MOT
( 00 ) MOT
tho' he afterwards confirmed it by Experiments. Theft he repeated again and again, and {till found the Spaces pafs'd over as the Squares of the Times : But it mull be obferved, that the Spaces are not to be taken in the Length, bur the Height of the Plane, as will be ftiewn hereafter. See Descent.
The fame Experiments were tryM, tho' in a different manner, by Ricciolui and Gnmahius, who let fall feveral ftony Balls t»f the fame Bulk and Weight* 8 Ounces each, from various Altitudes j meafuring the Times of Dekent by the Vibrations of a Pendulum. The Refult of their Experiments is feen in thefollowing Table.
Vibrati- ons.
Time
Space at the End of the Time.
Space pafs'd over in each Time.
Rom. Feet.
Rom. Vest.
5
o
5c
iO
10
10
i
40
40
50
>5
2
S.C
90
50
20
3
20
Ida
7°
6
4
10
250
90
L
c
15
15
12
2
o
(SO
45
18
3
135
75
.24
4
240
105
Tbeor. XIV. If a heavy Body fall thro' a Medium void of Refinance, and from a Height not very great ; the Space it puifes over is fubduple of that which it would pafs over in the lame time, with the Velocity it has acquir'd ar the end of its fail.
Dem. Let the right Line A B (Tab. Mechanics, Fig. 31.) reprefent the whole Time ot a heavy Body's Defcenr$ and let this be divided into any Number ol equal Parts 5 t the AhfifTes A P, A Q, A S, A P, draw the right Lines P M, QJ, S H, B C, which may be a? the Velocities ac- quired, in thofe Times, in the Defcent. Since then A P : A Qj : V M : QJ, A P ; A S : : P M : S H, &c. {Eucl. VI. 2 I if then the Alritude of the Triangle A B C be con- ceiv'd to te divided into 1 qual Parts infinitely fmall; the Mmon being uniform in aMoment of Time infinitely fmall 5 the little Area ? f> M m = ? p. P M as the Space pafs'd over in the little Moment of Time V p. {Tbeor. 2.) There- fore the Space pafs'd over in the Time A B r will be as the Sum of all the little Areas, i.e. as the Triangle ABC. But *he Space that would be defcribed in thefameTrme A B wirh the uniform Velocity B C, being as the Rectangle A B C D, {Tbeor.i.) it will be to the other Space as 1 to 2. (Eucl. I. 41.)
Cor. The Space therefore pafs'd over in half the Time A B, with the Velocity B C, is equal to the Space which the heavy Body paflcs over from a State of Reft in the whole Time A B.
Trcblemi. The Time wherein a heavy Body falls from any given Altitude being given 5 to determine the Spaces it paffes over in each part of that Time.
Refol. Let the given Altitude be = n, the Time = r, the Space pafs'd over in any part of that Time x. Then by Cor. ot Tbeor. 13.
The Space therefore pafs'd over in the firft part of Time is a : i 2 , and therefore that pafs'd over in the fecond part of Time =3*1 ;f* ; that pafs'd over in the third part= 5 a : t ,&c. (Ibid.')
E. gr. In the above-mentioned Experiments of Ricciolus ) the Ball defcended 240 Feet in four Seconds. The Space therefore pafs'd in the firft Second = 240 : 16 = 1 5 5 that in the next Second = 15.3 = 45 : that in the third =: 15.5
= 75,^- ,
Trob. 2. The Time ot a heavy Body's Defcent in a Me- dium void of Refinance thro' any given Space, being given, to determine the Time wherein it will pafs over another given Space, in the fame Medium.
Refol. and Dem. Since the Spaces are as the Squares of the Times, {Tbeor. 1 3.) to the Space the heavy Body moves in the given Time, the Space required in the Queftion, and the Square of the given Time, feek a fourth Propor- tional , this will be the Square of the Time required : Its fquare Root therefore being extracted, will yield the Time required. E. gr. In Ricchlus's Experiments the Ball fell 240 Feet in 4 Minutes 5 'tis demanded then how much Time it will take up in falling 135 Feet? This Time will be found = / (135: 16 ::240) —-/ O35 \i$) V9 = 3-
Frob. 3. The Space a Body falls in any given Time in a Medium void of Rcfiftance being given, to determine the Space it will fall, in any other given Interval of Time.
Rejol. and Dem, Since the Spaces are as the Squares ot the Times, {Tbeor.i^.) find a fourth Proportional to the Square of the lime wherein the Body falls thro 1 tho given Space, the Square ot the Time wherein it is to tall rhro 1 the Space required, and the Space required ; this fourth Proportional will be the Space required.
