the second and fourth. Like reasoning will evidently hold for a series of perfectly elastic bodies. Further, if the number of bodies in the geometrical progression be increased without limit, the quantity of motion communicated to the last, from a given quantity of motion in the first, however small, may also be increased without limit.

152. PROP. If bodies strike one another obliquely, it is proposed to determine their motions after the stroke.

Let the two bodies в, b,

B

E

L

G

C

H

G

F

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 repre. sent the plane in which the bodies touch in the point of concourse; to which draw the perpendiculars BC, D, 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, br, 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 im pulse; so that the velocities with which the bodies meet, are as вc and bD, or their equals Ea and Fa. The motions there. fore of the bodies в, b, directly striking each other with the velocities EA, FA, will be determined by art. 145 or 149. 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 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. For the full consideration of this branch of the inquiry the student is referred to the Treatises of Mechanics by Gregory and Bridge.

Problems for Exercise on Collision.

EXAM. 1. A cannon ball weighing 12lbs. moving with a velocity of 1200 feet per second, meets another of 18lbs. moving with a velocity of 1000 feet per second. Required the velocity of each after impact, supposing both to be nonelastic.

B

EXAM. 2. B and b are as 3 to 2, and the velocity of в is to that of b as 5 to 4. They are perfectly hard, and move before impact in the same direction; what are the velocities lost by B and gained by b?

EXAM. 3. B and b are perfectly elastic, and move in opposite directions.. B is triple of b, but b's velocity is double that of B. How do those bodies move after impact?

EXAM. 4. A body whose elasticity is to perfect elasticity as 15 to 16, falls from the height of 100 feet upon a perfectly hard horizontal plane. It then rebounds and falls again, and so on, always in a vertical direction. It is required to find the whole space described by the body before its motion ceases, as well as the entire time of its motion.

EXAM. 5. Investigate what must be the force of elasticity, so that the sums of the products formed by multiplying each body into any assumed power, n, of its velocity, may not be altered by the impact of the two bodies.

THE LAWS OF GRAVITY; THE DESCENT OF HEAVY BODIES; AND THE MOTION OF PROJECTILES IN FREE SPACE.

153. PROP. ALL the properties of motion delivered in art. 132, its corollaries and scholium, for constant forces, are true in the motions of bodies freely descending by their own gravity; nathely, that the velocities are as the times, and the