I To demonstrate that the d distances descended are pro. Iportionlet.ltothesquares_aolfre h time proposed of falling o 1 e m r tance A B, be represented by , the right line P Q ; which to be divided into an indefinite number of very small, equal particles, repre Q an sented, each, by the symbol in; and let the distance de scended in the first of them be A c; in the second c d ; in B the third de ; and so on.
Then the velocity acquired being al ways as the time from the beginning of the descent, it will, at the middle of the first of the said particles, be represented by one:half m ; at the middle of the se cond, by 11 m ; at the middle of the third, by 31 en, &c. which values constitute the series ni 3m 5m 7m 9m 2 2 2 2 2 But since the velocity, at the middle of any one of the said particles of time, is an exact mean between the velocities of the two extremes thereof, the corres ponding particle of the distance, A B, may be therefore considered as described with that mean velocity : and so, the spaces c d, de, e f, &c. being respec tively equal to the above-mentioned ni 3m .5in7m quantitiesv it follows,, by the continual addition of these, that the space A c, A d, A e, A f, &c. fallen through from the beginning, will be ex pressed by m 474 9m 16m. 25m1 &c. which are 2 2 2 2 2 evidently to one another in proportion, as, 1, 4, 9, 16, 25, &c. that is, as the squares of the times. Q. E. D.
Corollary. Seeing the velocity acquir ed in any number (n) of the aforesaid equal particles of time (measured by the space that would be described in one sin gle particle) is represented by (n) times en, or n m ; it will therefore be, as one particle of time is to n such particles, so is is in, the said distance answering to the former time, to the distance, 'Pm, cor responding to the latter, with the same celerity acquired at the end of the said n particles. Whence it appears m that the — (found above) through 2 which the ball falls, in any given time n, is just the half of that (Wm) which might be uniformly described with the last, or greatest celerity in the same time.
Scholium. It is found by experiment, that any heavy body, near the earth's sur face (where the force of gravity may be considered as uniform) descends about 16 feet from rest, in the first second of time. Therefore, as the distances fallen
through, are proved above to be in pro portion as the squares of the time, it follows that, as the square of one second is to the square of any given number of seconds, so is 16 feet to the number of feet, a heavy body will freely descend in the said number of seconds. Whence;tip number of feet descended in any given time will be found, by multiplying the square of the number of seconds by 16. Thus the distance descended in 2, 3, 4, 5, &c. seconds, will appear to be 64, 144, 256, 400 feet, &c. respectively. More over, from hence, the time of the descent through any given distance will be ob tained, by dividing the said distance in feet, by 16, and extracting the square root of the quotient ; or, which comes to the same thing, by extracting the square root of the whole distance, and then tak ing one-half of that root for the number of seconds required. Thus, if the dis. tance be supposed 2,640 feet then, by either of the two ways, the time of the descent will come out 12.84, or 12.50 seconds.
It appears also (from the corol.) that the velocity per second (in feet) at the end of the fill], will be determined by multiplying the number of seconds in the fall by 32. Thus it is found that a ball, at the end of ten seconds, has acquired a velocity of 32Q feet per second. After the same manner, by having any two of the four following quantities, viz. the force, the times, the velocity, and dis tance, the other two may be determined: for let the space freely descended by a ball, in the first second of time (which is as the accelerating force) be denoted by F ; also let T denote the number of se conds wherein any distance, D, is de scended; and let V be the velocity per second, at the end of the descent; then will V = 2 T = 2 vi =_- 211 T =_- = 2 D 2F V =FTT=VV=TV 4F 2 2T 4D All which equations are very easily de, duced from the two original ones, D F 1', and V= 2 F T, already demon strated ; the former in the proposition it self, and the latter in the corollary to it ; by which it appears that the measure of the velocity at the end of the first second is 2 F ; whence the velocity (V) at the end of (1') seconds must consequently be expressed by 2 F x T or 2 F T.