E. ^- Suppote a Bail to fall 290 Feet in four Minutes time j and 'tis inquir'd what time it will (pend in falling 135 Feet? The dnfwer will be found = ^ (135.11? : 240) = (135: 15 =/ 3-
T&eor. XV. If a Body proceed with a Motion uniformly retarded, it will pafs over half the Space which it would defcribe in the fame time by an equable Motion.
Dem. Suppofe the given Time divided into any Number of equal farts j and draw the right Lines B C, S H, QJ. t P M thereto, which are to be as the Velocities correfpon- ding to the Parts of Time 0, B S, B Q^, B P, B A 3 foas letting fall the Perpendiculars H E, ! F, M G,'hj right Lints C E, E F, C G, C B may be as the Velocities loft in the Times H E, F f, GM.AB; that is, B S, BQ_, BF, B A. Since C E : C F : ■ E H : F I, C G : C B : : G iVl
- BA, {Tbeor. 13.) A B C will be a Triangle, {End. III.
17J IfBA, therefore, be a Moment of Time infinitely fmall, its Motion will be uniform ; and, therefore, the Space defcribed by the moving Body will be as the little Area H be C (Tbeor. 2.) The Space therefore defcribed in the t ime is as the Triangle ABC, viz. as the Sum of all the little Areas B b c C. Now the Space flefcrrbed by the Body moving uniformly with the Velocity B C in the Time A B, is as the Rectangle A B CD, {Tbsor. 3.) there- fore the former is half of this. {Eucl. I. 41.) Q^E.D.
Tbeor. XVI. The Spaces defcribed by a Motion uniformly retarded, in equal Times, decreafe according to theuneqal Numbers 7, 5, 3, 1.
Dem. Suppofe the moving Body in the firft Inftant of Time to pafs over feven Feet ; I fay, that in the fecond if it be equally retarded, ir will pafs over 5 ; in the third 3; and in the fouith 1. For let the equal Partsof the Axis of L he Triangle B S, SQ, Q_P, PA (Tame %.) be as the Times; the Semi-ordinau-s B C, SB, QJ^PMas the Velocities ar the beginning of any Time j the Trapezia B S H C, S QJ H. Q^P M I, and the Triangle P A M as the Spaces defcribed in thofi times, as it appears they will be from Tbeor. 16. Let then BC = 4 and B S =SQ = Q^P. =PA=i. Then will SH= 5l QJ = 2, p M a± 1, {Tbeor. 13) B S H C = (4 + 3J 1:2 = £. SQ.IH = (3+2) 1:2=1. CLPMI= (2 + 1) t i z = i p A M = %. Confequently the Spaces defcribed in equal Times are as |, |, 4, 4, that is as 7, 5, 3, 1, £. E. D.
For tbe Caitfe, Sec. of the Acceleration of Motion, fee Gravity and Acceleration.
For tbe Caufe, &c. of Retardation, fee Resistance and Retardation.
Laws of tbe Communication of Motion.
The Laws wherein Motion is communicated by the Colli- fion and Percuflion of Bodies are very different, as the Bo- dies are either Efajiic or iJnelaj-.ic, and as the Direction of the Stroak is oblique or HiretJ,
What relates to the Collifion of Bodies not Blaflie t when the Stroak or Shock isdirecr, will come under the follow- ing Laws.
Tbeor. XVII. A moving Body {Inking again!! a Body at reft, will communicate Motion thereto, and both will pro- ceed in the direction ot the firlt 5 and the Momentum, or Quantity of Motion in the two, will be the fame after the Stroak, as in the finglc one before it.
Dem. For 'tis the Action of thefirft that gives the latter all the Motion it has j and 'tis the fte-action of the latter that takes off any part of the Motion of the firft. Now, as Action and Re-aclion are always equal, the Momentum ac- quir'd by the one muft be jult equal to that loft by theo- rher; fo that there is neither lofs nor gain from the> Stroak.
Corol. The Velocity after the Stroak is found by multi- plying the Mafs of the firft Body by its .Velocity before the Stroak, and dividing the Product by the Mafs of the fecond Body.
Corol. Hence if a Body in Mo(7<mftrikeon another moving in the fame direction, but more flowly, both will continue their Motion in their firft Direction j and rhe Momenta, ot Sum of Motion will be the fame after as before the Stroak.
Corol. If two equal Bodies move againft each other with equal Velocities, after the Stroak they will both remain at Reft.
Simple Motion is that produced from fome one Power.
Compound Motion is that produced by feveral con- fpiring Powers : Powers being faid to confpire, when the Direction of the one is not oppofite to that of the other ; as when the Radius of a Circle is imagined to whirl, round
on
