FORCE. | 
Lega ede mata fg ea +es 
> | 
+ C; and fince when waz oe. ume, i —, 
[) = —— 
- = 
a. 
— 
eed ard v 
pe ase r 
To nee the time i we ‘muft refer to the general equa- 
x 
tion dt = “a? and fubftitute the value of > v, above 
dx 
x 
found, in the expreffion - Jie afr 
a—x 
a a—x _ 
iGo i ye deadtutm ff] (3 afr 
*—*) y= a8 Ve —*; and multiplying the 
“ 2fr ae 3 plying 
latter fra€tion by a— x = 
— 
— 
—~ 
ae — 
2f * Yax—x 
To integrate this fraétion, a x=ta-—x, 
fa~s 
F@==) 
the 
integral is then that of the fraGion 
dx, which may be divided into = 
"(4a — 2°), and I= 
( cos. 
when « =e, ¢=0, » =4a; {ubftituting for z its value 
eon ts Vas Gee t+ £4 X arc 
G— 2X 
v4 IG — 2) 
—2adz ha. 
VE (a? — x° =) 
2). The conftant quantity C = o, becaufe 
arc 
cos. 
thod of finding this — ae ae fee 
praeie eae on Reétilinear Motio 
cample.— Let a body. begin efor “from a ftate of 
reft at A ( qe 13.) towards C. orce be fuch as, if 
continued tingform, would caufe A . defcend one foot i in one 
fecond, required the velocity when it arrives at O = 3 A C. 
F (force at O) : f (force atA = 1) :: AC*?: CO? 
a’:(@ — x)? 
a’ 
P= Gasp 
a 
Fids= -dxavde 
(¢—x)? 
2 2 
o a 
—_—= + C; whens = 0,v=0 
2 a-x 
v* a a 
— = — a= (when = $a) —-—- —- a =a 
2 a—x 4. 
y* 
=2a,andv= V 24. 
Suppofe a material particle at A (fig. 14.) to be folicited 
by two forces, one pro ees aes Hits A towards B, 
uniformly accelerated ; 
the my move it to- 
wards D, and acting in an inverfe o ro) 
the diftance of 
the particle A from B 3; requited the circumfaneag ef the 
rina otion which will take place from the anne of thefe 
= the {pace defcribed at the 
Phe the’ capeairee force of rez 
ae ‘B, and g the conftant force which 
ireCti bat m be the a 
or—~ 
3 
I 
by the nature of the queftion— ; =—-; it laa = 
; Y 
mm - 
The force f, which acs on the particle at: the end 
of the time #, is the difference of the two forces, or f= y — g. 
Hence f = = -—gifdsnudusds=udt: by 
means of chefs equations, ¢ two out of ad four queonnes 
sy ft, 0, f, may be elim 
The firft and fecond equations give vd v= Ge °° s) 
m. hyp. log. (a+e) —gs 
o3; hence Cams 
: . v* 
ds sand by integration —— 
—_— 
+ C3; at the pot A 9 = 0 5 
hyp. log. a 
ats 
Therefore v = + {* 2m x hyp. log. (ora <**) 
fue 
ain the relation between s -and #, value of 
v thus fad, mut be fubftituted in the differential equation 
a » and the integral found as in the former ex- 
ample. 
This problem relates: to the ‘eae of a pifton moving ina 
The 
v= 
a conftant force g, an 
be the lefs, as the = paeauiihg it is greater, or as the 
pifton is further from 
To obtain the maximum of velocity, dv muft = 6, or 
i —— —g=o,anda +s= “=BN; beyond which 
ae v aoa: the motionis retarded, and ceafes when 
= gs, when it is again accelerated, © 
m hyp. log. - - 
and but be the effe& of fri€tion me body. would thus ofcil- 
late for 
We are next to inveftigate a see formula. for cu 
linear nee on, and to confider the method of its a conte 
cation, 
San or prolongation of the fide a ate fall polygon. 
eftimate the sagen s at any moment 
ak ive the forces to ce in the fame manner a 
in the cafe of recline: motion and then the velocity » 
may be reprefented by <—~ : 
E2 What- 
