ACCELERATED MOTION. 



l.KHATF.D MOTION. 



architectural and sculptural rarirJimmite generally ; in the enrichment 

 ul modillion. ( mouldings, nnd of VMM, M wall M of foliated capitals ; 

 ad we gather from Virgil, that the acanthus wu by the ancients 

 bo employed M an ornament in embroidery. In the first book of 

 the ' JEnmd,' TWC 848, aod again at 711, a Tell or vert is said to be 

 inUrworen or embroidered witJh the crocus-coloured or saAVon acant IIIM. 



Pliny the elder, in hi ' Natural History. 1 describe* the acanthux in 

 stioh a manner that it can only be recognised in the brank-uroine ; And 

 hi* nephew, in speaking of the miccessful cultivation of the same plant 

 a* an ornament to his garden, leaven little doubt that the break-ursine 

 is frhn*il with the common architectural and sculptural acanthus. 

 It is stated, however, that the brank-ursine (Ata*tkm mollit) does not 

 grow in Greece, and it has been miggntod that the plant from u hu-li 

 the Greek architectural ornament was taken was the Afoul* ipinota, 

 which grow* there, and is rtill called the btarfa. 



ThU ornament, in the ancient Greek and Roman models, in very 

 rliaracteristic of the styles of architectural enrichment of thow- nation* ; 

 in the Roman it i full, and somewhat Itixurinnt. and in the Om-k 

 > restrained, but nimple and graceful. 



lloman. 



(ircc-k. 



ACCELERATED MOTION, ACCELERATING FORCE, ACCE- 

 LERATION. When the velocity of a moving body is continually 

 increased, so that the lengths described in successive equal portions of 

 time are greater and greater, the motion is said to be accelerated, which 

 is the same thing as saying that the velocity continually Increases. 

 [Vltocrrr.] Wo see instances of this in the fall of a stone to the 

 earth, in the motion of a comet or planet as it approaches the sun, and 

 also in the ebb of the tide. As it in certain that matter, if left to itself, 

 would neither accelerate nor retard any motion impressed upon it, we 

 must look for the cause of acceleration in something external to matter. 

 This cause is called the accelerating force. [See IXERTTA ; FORCE ; 

 CAUSE : to the remarks in the last of which articles we particularly 

 refer the reader, both now and whenever the word cause U mentioned.] 

 At present, the only accelerating force which we shall consider is the 

 action of the earth, producing what is called migltt, when not allowed 

 to produce motion. 



It is observed, that when a body falls to the ground from a height 

 above it, the motion is uniformly accelerated ; that is, whatever velo- 

 city it moves with at the end of the first second, it has half as much 

 again at the end of a second and a half ; twice as much at the end of 

 two seconds ; and so on. At least this is so nearly true, that any small 

 departure from it may be attributed entirely to the resistance of the 

 air, which we know from experience must produce some such effect. 

 And this is the same with every body, whatever may be the substance 

 of which it is composed, as is proved by the well known experiment of 

 the guinea and the feather, which fall to the bottom of an exhausted 

 receiver in the same time. The velocity thus acquired in one second 

 is oiled the measure of the accelerating force. On the earth it is 

 about 32 feet 2 inches per second. If we could take the same body to 

 the surface of another planet, and if we found that it there acquired 

 40 feet of velocity in the first second, we should say that the accele- 

 rating force of the earth was to that of the planet in the proportion of 

 32$ to 40. By saying that the velocity is 32* feet at the end of the 

 first second, we do not mean that the body falls through 82$ feet in 

 that second, but only that if the cause of acceleration were suddenly to 

 cease at the end of one second, the body would continue moving at 

 that rate. In truth. It fall* through only half that length, or 16y, , in 

 the fint second. It may be proved mathematically, that if a body, 

 netting out from a state of rest, have its velocity uniformly accelerated, 

 it will, at the end of any time, have gone only half the length which it 

 would have gone through had it moved, from the beginning of the 

 time, with the velocity which it has acquired at the end of it. Thus, 

 if a body have been falling from a state of rent during ten seconds (the 

 resistance of the Mr having been removed), it will then have a velocity 

 of 32| x lOor 321] feet per second. Had it moved through the whole 

 tea seconds with this velocity, it would have passed over 321} x 10 or 

 321i feet It really has described only the half, or 1808$ feet. We 

 may give an idea of the way in which this proposition i* established, a* 

 follows : The are* of a rectangle [Rr.cTAXOLIJ that is, the number o] 

