Page:Cyclopaedia, Chambers - Volume 2.djvu/237

 MOT

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MOT

Regardful of Motion 5 notwith (tan ding that they feem'd fo fenfible of its Importance, that they defined Nature by the firft Principle of Slotion and Reft of the Subttance where- in it is. SeeNA-ruRE.

Among all the Antients^ there is nothing extant of Mo- tion, excepting tome things in Archimedes' & Books de JEqu,

Den

For V :v : .- Sf./T

Therefore V/T = v S t

And S :/:; VT:.i

In Numbers 12 : 16 :: 4.:

Coral. If S=f, VT=»f,'


 * 2 : 8 : : 12 : 16

fo that V:? : : f T.

that

ponder nntilus. To Galulco, a great part of the Doclrine of is, if two Bodies moving equably, defcribe equal Spaces; Motion is owing. He firft difcover'd the general Laws of their "Velocities will be in a reciprocal Ratio of their Motion, and particularly of the Defcent of heavy Bodies, Times.

born at large, and on inclined Planes; the Laws of the Motion of Projeftiles ', the Vibrations of Tendulums, and iUetched Chords j with the Theory of Refinances, $$c. which were things the Antients had no Notion of. See Descent, Pendulum, Projectile, Resistance,

In Numbers, if we fuppofe S = 12., and/— n. Be- caufe S = V T and/ =5 vt; if V = 2 and v =a 3, T= 6 and t ;= 4.

So that we have V : v =s t : T 2:3 = 4:6-. Cor. 2. Further, if f==T, then will V =v; and there-

His Difciplc, TorriceSi, poli/h'd, and improv'd on the fore Bodies which move equably, defcribe equal Spaces

Difcoveries of his Matter 5 and added to them divers Ex^ periments concerning the Force of Percuflion, and the Equi- librium of Fluids. See Fluid. M. Huygens improved very confiderably on the Dodlrine of the Pendulum ; and both he and Horeiti on the Force of Percufllon. Laftly, New- ton, Leibnitz, Vartgmtt, Mariotte, &c. have brought the Doctrine of Motion ftill nearer to Perfection.

The general Laws of Motion were firft brought into a tio of V to v, and of M to m. Q. E. LT Syftem, and Analytically demenftrated together, by Dr. Cor.i. If 1 = i 7 then will V M — aw; and therefore Wallis, Sir Chrijiofker Wren, and M. Httygens all much about V : v = M. That is, if the Momenta of two Bodies mo- th e fame time ; the firft in Bodies not Elaftic, and the virig equably, be equal j their Velocities will be in a reci- twolaftin Elaftic Bodies. LafHy, the whole Doctrine of procal Ratio of their Mattes.

Motion, including all the Difcoveries both of the Antients Cor. 2. And therefore if M = w s V = i»; that is, if the and Moderns on that head, was given by Dr. fflallii in his Momenta and Maffes of two moving Bodies be equal, their

equal Times, and have fheir Velocities equal.

Theor. III. The Momenta, or Quantities of Motion, of two Bodies moving equably, 1 and/, are in a Ratio com- pounded of the Velocities V and v, and the MafFes or Quantities of Matter M and m.

Dem. For I = V M,and i = v m ; therefore I : * : : V M : that is, the Ratio of I to i is compounded of the Ra-

M^hanica, or de Mom, published in itftfo

Motion may be confider'd either as Equable, and Uni- form ; or as Accelerated, and Retarded. Equable Motions again may be confider'd either as Simple, or as Compound. Compound Motion may again be confider'd either as ReBi li- near, Or Curvilinear.

And all thefe again may be confider'd either with regard to themfelves, or with regard to the manner of their Pro- duction, and Communication, by Pcrcuffion, lye.

Equable Motion, is that wherein the moving Body pro- ceeds with the fame unvary'd Velocity.

The Laws of Equable Motion are as follow ; the Reader being only to obferve, by the way, that by Mafs we mean Quantity of Matter or Weight, exprefs'd by M ; by Mo- mentum, the Quantity of Motion or Impetus, exprefs'd by J ; by Time, the Duration of Motion, exprefs'd by T ; by Velocity, its Swiftnefs, noted by V ; and by Space, the Line it defcribes, noted S. See Moment, Mass, Velocity.

Thus if the Space be =/, and the Time =fj the Ve- locity will be alfo expreis'd fit: And if the Velocity

Velocities are alfo equal.

