6*6 
MECHANICS. 
Cor. 3. The velocities generated or deftroyed in any 
times are direffly as the forces and times, and recipro¬ 
cally as the bodies or mafTes. For, fince the compound 
ratios of the bodies and their velocities are as tbofe of the 
forces anti times, the velocities are as the forces and times 
divided by the bodies. 
Prop. XLIV. In motions uniformly accelerated, when the 
force and body are given, the /pace defcribed during a certain 
lime is the half of that which the body , moving uniformly with 
the laji-acquircd velocity, would defcribe in an equal tune. 
Let F reprefent any conitant accelerative force, S any 
fpace, T the correfponding time, and V the velocity ; 
then VOCFxT. For, if F be given, VCCT; for the 
fame force muft generate equal velocities in equal times ; 
therefore, as T increafes, V increafes in the fame ratio, or 
VCCT. Alfo, if T be given, VqcF ; for, as the velocity 
acquired, in a given time, depends on the force only, it 
rmift vary as the force; therefore, if neither F or T be 
given, VctFxT. 
If a body moves with an uniform velocity, SOCTxV. 
For, if V be given, SCCT; and, if T be given, SCCV ; 
therefore, if neither be given, SCX VxT, by the compoli- 
tion of ratios. Hence, if one fide of a right-angled pa¬ 
rallelogram reprefent T, and the other V, the area will re¬ 
prefent S. 
If a body defeend from a ftate of reit, and be added upon 
by a conftant accelerative force, then will SocT 2 , or V 2 , 
or Tx V. For, let A £ (fig. 71. Plate V.) reprefent any 
time, which divide into A in, mti, no, op, See. Ereft any 
perpendicular B C, and join AB ; alio ere£t the perpen¬ 
dicular mv, nw, ox,py, See. which will reprefent the ve¬ 
locities at the er.d of the times, A m, A n, Ao, A p, See. 
for, by the lalt paragraph but one, as F is given, TCCV ; 
and, as the triangles A mv, An to, A ox, Apy, See. are 
fimilar, A tn : m v :: An : nv :: Ao : ox :: A p : py 
:: &c. therefore, as the antecedents reprefent the times, 
the confequents muft reprefent the velocities. Draw any 
other perpendicular L M. Now the fpace defcribed in 
the time Am with the uniform velocity mv is (by the lalt 
art.) reprefen ted by cn Am vb ■, the fpace in the time 
tmn with the velocity nw, by the cm m n w c ; See. See. 
Therefore the fpace defcribed in the time A p is repre- 
l'ented by the fum of all the ens; but, as the velocity of 
a body, when added upon by a conftant accelerating force, 
continually increafes, we muft diminifh the times A m, 
m n, no, op, See. in which the velocity was luppofed uni¬ 
form in infinitum, in order that it may approach to an 
uniformly-accelerating velocity as the limit; in which 
cafe the limit, if the fum of all cus on Ap is the triangle 
Apy, which therefore reprefents the fpace defcribed in 
the time Ap-, for the fame reafon the triangles A LM, 
ABC, will reprefent the fpaces defcribed in the lines AL, 
A B, and L M ; B C will reprefent the velocities acquired. 
Now, as thefe triangles are fimilar, AI.M : ABC :: 
ALL AB::LM 2 Be 2 ; that is, SaT 2 or V 2 And, 
as TCC V, therefore T 2 CXTx V : hence, SCCTx V. 
It A B C D (fig. 7a.) reprefent any time, and B C the 
velocity acquired, the triangle ABC reprefents the fpace 
defcribed in failing from a Itate of reft, by the laft article ; 
now in the fame time A B, if the body moves uniformly 
with the laft-acquired velocity B C, the fpace will be re- 
prefented by the a ABCD; this latter fpace therefore 
is double the former. 