 unart ftet it contains, U found by multiplying together the numbers of 

 Hntarfert in the side*. Thun, If A 8 be 4 feet, and A c 5 feet, the num 

 her of square feet In the area 1s 4 x 5, or 20. Again, the number of 

 f- deribd by a body moving with a uniform velocity, for a certain 



number of seconds, is found by multiplying the number of seconds by 

 the number of feet per second or the velocity. If, then, A B contain as 

 many feet as there are seconds, and A c as many feet as the body move* 

 through per second ; so many feet as the body describes in its n 

 so many square feet will there be in ABUT. That i-. if \- 1,-t % i; 

 rrprrte*! the time of motion, and AC the vtl.-Mv -. ihe area A B i< 

 the length dpcril>ed in the time A B, with tli< 



Not that ABDC is the length described, or AB tlu- time of describing 

 it; but A B contains a foot for every second of the time, and ABDC 

 contains a squaie foot for every foot of length described. Similarly, U 

 at the end of the time just considered, the body suddenly receive an 

 accession of velocity D F, making its whole velocity B r per second ; 

 and if with this increased velocity it move for a tiim- whirh contains as 

 as B E contains feet, the length described in thi.i xecoud 

 portion of time will contain an many feet as B E o r contains square- 

 Feet ; and the whole length described in both portions of time will be 

 represented by the sum of the urea* ABDC and n i OF. And similarly 

 for another accession of velocity u i, and an additional time represented 

 by E n. Now, let a body move for the time represented by A n ; at 

 the beginning of this time let it be at rest ; and by the cud let it h.iv.- 

 acquired the velocity M : so that had it moved from the beginning 

 with this velocity, it would have described the length represented by 

 AM N r. Instead of Miining tin- vrliK-ity to be per). casing, 



let us divide the tiim- \ M into a number of equal parts say four, A B, 

 ur. FH.HM and let one-fourth of the velor. munioaUd at 







the beginning of each of these times, so that the body sets off fiviu A. 

 with tie velocity A c, which continues through the time represented by 

 A n, and causes it to describe the length represented by A B D c. Wi- 

 know from geometry that B l>, Eu, and u K, are respectively one-fourth, 

 one-half, and three-quarters of M s, which is also evident to the eye, 

 and may be further proved by drawing the figure correctly, which we 

 recommend to such of our readers as do not understand geometry. 

 Hence, o E or B F is the velocity with which the body starts at the end 

 of the time A B ; E I at the end of A E ; and H q at the end of A n. 

 Consequently, the whole length described is a foot for every square 

 foot contained in ABDC, E B F o, E i K u, and n Q x M, put together. 

 But this is not a uniformly accelerated velocity, for the body first 

 moves through the time A B, with the velocity A c, and then suddenly 

 receives the accession of velocity D F. But if, instead of dividing A M 

 into four parts, we had divided it into four thousand parts, and sup- 

 posed the body to receive one four-thousandth part of the velocity M N 

 at the beginning of each of the part* of time, we should be so much nearer 

 the idea of a uniformly accelerated velocity as this, that instead of 

 moving through one-fourth of its time without acquiring mor 

 city, the body would only have moved one four-thousandth part of the 

 time unacoelerated. And the figure is the same with the excei>i 

 there being more rectangles on A M, and of less width. Still nearer 

 should we be to the idea of a perfectly uniform deceleration if un- 

 divided A M into four million of parts, and so on. Here we observe 

 1, that the triangle A N U is the half of A P N M ; 2. that the sum of tin- 

 little rectangles A c D B, B F o K, Ac., is always greater than the triangle 

 A x si, by the sum of the little triangles A c D, D F o, &c. ; 3, that the 

 sum of the last-named little triangles in only the half of the la>- 

 augle II Q K M, as is evident from the inspection of the dotted part of 

 the figure. But by dividing A M into a sufficient number of parts, wo 

 can moke the last rectangle n q N u as small as we please, consequently 

 we can make the sum of the little triangles as small as we please ; that 

 is, we ran make the sum of the rectangles A c D B, &c., as near as we 

 please to the triangle A N M. But the more parts we divide A M into, the 

 more nearly is the motion of the body uniformly accelerated ; that if, 

 the more nearly the motion is uniformly accelerated, the more nearly i' 

 ASM the representation of the apace described. Hence we must infer 

 (and there are in mathematics accurate methods of demonstrating it). 

 that if the acceleration were really uniform, A x it wuiild really have a 

 square foot for every foot of length described by the body ; that i. 