Theor. IV. The Velocities V and o of two Bodies mo- ving equably, are in a Ratio compounded of the direct Ra- tio of their Momenta I and i, and the reciprocal one of

their Maffes M and m.

Dem, Since I : i : : V M : v m

1 1

iVM

V : v = I m : i M Q.E.D.

In Numbers 4:2: :i8.5 : 10.7 =4.1 =2.1 =4.

Cor. 1. IfV==z>, thenIw=iMj and therefor =M : m j tfciat is, it two Bodies move equably, and wit fame Veloc.ty, their Momenta will be in the fame Ratio with their Mifles.

Cor. 2. It M=w, I=i; and therefore if two Bodies, that have the fame Maffes, move equably, and with equal Velocity, their Momenta are equal.

I : i

h the

Theor. V. In an equable Motion, the Maffes of the Bo- = v, and the Mafs = ?Bj the Momentum will likewife dies M and m are in a Ratio compos'd of the direct Ratio

of their Momenta, and the reciprocal Ratio of their Velo- cities V and v.

Dem. Since I:/;:VM:n»

be =3 v m.

Laws of Uniform or Equable Motion.

Theor. I. The Velocities V and v of two Bodies moving equably, arc in a Ratio compounded of the direct Ratio of the Spaces S and /, and the reciprocal Ratio of the Times T and t.

Therefore loi«=:iVM

Demonft. For V = S. therefore V : v ;

T and i


 * 8jf

Tr

ri/' fc

M : m = I v : 1 V. In Numbers 7:5: : 28.2 : 10.4: : 7-r : 5 : 1: : 7 : 5, Cor. If M=w, then willIt> = ;V; and therefore I : i = V : •a. That is, if two Bodies moving equably, have their Maffes equal, their Momenta will be as their Velo- cities.

\:v.

lit


 * St : fT

Q, E. D. Schol. This and the following Theorems may be il- luilrated in Numbers; thus fuppofe that a Body A, whofe Mafs is as 7, that is, 7 Pound, in the time of 5 Seconds paffes over a Space of 12 Feet; and another Body B, whofe Mafs is as 5, in the time of 8 Seconds paffes over a Space of is Feet. We /hall then have M = 7, T = 5, S = m »» = 5> t=8, /=!«. And therefore V = 4, v = 1. The Cafe then will Hand thus : V:»::St:/T. 4 : 2 : : 12. 8 : 16. 5 : :4 : 2. _ Coral. If V=ii, then willSt=/T; therefore S :/
 * : T : t. That is, If two Bodies move equably, and with

the fame Velocities, their Spaces are as the Times.

Schol. The Corollaries may be illullraied by Numbers, in like manner as the Theorems. Thus fuppofe S = 12 T = tf,/=8, t = 4. Then will the V= 12:5 = 2, and s- = 8 : 4 =2.

Confequently by reafon V = o

S:/=T : r

12 : 8 =:<S; 4.

Corn;. 2. If V=t, and alfo r = T; then will S=/, and fo the Bodies moving equably, will defcribe equal Spaces in equal Times.

Theor. II. The Spaces Sand/, over which two Bodies pafs, are in a Ratio compos'd of the Ratio of the Times Spaces'are as their Ti' T and t, and of the Velocities V, £?<:.

In Numbers, fuppofe I =: 12, i = 8, M=4, m = 4 ; then will V = I2 : 4 = 3, and 0=8:4=2. Therefore I : i = V j v

Theor. VI. In an equable Motion, the Momenta I and i are in a Ratio compounded of the direcl Ratio's of the Maf- fes M and »;, and the Spaces S and/, and the reciprocal Ratio of the Times T and t.

Dem. Becaufe V : v : : S( :/T And 1 1 it: VM:o»,

Therefore VI:»;:;VMS«: m/T

»/T

H-MSt:. Q. E. D.

Cor.,. Ifl=;, then will MSt = mfT; and therefore M :m =/ T:S '' S: /='»T:MtandT:t = MS:M/; that is, if two Bodies moving equably, have their Momen-

t C ? U A i' Their Maffes are ln a Raeio compounded of the dirett Ratio of their Times, and the reciprocal one of their Spaces. 2. Their Spaces are in a Ratio compounded of the direft Ratio of the Times, and the reciprocal one of their Maffes. 5. Their Times are in a Ratio, com- pounded ot their Maffes and their Spaces.

Cor. 2. Further, if M= w ; then will /T = Sr; and therefore S:/=T:t; that is, if two Bodies moving e- quably, have their Momenta, and their Maffes equal, their

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