MOTION. 
sed, because it produces an equal effect ; 
therefore, in this sort of bodies, there is a 
power of action twice as great as in the 
former non-elastic bodies, for these bodies 
not only strike each other by impulse, but 
likewise by repulse, they always repelling 
each other after the stroke. But we have 
shewn- tliat the force witli which non-elastic 
bodies strike each other is as V j there- 
fore the re-action of elastic bodies is the 
same ; that is, the velocity with which 
elastic bodies recede from each other after 
the stroke, is equal to the velocity with 
which they approached each other before 
the stroke. Wlience if x and y be the velo- 
cities of two bodies A and B, tending the 
same way, after the s troke, si nce V — V ■= 
y — a;, we have a: -)- V — V —y, whence 
the motion of A after the stroke, will be Q a;, 
and that of B will beQx-l-QV — QFj 
and the sum of these motions will be equal 
to the sum of the motions before the 
stroke, viz. Qx-j-Qx-j-QV — Q V zx: 
Q V -j- Q F. Whence, by reducing the equa- 
tion, it will be Qx Qx = Q V — Q V 
. . QV-QV + 2QF 
-4- 2 Q F ; and X = Q^pQ 
= the velocity of tlie body A. 
Again, the velocity ofBisx-|-V — F= 
QV— QV4-2QF 
Q-f Q 
9QY-QV+QV „ 
Q + Q • ® 
bodies tend the same way before, the stroke ; 
and it is evident from the equation above, 
that so long as Q V -(- 2 Q F is greater than 
Q V, the velocity (x) of A after the stroke 
will be affirmative, or the body A will move 
the same way after the stroke as before ; 
but when Q V is greater than Q V -|- 2 F Q, 
the velocity (a) will be negative, or the 
body A will be reflected back. 
If the body B be at rest, then V=zo ; and 
X = , which shews the body A 
a-}- Q ’ •’ 
will go forwards or backwards, as QV is 
greater or lesser tlian Q V, or A greater or 
lesser than B. 
If Q = 3, Q = 2, V = 10, (fig. 5.) aiid 
F = o; then after the stroke the velocity of 
. QV — QV .30 — 20 
-A will be X — 
4-V— F = 
! we suppose the 
Q + Q 5 
and the velocity of B will be y z= 
_ 60 
Q4- Q~ 5 
10 
2QV 
= — = 12. If the bodies are both 
in motion, and V — 5, the rest is the same 
as before j then 
QV — QV-4- 2QF 
Q + Q 
= 6 
= velocity of A (fig. 6.) after the stroke, 
^2QV— QF-fQF ,, , 
and . ^ ' = 11 = velocity 
Q + Q" 
of B after the stroke. 
If the bodies A and B move towards con- 
trary parts, or meet each other, then will 
the relative velocity, to which the force of 
the stroke is proportional, be V + F j and 
so the velocities of A and B after the stroke 
will be X and x + V + F ; and so the 
motion of A will be Q x and Q x + Q V + 
Q F ; the sum of these motions is Qx + Qx 
+ Q V + Q F= QV — Q Fz= the mo- 
tion towards the same part before the 
stroke. Whence we have x =z 
and therefore the ve- 
locity of B will be 
QV — QV — 2QF 
V+ F 
2QV = QF— QF 
Q+Q 
Q + Q 
If Q V + 2 Q F be greater than Q V, the 
motion of the body A will be backwards j 
otherwise it will go on forwards as before. 
If Q = 3, Q = 2, V = 10, and F = 5 ; 
then will the velocity of A (fig. 7.) be 
QV— QV— 2QF —10 
Q + Q ~ 5 ~ 
so the body A will go back with two de. 
grees of velocity. The velocity of B, after 
the stroke, will be + 
a Q 
= 13. 
If the bodies are equal, that is, if Q = Q, 
(fig. 8.) then x = _ y. 
shews, that when equal bodies meet each 
other, they are reflected back with inter- 
changed velocities ; for in that case also the 
2Q V 
velocity of B becomes — rr- = V. 
2Q 
If the bodies are equal, and one of them 
at rest, as B (fig. 9.) then since Q = Q, and 
F = 0 , we have the velocity of A after the 
stroke x 1 = o ; or the body A will abide at 
rest, and the velocity of B will be = V, tlie 
velocity of A before the impulse, as ap- 
pears by the example in the figure refer- 
red to. 
If several bodies, B, C, 1), E, F, (fig. 10 .) 
are contiguous in a right line, and another 
equal body A strike B with any given velo- 
city, it shall lose all its motion, or be quies- 
cent after the stroke ; tlie body B whifch re- 
ceives it will communicate it to C, and C to 
D, and D to E, and E to F ; and because 
action and re-action between the bodies B, 
C, D, E, are equal, as they were quiescent 
