A C C 



F X T : f X t.confequently F x T" : f x t • : : VT : vt : : S : s. 



Tlierefore, T^ : t" 



JT 



^: A~, andT 

 i: t 



J 



'F 



4. If a body fall thipugh any fpacc in any time, it ac- 

 quires a velocity equal to double that fpace ; ;. e. in an equal 

 time, with the laft acquired velocity, uuiformly continued, 

 it would pai's through double the fpace. Thus, if a body 

 fall through 16 ^r i<^<-'t in the lirfl: fecond of time, it will have 

 acquired a velocity of 32 -i in a fecond : i. e, if the body 

 move uniformly for one fecond, with the velocity acquired, 

 it will pafs over 32 -i feet in this fecond ; and if in any time 

 the body fall through ico feet, then in another equal time, 

 if it move uniforml)- with the velocity laft acquired, it will 

 pafs over 200 feet, &c. 



To thofe who difapprove of Galileo's dcmonftration of 

 the laws of accelerated motion, the following method of 

 illuflrating and evincing them, may poflibly be more fatis- 

 faftory. Let the whole time of a body's free defc-ent be 

 divided into any number of parts, each of which is called i ; 

 and let a denote the velocity acquired at the end of the firfl 

 part of time ; then 2 <?, 3 «■, 4 a, &c. will reprcfent the ve- 

 locities at the end of the 2d, 3d, 4th, &c. parts of time, 

 becaufe the velocities are as the times ; and for the fame 

 reafon, | a, I a, 5- a, &c. will be the velocities at the middle 

 point of the ill, 2d, 3d, &c. parts of time. But as the ve- 

 locities increafe uniformly, the fpace defcribed in any one of 

 thefe parts of time may be confidered as uniformly defcribed 

 with the velocity in the middle of that part of time ; and 

 tlierefore, multiplying each of thofe mean velocities by their 

 common time i,we ihall have the fame fractions la, -' <7, ia, 

 &c. for the fpaces pafTed over in the fucceifive parts of the 

 time ; /. e. the fpace \a in the firfl time, |-a in the fecond,. 

 -l-fl in the third ; and adding thefe fpaces fuccefilvely to one. 

 another, we fliall obtain ia, ^a, -|(j, '/(7, &c. for the whole 

 fpaces defcribed from the beginning of the motion to the end 

 of the firll, fecond, third, fourth, &c. portions of time, viz. 

 I a in one fpace of time, -jrt in two fpaces, Ja in three fpaces, 

 '/a in four, &c. and the fpaces will be as the numbers I, 4, 

 9, 16, &c. which are as the fquares of the times. 



From this mode of dcmonftration, all the pn.perties above 

 mentioned will evidently follow ; fuch as, that the whole 

 fpaces, la, 4n, -%a, are as the fquares of the times i, 2, 3, 

 &c. and thefeparate fpaces ia, i-a, ^a, &e. defcribed in the 

 fuccefilve times, are as the odd numbers i, 3^ 5, &c. And 

 that the velocity a, acquired in any time I, is double the 

 fpace I a defcribed in the fame lime. 



From the properties above di . lonilrated, we obtain the 

 following practical theorems or formulic for ufe. Let ^■^ 

 denote the fpace pafled over in the firft fecond of time by a 

 body urged by any conftant force, denoted by I, and / de- 

 note the time or number of feconds in which tlit body paiTes 

 over anv other fpacc s, and v the velocity acquired at the 

 end of that time : then we Ihall have 'v=2gt, and s=:gt' : 

 and from thefe two equations we obtain the following gene- 

 ral lonnuLe : viz. 



I. ;: 



~ Zg~ 11 " -J 



^8 



3- " = 81' 



^•tS""" ^2~"*.. 



47' 



V 



u 



2 



4^ 



A C C 



Hence it appears, that when the ronflant force I i« the 

 natural force of gravity, then the dillancc f; dcfccnded in 

 the hrll fecond, m the latitude of London, is i^),', fctt : 

 but if it be any other conflant force, the value of _!j will be 

 different in proportion as tlie force is greater or Ids. See 

 Hutton's Did. Art. Atccleral'ton, where two propofitiona 

 are introduced, which were comnuiiiieatcd to the author by 

 Mr. Abram Robertfon of Chrill Church College, Oxford, 

 ni which the laws of accelerated motion are dinioiiftrated in 

 a manner fomewhat different from that which is above given. 

 See farther on tliis fiibjea, Latvs of the Descent of Bodies, 

 and La'ivs of Motion, uniformly accelerated and retarded. 



Having above illuRrated the laws of accelerated motion, 

 when the accelerating forces are conftant, and deduced the 

 formula? for exprefling them in finite determinate quantities, 

 we fliall now fubjoin thofe that pertain to the cafes of va- 

 riably acceh rated motions. Here the fomiul.e will lie fluxi- 

 onar)' expreflions, the fluents of which, adapted to parti- 

 cular cafes, will give the relations of time, fpace, velocity, 

 &c. Let t denote the time of motion, v the velocity gene- 

 rated by any force, s the fpace pafled over, and 2g the va- 

 riable force at any part of the motion, or the velocity which 

 the force would generate in one fecond of time, if it fliould 

 continue invariably like the force of gravity during that one 

 fecond, and the value of this velocity 2g will be m propor- 

 tion to 32! feet, as that variable force is to the force of 

 gravity. Then, becaufe the force may be fuppofed con- 

 ftant during the indefinitely fmall time/, and the fpaces and 

 velocities, in uniform motions, are proportional to the times, 

 we fliall have two fundamental proportions, viz. v. s •.: i " : 

 /, or J =1 ; 0) : and 2 g : v : : \" : t, ox v =-2 g t : from 

 which are deduced the following formula', in which the va- 

 lue of each quantity is exprcfled in terms of the reft. 



Thefe theorems are equally applicable to the deflruftion of 

 motion and velocity, by means of retarding fiircts, as to the 

 generation of them by means of accelerating forces. Hutton's 

 Did. uii fupra. Parkinfon's Syftem of Mechanics, &;c. 



E-50- 



Themotionof a body afcending, or impelled upwards, is 

 diminiflied or retarded from the fame principle of gravity 

 ailing in a contrary direftion, in the fame manner as a 

 falling body is accelerated. Sec Rktard.ition. 



A body, thus projcdled upwaids, rifes till it has loft all its 

 motion ; which it does in the fame time that a falling body 

 would have acquired a velocity equal to that with which the 

 body was thrown up. Hence, the fame body thrown up, will 

 rife to the fame height from which, if it fell, it would have 

 acquired the velocity with which it was projecled upwards. 

 And hence the heights to which bodies thrown up with 

 dilferent velocities afceiid, are to oiic another as the fquares 

 of thofe velocities. 



Acceleration of todies on In^/ined planes. The fame 

 general law obtains 111 this cafe, as in bodies falling perpen- 

 dicularly ; viz. that the velocities are as the times, and the 

 fpaces defcendcd down the planes as the fquares of the times 

 or of the velocities. Cut the velocities arc lefs, according 



10 



