MECHANICS. 
639 
both cafes, by ufing the Upper fign when the bodies move 
in the fame direftion, and the under fign when they move 
in oppofite directions: 
,_B V — B' V ± 2 B' V' 
V ~ B'+B' ' 
± BV'rh B'V'-j-iBV, 
— B + B' 
From the preceding equation the following Corollaries 
may be deduced : 
Cor. i. The velocity gained by the body that is (truck, 
and the velocity lolt by the impinging body, are twice as 
great in elaftic as they are in hard bodies ; for in hard bodies 
tlie velocities gained and loft were v — V', and V — v ; 
whereas in elaftic bodies the velocities gained and lolt 
were zv — zV, and iV— zv. 
2. If one of the bodies, fuppofe B', is at rejl, its velocity 
VB—VB' 
V'=zo, and the preceding equation becomes v — B-{-B' 5 
*BV 
B+B' 
3. If one of the bodies B' is at reft, and their mafies 
equal, we have B=B', and V'=o, by fubftituting which in 
the preceding formulae," we obtain v'—o, and tC'—V ; that 
is, the impinging body B remains at reft after impaft:, and 
the body B' that is (truck when at reft moves on with the 
velocity of the body B that (truck it, fo that there is a com¬ 
plete transfer of B's velocity to B'. 
4. If B f is at reft, and B greater than B', both the bodies 
will move forward in the direftion of B’s motion ; for it 
is obvious from the equations in Cor. 2. that, when B is 
greater than B', v’ and v" are both pofitive. 
5. If B' is at reft, and B lefs than B', the impinging body 
B will return backwards, and the body B' which is (truck 
w ill move forward in the direftion in which B moved be¬ 
fore the ftroke. For it is evident that when B is lefs than 
B', v' is negative, and v" pofitive. 
6 . If both the bodies move in the fame direElion, the body B' 
that is (truck will after impaft move with greater velocity 
than it had before it. This is obvious from the formula 
in Cafe x. of this propofition. 
7. If the bodies move in the fame direftion, and if B=B', 
there will at the moment of impact be a mutual transfer of 
velocities; that is, B will move on with B'’s velocity, and 
B 1 ' will move on with B’s velocity. For in the formulae in 
Cafe 1. when BtnB', w'e have v’—W and w"=V. 
8. When the bodies move in oppofite dire&ions, or mutually 
, approach each other, and when B=B' and V=V', both the 
bodies will recoil or move backwards after impaft with the 
fame velocities which they had before impact. For in the 
formulae in Cafe 2. with the inferior (igns, when B =2 B' 
and V—V, vve have v'— —V and v"z=M'. 
9. If the bodies move in oppofite direftions, and V=V, 
we have b'=Vx , and u"=Vx^——. Hence it 
B+B BXB' 
is obvious, that if Bez^B', or if one of the impinging 
bodies is thrice as great as the other, the greatelt will be 
(topped, and the fmallelt will recoil with a velocity double 
of that which it had before impaft. For (ince B — 3 B', 
by fubftituting this value of B in the preceding equations, 
we obtain v'—o, and v"—zV. 
10. If the impinging bodies move in oppofite direftions, 
and if B —B', they wiil both recoil after a mutual ex¬ 
change of velocities. For, when B=B', we have v'~ —V', 
and v"—\. 
11. When the bodies move in oppofite direftions, the 
body which is ftruck, and the body which (hikes it, will 
flop, continue their motion, or return backwards, accord¬ 
ing as BV—B'V is equal to,, or greater or lefs than, 2 B'V'. 
12. The relative velocity of the bodies after impaft, is 
equaL to their relative velocity before impaft, or, which is 
the fame thing, at equal inftants before and after impaft, 
the diftance of the bodies from each other is the fame. 
For in the different cafes we have v'—zv —V, v"—zvtfVf 
But the relative velocity before impaft: is in the different 
cafes V + V', and the relative velocity after impaft is 
v‘ —z/=VrpV'. 
