MECHANICS. 



11 



falls through in the fourth, fifth, sixth 

 seconds, &c. respectively. The spaces, 

 therefore, through which a body tails in 

 the successive seconds, or any other 

 equal portions of time, are as the odd 

 integers, 1,3, 5, 7,&c. 



(31.) From these investigations it 

 appears, that the calculation of the ve- 

 locity which a falling body acquires 

 in any given time, depends on that 

 which it acquires in one second, which, 

 therefore, it is absolutely necessaiy 

 to know in order to be able to com- 

 pute any other velocity. In like man- 

 ner, in order to be able, to compute 

 the space through which a body will 

 fall in any given time, it is necessary to 

 know the space through which a body 

 would fall in one second. The velocity 

 acquired in one second, and the space 

 fallen through in one second, are there- 

 fore the fundamental elements of the 

 whole calculation, and are all that are 

 necessary for the computation of the 

 various circumstances attending the 

 phenomena of falling bodies. 



(32.) But even these two elements 

 are not independent. If we knew either, 

 we should immediately detect the other. 

 This circumstance arises from a very 

 remarkable relation which is found to 

 subsist between the space through 

 which a body falls in any time, and the 

 velocity which it acquires in that time. 

 If, after a body has fallen by the action 

 of the force of gravity for any time, say 

 one second, the action of the soliciting 

 force were suddenly suspended, what 

 would be the consequence ? No further 

 velocity would be communicated to the 

 body, since the cause from which its 

 constant accession of velocity proceeded 

 is suspended ; but, on the other hand, 

 the body will not lose that velocity 

 which it has already acquired. It will 

 consequently continue to fall, but in- 

 stead of descending with an accelerated 

 motion, it will descend with the velocity 

 which it has acquired, continued uni- 

 form through the whole of its descent, 

 describing equal spaces in equal times. 

 In this case, it will be found that the 

 space through which it falls in each 

 second 'after the first will be exactly 

 equal to twice the space through which 

 it fell in the first second by the force of 

 gravity.* Now if the velocity be esti- 



By art. (26.) we found V = g T : and in note on 

 art. (29.) we obtained S = $g T*. Eliminating g by 

 these equations, we obtain B=JVT. But V T 

 is the space which would be described with the 

 uniform velocity V in the time T, and is, therefore, 



mated by the space described uniformly 

 in one second, it will follow that the 

 velocity acquired in one second is equal 

 to twice the space through which a 

 body will fall freely by the action of 

 gravity in one second. Thus the space 

 which we have expressed by m is equal 

 to half of that which we have expressed 



(33.) The two formulae expressing 

 algebraically the relation between the 

 space, time, and the velocity acquired, 

 become therefore V=g T, S = kg T 2 , 

 where g represents the velocity acquired 

 in one second; or V=2 ra T S = m T 3 , 

 where m represents the space through 

 which a body falls freely in one se- 

 cond. 



The following TABLE exhibits the 

 relation of the spaces, velocities, and 

 times, conformably to the laws which 

 we have just laid down. The space 

 fallen through in the first second is 

 taken as the unit of length : 



and, in the same manner, the table 

 might be continued to any extent. 



(34.) To submit the several laws 

 which we have now explained, as go- 

 verning the descent of heavy bodies, to 

 direct experiment, would be attended 

 with considerable difficulties. A body 

 will fall, in a single second, through the 

 height of about sixteen perpendicular 

 feet.* In two seconds it will therefore 

 fall through sixty-four feet; and in 

 four seconds through about 256 feet. 

 Thus if our experiments are limited 

 to four seconds, it would be necessary 

 that we should command an height 

 of 256 feet. But further; in observing 

 the velocity, considerable difficulty 

 would arise from its magnitude. The 

 velocity acquired in one second would 

 be one of 32 feet per second ; and, 

 therefore, the velocity acquired in four 



twice the space S through which the body falls in. 

 the time T. 



* Accurately 16 feet and one inch or 193.09 inches 

 in the latitude of London 



