MOTION. 
velocity in the same manner accelerated, 
till it comes to the earth’s surface. 
Since the momentum (M) of a body is 
compounded of the quantity of matter (Q), 
and the velocity (V), we have this gene- 
ral expression M = Q V, for the force 
of any body, A ; and suppose the force of 
another body, B, be represented by the 
same letters in italics, viz. M=iQ V. 
Let the two bodies, A and B, in motion, 
impinge on each other directly; if they 
tend both the same way, the sura of their 
motions towards the same part will be Q V 
Q V. But if they tend towards contrary 
parts, or meet, then the sum of their mo- 
tions towards the same part will be QV — 
QV ; for since the motion of one of the bo- 
dies is contrary to what it was before, it 
must be connected by a contrary sign. Or 
thus ; because, when the motion of B con- 
spires with that of A, it is added to it ; so, 
when it is contrary, it is subducted from it, 
and the sum or difference of the absolute 
motions is the whole relative motion, or 
that which is made tow'ards the same part. 
Again, this total motion towards the same 
parts, is the same both before and after the 
stroke, in case the two bodies, A and B, 
impinge on each other ; because, whatever 
change or motion is made in one of those 
bodies by the stroke, the same is produced 
in the other body towards the same part ; 
that is, as much as the motion of B is in- 
creased or decreased towards the same 
part by the action of A, just so much is the 
motion of A diminished or augmented to- 
w'ards the same part by the equal re-action 
of B, by the third law of motion. 
In bodies not elastic, let a; be the velocity 
of the bodies after the stroke (for, since we 
suppose them not elastic, there can be no- 
thing to separate them after collision ; they 
uiust therefore both go on together, or with 
the same celerity). Then the sum of the 
motions after collision will be Q a; -j- Q x ; 
whence, if the bodies tend the same way, 
we have Q V + Q K = Q X -}- Q X, or if 
they meet, QV — QF=Qx-^Qx; and 
. QV + QF 
accordingly, -y--j-^ = x, or 
QV— Q F 
Q + Q 
If the body (B) (Plate XI. Miscel. fig. 1.) 
be at rest, then F = o, and the velocities 
of tlie bodies after the stroke will be 
Q V 
Thus if the bodies be equal (viz, Q = Q, 
lig. 1.) and A with 10 degrees of velocity 
impinge on B at rest ; then ~ 
= 5 = .r. IfQ = Q, and V : F:: 10 : 6, 
... . QV-(-QF 16 „ 
(fig. 2.) we have z= = 8 = x, 
the velocity after the stroke. 
If the bodies are both in motion, and tend 
the contrary way ; then when Q = Q (fig. 3) 
and V= F, it is plain ^ ^ = o =x ; 
that is the bodies which meet with equal 
bulks and velocities, will destroy each 
other’s motion after the stroke, and remain 
at rest. If Q = Q, (fig. 4.) but V : F:; 6 ; 14, 
= “ = — 4 =: x; which 
shews that equal bodies meeting with un- 
equal velocities, they will, after the stroke, 
both go on the same way which the most 
prevalent body moved before. 
If the velocity — ^ ^ be multiplied 
by the quantities of matter Q and Q, we 
shall have — — ^ — = the momentom 
f a ft ft .1 . QVQ^ Q^F 
of A after the stroke ; and ,, , „ 
= the momentum B : therefore Q V — 
Q'V±QQF__QQV±QQF_ 
Q-fQ ~ Q-fQ “ 
QQ 
X V ^ F = the quantity of the 
motion lost in A after the stroke, and con- 
sequently is equal to what is gained in B, as 
may be shewn in the same manner. 
But since a part of this expression (viz. 
is constant, the loss of motion will 
ever be proportional to the other part 
V F. But this loss or change of motion 
in either body is the whole effect, and so 
measures the magnitude or energy of the 
stroke. Wherefore any two bodies, not 
elastic, strike each other with a stroke al- 
ways proportionable to the sum of their 
velocities (V -}- P) if tliey meet, or to the 
difference of their velocities (V — F) if they 
tend the same way. 
Hence,, if one body (B) be at rest before 
the stroke, then F — o; and the magnitude 
of the stroke will be as V ; that is, as the ve- 
locity of the moving body A ; and not as the 
square of its velocity, as many philosophers 
were accustomed to maintain. 
In bodies perfectly elastic, the restituent 
power or spring by which the parts displac- 
ed by the stroke restore themselves to their 
first situation, is equal to the force impres- 
