ACCELERATING FORCE.] MECHANICAL PHILOSOPH Y. DYNAMICS. 



735 



acceleration or retardation, as the measure or representa- 

 tive of a constant force, continuously acting on th 

 moving body. 



The symbol for this force is/, which, viewed only ir 

 its effects, denotes merely the velocity generated (or de 

 stroyed) in a second of time. In the illustration above 

 for instance, /=!/. 



It is of importance that the student do not attach an 

 other meaning to this symbol /, than that here assignee 

 to it. The nature of the occult influence called dynamica 

 force or, as it is more frequently named, accelcrative 

 force nobody can explain. In the present inquiry, we 

 have to do only with its effect ; and this we see is merel' 

 augmentation (or diminution) of velocity ; and there 

 fore, like velocity itself, it is expressed by space that is 

 by linear measure : it is not the/orce that is accelerated 

 but the velocity. 



If the force be constant or uniform, the acceleration 

 of the velocity is also constant or uniform, as supposec 

 above ; but if the force be variable, the acceleration o 

 the velocity is also variable. The only office of accelera- 

 tive force is to produce accelerative velocity; and it is 

 only by this latter phenomenon that we become cognizant 

 of its activity, and can estimate its intensity. 



Although, as remarked at the outset, we are altogether 

 unacquainted with the essence of the thing called force, 

 y'.-t we may have abundant means of ascertaining whether 

 its effects are constant or variable ; and it must be care- 

 fully borne in mind that, in speaking of a constant force, 

 or of a variable force, we have reference solely to the 

 constant effect, or the variable effect. The force itself, 

 whatever it may be, may in reality remain entirely un- 

 changed ; and yet, if it affect a body differently at dif- 

 ferent distances from its supposed seat of motion, we 

 should call it a variable force. For example, there is a 

 force familiar to everybody called gravity; and there 

 can be no doubt that this force, like the magnitude of the 

 earth itself, remains unchanged : yet as we find that the 

 higher above the surface a body be taken, the less it 

 will weigh, and the less will its velocity be accelerated, 

 wo say exclusively in reference to these variable effects 

 that gravity is a variable force. It may be observed, 

 however, that as this variation of weight and accelera- 

 tion is insensible at moderate distances from the earth's 

 surface, in all the ordinary applications of dynamica to 

 terrestrial matters, gravity may without error be re- 

 garded as a constant force. 



It is of importance that the student have correct con- 

 ceptions of the sense in which the terms velocity and 

 force are employed in these inquiries, and also that 

 he should clearly perceive that space is the only concrete 

 quantity concerned in the present portion of dynamics. 

 The two propositions following, contain the whole theory 

 of the rectilinear motion of a body under the influence 

 of a constant accelerating force. 



1. If a constant accelerating force / act on a body 

 now at rest, during the time t, producing in it at the end 

 of that time a velocity v, then v = ft. For since / ex- 

 presses the velocity generated in each second, it follows 

 that in the t seconds, the velocity v will amount to ft, 

 .'. e = ft. 



2. If s be the space through which the body is moved 

 from rest by the constant accelerating force/, in the 

 time t, then i = J ft 2 . 



Suppose the time t to be divided into n equal parts ; 

 then since equal velocities are generated in equal times, 



the velocity generated in the time is the nth part of 

 that generated in the time t ; that is, by last proposition, 



it is Consequently, 



t 2t 3t nt 



at the end of the times > > or t 

 n n n n 



ft 2ft ?,ft nft 



the velocities will be > > 



n n 11 n 



or/*.. (A) 



and at the commencement of the same intervals, the 

 velocities will be 



0, ??, ...fc 1 )/'... (B) 



If the several velocities (A) were uniform during the 

 several equal intervals of time, then the whole space S 

 described would be 



_ft 



~ n 





 n 



n 



n 



And if, in like manner, the several velocities (B) were 

 uniform during those intervals, the whole space S' de- 

 scribed would be 



0. 



* ,ft 



L + W.L 

 n ' n ' n 



(nl)ft n 

 n ' t' 



, it is obvious that the space s actually described, 

 is intermediate between the two spaces S, S' ; being less 

 than the former, and greater than the latter. And this 

 is evidently true, however short the several equal intervals j 

 that is, however large the number n may be. 



But the shorter the intervals be made, the closer do 

 the values S and S' approach to each other, and there- 

 fore to the intermediate value s, till at length, when the 

 intervals are diminished down to zero, by n becoming 

 infinite, all three values S, s, S' unite and become 

 identical. 



The determination of * is therefore reduced to the 

 following algebraical problem ; namely to determine 

 the value of S or of S' when n becomes infinite. 



Now the general expression for the value of S is 



(1+2 + 3+ ... 



In like manner, the general expression for the value 

 of S' ia 



{0+1 + 2 + 



When n is infinite, since in that case - = 0, each of 



;hese expressions becomes yt 1 : consequently = \fp. 

 tt appears then, from these two propositions, that if a 

 xjdy at rest bo acted upon by a constant and uniform 

 accelerating force /, during t seconds, the velocity ac- 

 quired and the space described will be expressed by the 

 equations 



c=/tand*= i/P. ... (1) 



From these two important equations we may proceed 

 ,o deduce some inferences. 



1. The space described in any time, reckoning from 

 he commencement of the motion, is half the space that 



wyuld be described in the same time, if the body were to 

 move from rest with a uniform velocity equal to the last 

 acquired velocity. 



For since by (l)s = $(ft) t, and v =ft .'. s = vt ; but 



vt IB the space described in t seconds with the uniform 



velocity v : hence the space i, actually described under 



he influence of a uniform accelerating force, is half the 



pace that would bo described in the same time, if the 



xxJy were to commence with the final velocity, and move 



uniformly. 



2. The spaces described in equal successive portions of 

 ime, from the commencement of the motion, are to one 



another as the odd numbers 1, 3, 6, 7, &c. 



For let t in (1) be successively 1, 2, 3, 4, &c., then 

 he spaces from beginning are \f . 1, \f. 4, |/. 9, \f . 16, 

 tc. ; therefore the spaces for the successive portions of 

 ime given by subtracting these spaces from the be- 

 diming, each from the following, are /. 1, |/. 3, \f . 5, 

 /. 7, <tc. ; so that the spaces described from the be- 

 jinning are as the squares of the times ; and the spaces 