If the times be reprefented by 0,1, 2, 3, 4, 5, 6, See, 
the fpaces will be as the fquares, or as o, 1, 4, 9,16, 25, 36, 
Si c. and, by fublradtion, the fpaces puffed over in fuc- 
cetlive equal portions of time are as the odd numbers 1, 
3> s, 7, 9, xi, See. 
it is found by experiment, that a body falls 16-^ feet 
in the firft fecond of time; it therefore acquires tuch a 
velocity as, if continued uniform for 1", would carry it 
over 32-t feet, by the laft article ; this, therefore, repre- 
lents the velocity acquired in the firft fecond. Let S re¬ 
prefent any other fpace, T the time, and V the velocity; 
and put m— i6-Jj feet. Then, as the fpaces vary as the 
fquares of the times, m : 1" :: S : T 2 , S=z«T 2 , and 
j“T~ 
T^t/— • Hence, if T be given, we find S; and, if S 
m 
be given, we find T. Alfo, the fpaces vary as the fquares 
—1 Va 
of the velocities ; hence, m : zm > 2 :: S : V 2 , S=c—» 
and Vzry^wS. Therefore, if V be given, we find S ; and, 
if S be given, we find V. Laltly, fince the time varies as 
V 
the velocity, 1" : zm :: T : V~zmT, and T=—. Hence, 
Zlll 
if T be given, we find V ; and, if V be given, we find T. 
Ex. 1. To find the fpace fallen through in 7".—Here 
T=7, and SmwT 2 =r.i6j < -2X+9=788 T 1 2 feet. 
Ex. z. To find the time of falling xoo feet.—Here 
S=ioo, and T=f 
x 6 t V 
Ex. 3. To find the Ipace correfponding to the velocity 
of 80 feet per fecond acquired in falling.—Here V=8o, 
80 2 , 
and S=z —r-^99'5 feet. 
64.3 
Ex. 4. Given the fpace 120 feet fallen through, to find 
the velocity.—Here S=i2o, and V-^Xi^Xno^ 
87*9 feet. 
Ex. 5. Given the time of defeent 9", to find the velo-, 
city.—Here T=9", and V=aXX 9 = 28 9*5 feet. 
Ex. 6. Given the velocity 80 teet; to rind the time.— 
80 
Here V=8o, and T— , r —2 ' 4 * 
2. ^ g 
Let F— any variable force, f=:timeof a body defend¬ 
ing in a right line by that force, v— velocity, x— fpace ; 
then zmFx, 4- being ufed when v and * increafe 
together, and — when one increafes and the other decreafes. 
m * $C a SC 
For, by Mechanics, »CCFx<> and t CC - ; hence,»aFx~, 
and vvO.Fx-, let, therefore, vv — dFx. Now, when a 
body falls on the earth’s furface, v 2 t=z^mx, x being the 
ipace defcribed ; but, if x be the fpace to be defcribed, 
and a the wdiole fpace, the » ? = 4 « X j hence, 
vv-=z±.zmx ; but here F=i, the force of gravity ; there¬ 
fore d—±.zm. Hence vv =±«»FL Alfo, by uniform 
1 v 
motion, v : :: 
Let F=x% x being the diftance of the body from the 
- , * „ • j v z zm 
centre of force; then v vzx. —2 m x x, and —— —-X 
2 iz-j-i 
zm 
^•■+1 4 -C ; but, when v—o, x—a, and 0 =.——— Xa” + + C, 
1 2 n+i 
zm . v 2 z m 
C—-x«’*+ 1 j hence the correct fluent is —;— 
n+i ’ 2 
4 m 
Xa'f 1 -x'F 1 , and v —^'—— X — x"**. Hence, 
Sc 
f V a ' ,+ l —*" +1 > whole fluent gives t ; 
v * n+ i _ . . 
but this can only be found in particular cafes.. This is 
the theory of the motion of bodies falling in light lines 
towards the centre of force, and adted upon by variable 
forces. 
Motion of Bodies falling down Inclined Planes. 
Prop. XLV. The motion of a body defending down an in¬ 
clined plane, is uniformly accelerated. — In every part of the 
fame plane, the accelerating force has the lame ratio to 
the force of gravitation acting freely in a perpendicular 
direction, and is therefore equally exerted in every in- 
Itant 
