PROJECTILES. 



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 lg m ; at the middle of the third, 

 by 3^ in, &c. which values constitute the 

 series m 3m 5m 7m 9m 



~2* ~2 ~2 ' ~2* ~2 &C ' 



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 Ac, c d, de, ef, &c. being respec- 

 tively equal to the above-mentioned 



m 3m 5 in 7 in 

 quantities-^- 5 - T ' ~ &c. it follows, 



4i ~i A Z 



by the continual addition of these, that 

 the space Ac, A rf, A e, A/, Sec. fallen 

 through from the beginning, will be ex- 

 pressed by 



m 4m 9m 16m 25m , 

 ~2 IT' T' ' ' &C ' whlch are 

 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 (?i) times 

 m, or n m ; it will therefore be, as one 

 particle of time is to n such particles, so 

 is n niy the said distance answering to the 

 former time, to the distance, n*m, cor- 

 responding to the latter, with the same 

 celerity acquired at the end of the 

 said n particles. Whence it appears 



that the space (found above) throug'h 



which the ball falls, in any given time n t 

 is just the half of that (n-m} 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 the 



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 fall, 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 320 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 F T = 2 



T == _/D = V = 2 D 



~F~ 2~F T" 



= 2_D 

 T 



F 



F = D -= V = V V 

 T T 2T 4D 



All which equations are very easily de- 

 duced from the two original ones, D = 

 F T T, 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 (T) seconds must consequently 

 be expressed by 2 F x T or 2 F T. 



Theorem 1. A projected body, whose 

 line of direction is parallel to the plane 

 of the horizon, describes by its fall a pa- 



