6:8 
MECHANICS, 
the laft cafe, we have the common velocity of the bodies in 
g V_j> v' 
the moveable plane ■ 0 — : ; and by adding to this V', 
13 b 
the velocity of the plane, vve (hall have v, or the abfo- 
BV-fB' V' 
lute velocity of the bodies, after isnpulfe, v~ ———— ; - 
B+B' 
Hence, the quantity of motion, after impadf, is equal to 
the fum of the quantities of motion before impadt, 
Cafe 3. If the impinging bodies mutually approach each 
other, we may conceive, as before, that the body B' is at 
rtlf upon a plane which moves with a velocitv V' in an 
oppofite direction to V, and that B moves on this plane 
BV-fBV' 
with the velocity V + V'. Then, by Cafe 1. — 
is + .b 
will be the common velocity upon the plane after im¬ 
pulfe; and, adding to this V', or the velocity of the plane, 
v.e fh a 11 haven, or the abfolute velocity of the bodies alter 
BV—B'V' 
iinpadf, v—~ —-——. Hence, the quantity of motion 
1j + w 
after impadt is equal to the difference of the quantities of 
motion before impadt. It is obvious that v is pofitive or 
negative, according as BV is greater or lefs than B'V'; 
fo that, when BV is greater than B'V', the bodies will move 
in the direction of B’s motion ; and when B V is lefs than 
B'V', the bodies will move in the diredtion of A’s motion. 
All the three formulae which we have given, may be 
comprehended in the following general formula, v — 
BV±B'V' 
-; for, when B' is at red, V' = o, and the formula 
B+B' 
affumes the form which it bas in Cafe r. 
Cor. 1. If B=.B', and the bodies mutually approach 
V_V' 
each other, the equation m Cafe 3. becomes v — -; 
2 
or the bodies will move in the diredtion of the quickeft 
body, with a velocity equal to one half of the difference 
of their velocities. 
Cor. 2. If V = V', and the bodies move in the fame di- 
B+B' 
redtion, the laft formula will become » = Vx ———, or 
B -f- B 
v — V ; for in this cafe there can be no impulfion, the one 
body merely following the other in contact with it. When 
the bodies mutually approach each other, and when V=V', 
B —B' 
vve have v X ———. 
-jr- 
Cor. 3. When the bodies move in the fame diredtion, we 
BV+BV' 
have, by Cafe z. v — ^ . Now, the velocity gained 
. ., , T7/ BV+B'V' „ BV —BV' 
by B' ts evidently v V', or B+g - ~ V = " B . } 
BV—BV' t . , „ 
hence, B+B' : B=V—V' : 5 but this laft term 
is the velocity gained by B, and V—V' is the relative ve¬ 
locity of the two bodies. Therefore, in the impadt 0/ two 
hard bodies moving in the fame dtreClion, B+B' : B as the rela¬ 
tive velocity cf the two bodies is to the velocity gained by B . It 
is obvious alfo, that the velocity loft by B is V— »=V— 
BV+B'V' B'V—B'V' , T7 , 
~£+i?- 01 ‘ B+B'"~’ hence » B+B = B = V ” V : 
B V _'. buf th j s term i s t h e velocity loft by B, 
B + B 
and V—V' is the relative velocity of the bodies; therefore, 
in the imp aft of two hard bodies, B+B' : B' as their relative ve¬ 
locity is to the velocity lojl by B. The fame thing may be 
lhown when the bodies move in oppofite diredtions, in 
which cafe their relative velocity is V + V'. 
Prop. XXVI. To determine the velocities of two elafic bodies 
after impulfe. —If an elaftic body ftrikes a hard and im¬ 
moveable plane, it will, at the inltant of collifion, be com- 
prefied at the place of contadl. But, as the elaftic body 
Miltantaneoully endeavours to recover its figure, and as this 
force of reftitution is equal and oppofite to the force of 
compreffion,’ it will move backwards from the plane in 
the fame diredfion in which it advanced. If two elaftic 
bodies, with equal momenta, impinge againft each other, 
the effeft of their mutual compreffion is to deftroy their 
relative velocity, and make them move with a common 
velocity, as in the cafe of hard bodies. But by the force 
of reftitution, equal to chat of compreflion, the bodies 
begin to recover their figure ; the parts in contadl ferve 
mutually as points of fupport, and the bodies recede from 
each other. Now, before the force of reftitution began to 
exert itfelf, the bodies had a tendency to move in one di- 
redlion with a common momentum ; therefore the body, 
whofe effort to recover its figure was in the fame diredfion 
with that of the common momentum, will move on in 
that direction, with a momentum or moving force equal 
to the fum of the force of reftitution and the common 
momentum ; while tiie ot'-er body, whofe effort to recover 
from ccmpreiTion is in a diredtion oppofite to that of the 
common momentum, will move with a momentum equal 
to the difference between its force of reftitution and the 
common momentum, and in the diredtion of the greateft 
of thefe momenta. After impulfe, therefore, it either 
moves in the direction oppofite to that of tire common 
momentum, or its motion in the fame direction as that of 
the common momentum is diminiflied, or it is (topped al¬ 
together, according as the force of reftitution is greater, 
lefs, or equal, to the common momentum. 
In order to 3pply thefe preliminary obfervations, let us 
adopt the notation in the two preceding propofitions, and 
let v be the common velocity which the bodies would have 
received after impulfe, if they had been hard, and v', v ", 
the velocities which the elaftic bodies B, B', receive after 
impadt. 
Cafe 1. If B follows B', then Visgreater than V'; and, 
when B has reached B', they are both compreffed at thi 
point of impadt. Hence, lince v is the common velocity 
with which they would advance if the force of reftitution 
were not exerted, we have V— v = the velocity loft by B, 
and v —V'= the velocity gained by B'in confequence of 
compreffion. But, when the bodies ftrive to recover their 
form by the force of reftitution, the body B will move back¬ 
wards in confequence of this force, while B' will move on¬ 
ward in its former diredtion with an accelerated velocity. 
Hence, from the force of reftitution, B will again lofe the 
velocity V— v, and B' will, a fecond time, gain the velo¬ 
city v —V'; confequently, the whole velocity loft by B 
is iV — zv, and the whole velocity gained by B' is 
iv — iV'. Now, fubtrailing this lofs from the original 
velocity of B, we have V—2 V— z v, for the velocity of B 
after impadt 5 and, adding the velocity gained by B to its 
original velocity, we have V' + rv —2 V' for the velocity 
of B'after impadt; hence we have. 
—V 
-zv — 2 v- 
-V 
-V. 
v^—Vf-zv — zWzzzzv^ .. 
Now, fubftituting in thefe equations, the value of v as 
found in Cafe z. Prop. XXV. we obtain 
BV-B'V+i 3 'V' 
v' -— 
B+B' 
BV'— B'V'+2BV 
» ----- • 
B+B' 
Cafe 2. When the bodies move in oppofite diredtions, 
or mutually approach each other, the body B is in pre- 
cifely the fame circumftances as in the preced ing cafe ; but 
the body B' lofes a part of its velocity equal to 2 vf-zV— V'. 
Hence we have, by the fame reafoning that was employed 
in the preceding cafe, v'=zzv — V', and r' = av + V'; 
and, by fubftituting inftead of v its value, as determined 
in Cafe 3. Prop. XXV. or by merely changing the fign of 
V' in the two laft equations in the preceding Corollary, vve 
obtain the two following equations, which will anfwer for 
both 