13. By rea Toning fimilar to that which was employed 
in Prop. XXV. Cor. 3. it may be fliown that B-J-B' : 2B 
as their relative velocity before impaft is to the velocity 
gained by B' in the direftion of B’s motion ; and 
is to 2B' as their relative velocity before impaft is to the 
velocity loft by B in the direftion of A’s motion. 
14. The vis viva, or the fum of the products of each 
body multiplied by the fquare of its velocity, is the fame 
before and after impaft ; that is, Bt/ 2 -j-B V' 2 =BV 2 + B'V-' 2 . 
From the formula; at the end of Cafe 2. we obtain 
„ „ B- p X BV 2 -j-B'V /2 , 
Bz; 2 —- c — k - and 
B +B'i 2 
4 -BB'x B V 2 B' V' 2 , . „ 
B v 2 =z -==:-> hence their fum 
B + B 7 ! 2 
Bz/2 ,p^_ B^ ! XBV 2 + B»V' 2 +4BB-X BV 2 + B'V' 2 
V ° B -f B'l 2 
bV 2 + B V' 2 XB—B' 2 -HBB '_ cv , , 
B + b' 2 
15. If feveral equal elaftic bodies B, B", B'", B"", &c. 
are in contaft, and placed in the fame ftraight line, and if 
another elaftic body |3 of the fame magnitude impinges 
againft B, they will remain at reft,except the laft body B"", 
which will move on with the velocity of | 3 . By Cor. 3. B 
wiil transfer to B" all its velocity, and therefore B will be 
at reft, in the fame way B" will transfer to B'" all its velo¬ 
city, and B" will remain at reft, and lo on with the relt 
but, when the laft body B"' 7 is fet in motion, there is no 
other body to which its velocity can be transferred, and 
therefore it will move on with the velocity which it re¬ 
ceived from B'", that is, with the velocity of ( 3 . 
16. If the bodies decreafe in fize from B to B (/ ", they will 
all move in the direftion of the impinging body / 3 ; and the 
velocity communicated to each body will be greater than 
that which is communicated to the preceding body. 
17. If the bodies increafe in magnitude, they will all recoil, 
or move in a direction oppofite to that o( /§, excepting the 
laft ; and the velocity communicated to each body will be 
lefs than that which is communicated to the preceding 
body. 
Prop. XXVII. Todetermine the velocities of two imperfeElly- 
elqjlic bodies after impulfe, the force of compreffion bang in a 
given ratio to the force of refiituhon or elajlicily. —Let B, B', 
be the two bodies, V, V', their velocities before compaft, 
v',v", their velocities after impaft, and 1 : n as the force of 
compreffion is to that of reftitution. It is evident from 
Cafe 1. Prop. XXVI. that, in confequence of the force of 
comprefiion alone, vve have, 
V » —velocity lolt by B ? from compre ffion. 
v —V“velocity gained by li j 
But the velocity which B lofes and B' gains by the force 
of compreffion will be to the velocity which B lofes and 
B' gains by the force,of reftitution or elafticity as 1 : n ■, 
hence 
1 ; n—V—v : nV—nv, the velocity loft by B ? from elaf- 
1 : n=zv — V'\nv — nV' the velocity gained by B S tieity 
therefore, by adding together the two portions of velocity 
loft by B, and alfo thole gained by B', W'e obtain 
1 J-sV—1 +nv, the whole velocity loft by B, 
x 4- nv — i-fnV’, the whole velocity gained by B. 
Hence, by Tubtrafting the velocity loft by B in conitquenc® 
of coliifion from its velocity before impact, we {hall have v\ 
or the velocity of B after impact ; and, by adding the velo¬ 
city gained by B' after coliifion toils velocity before im¬ 
pact, we fhall find v", or the velocity of B' after impaft, thus : 
V— 1 f nv the velocity of B after impaft. 
v "— V'-j-i fnv —1 + nW the velocity of B' after impaft. 
2 ' Now, 
