202 



DYNAMICS. 



COB. 1. It' / the nurmVr of intervals is very grc.it, 

 then /' will be very great in comparison with /, so that 



a / 

 the spue will be = nearly. 



COR. 2. Hence, too, in this case, the space will l>e 

 nearly promotional to /*, since a and 'I are constant. 



Co'n. :<. The . r which the body would go 



with the Inst acquired velocity in the time / would be 

 / a x t=t*a, nearly double the SJMICC already gone over, 

 if t is great. 



C \*E 4. A body acted on by pretsvre, alieayt acting 

 in the same direction, and irilli the same intensity. 



When once a body has acquired a finite velocity, by 

 the action of an unremitting force, and the action has 

 ceased, the motion (according to law 1st,) will In- per- 

 fectly uniform, and therefore all its circumstances cal- 

 culated as in Case 1. But in tlie present case, a consi- 

 derable time elapses before a considerable velocity is ac- 

 i|iiired, and it is interesting to investigate the circum- 

 stances of the motion during the lapse of that time. 



The action being renewed at the end of every instant, 

 the velocity will not remain the same longer than an 

 instant, or the motion will be perpetually accelerated. 



The velocity at any point is expressed by the space 

 passed over during die next instant, or by the space it 

 would pass over during the next unit of any magnitude, 

 were it to preserve its motion unaltered, or nearly by 

 the space actually passed over during the next mo- 

 ment. 



When the force is supposed to act with the same in- 

 tensity, the velocity added at the end of each instant 

 will be the same ; and hence if t denote the number of 

 instants in any finite portion of time, and r the velocity 

 acquired, it follows by the demonstration in last case, 

 that v==t. Also, since in this case I is infinitely great, 

 *==<* exactly. 



but !==/ ; hence t=v*. 



For the same reason, in this case, the space through 

 which die body will go with the last acquired velocity 

 in the time /, will be exactly = twice the space it has al- 

 ready gone over. 



COB. 1. In the case of any pressure, having ascertained 

 by experiment the space passed over in a certain time, 

 such as a second, we will have the space that would be 

 passed over during the next unit with the motion un- 

 altered, that is, the velocity acquired, and we will be 

 able, given any one of these three /, v, s, to find the 

 other two. 



Thus we will be able to find s when I is given, or t 

 when s is given, by the formula $==/*. 



We will find v when t is given, or t when v is given, 

 by the formula r==/. 



And we will find s when c is given, or v when s is 

 given, by the formula *==r*. 



Or an absolute equation may be derived, for each of 

 the six problems that may arise. Thus, let p=the space 

 passed over during the first unit of time, and conse- 

 quently Zp will be=the velocity acquired during that 

 time. 



Then 1 : t':: p : s=pl'. 



Hence J =t. 

 P 



I lence =/. 

 P 



Hence 2 VM = V. 



COR. 2. *==y /, and consequently =/, and ==v. 



Con. 3. If the times, counting from the beginning of 

 the motion, are taken as 1, 'J, 3, &c. the spaces will be 

 as 1, I, ;t, Jij, \c. and the -puces passed over during' 

 the successive equal intervals of time, will be as the 

 differences of these numbers, that is, as the odd num- 

 bers 1, 5, 7,&e. 



Con. 4. Since with the same pressure, the velocity ac- 

 cpiireil is proportional to the time, in comparing two 

 pi, s,ires, we will obtain the same ratio, whatever be 

 the time we assume. This shews the consistency and 

 propriety of the measure we adopted. 



Con. .1. If one of the two pressures produce the same 

 velocity in the same body in half the time, it is evident 

 it would, in the same time, produce double the veloeiu ; 

 hence the force would be double, and so on. There- 

 fore, when the mass and the velocity, or m and v are 



Application. 



TO v 



the same, then /== . Hence we liave J= ,or 



f= , if we measure f by the effect in an unit of time. 



2 1 



COR. 6. Again, since =-p 'f we substitute the lat- 

 ter in the expression of the foregoing corollary, we get 

 . 1ms 



2* 



Also, since /= , if we substitute the lat- 

 v 



COR. 7. 



ter, weget/= . 



Con. 8. Further, it may sometimes happen, tliat we 

 do not know the circumstances of a motion from its 

 commencement, mid we may wish to determine the 

 force by observing the increments on the time, space, 

 or velocity. 



Now, it is easy to shew that the above formula will 

 apply to the increments, reckoning from any point, as 

 well as from the beginning. 



Thus, let t/, t', s', denote the corresponding incre- 

 ments on v, t, s. 



Since v= t, hence = . 



Therefore/= r ^'. 



f s' 

 Since J=< 2 , hence = . 



2ms' 

 I hereforej=. 



v' v" 

 Since =v, hence ~r- 



S S 



Thereforey= -. 



COR. 9. When the mass remains the same, the force 

 will be measured by the velocity generated in the unit 

 of time ; and the expression for it will evidently be had 

 by dividing the above formula by m, or by expunging it. 



COR. 10. If the force continually increases, the velo- 

 city will increase faster than in the ratio of the times, 

 and the spaces faster than in the ratio of the squares of 

 the times. The reverse will hold, if the force conti- 

 nually diminish. 



CASE 5. A body acted on by pressure, alirny; in the 

 same direction, and continually varying its intensity ac- 

 cording to a certain lam. Here, as in last case, the mo- 

 tion being uniform only for an instant, it will follow by 



