OF ACCELERATIXG FORCt:S. 



29 



action of such a cause in the direction of its motion can 

 only increase or diminish that velocity. 



229. Definition. When the increase or 

 diminution of the velocity of a moving body 

 is uniform, its cause is called a uniform 

 force : the increments of space which would 

 be described in any given time with the ini- 

 tial velocities, being always equally increased 

 or diminished. 



Scholium. The power of gravitation acting at the 

 earth's surface, in a direction perpendicular to it, may, 

 without sensible error, be considered as such a force. 



230. Theorem. The velocity produced 

 by any uniformly accelerating force is pro- 

 portional to the magnitude of the force, and 

 the time of its operation, conjointly. 



For the time and the velocity flow equably (229, 4;). 

 Calling the accelerating force a, the time t, and the velo- 

 city u, at : 1) is a constant quantity ; or making this quan- 

 tity unity, atzzv. It may be shown by the composition of 

 niotion that a double action produces a double velocity. 



231. Theohem. The increment of space 

 described is as the increment of the time, 

 and as the velocity, conjointly. 



This is evident from the definition of velocity (45) ; call- 

 ing the space described .r, x'—vt'. If the velocity is varia- 

 ble, the increment must be considered as evanescent. 



232. Theorem, The space described by 

 means of a uniformly accelerating force, is 

 as the square of the time of its action ; it is 

 also equal to half the space which would be 

 described in the same time with the final 

 velocity ; and if the forces vary, the spaces 

 are as the forces, and the squares of the 

 times, conjointly. 



Since u— at (230), and x'zzvl' (231), x'^zzatl' alsoiz:a« 



(4fi), of which the fluents are x^ — (49)=: — . There- 

 fore X varies as tt, or as atl, and v being the velocity ac- 

 quired in the time t, tv instead of liw v?ould be described 

 with that velocity in the same time. 



Scholium. The space described by the fall of a heavy 

 body in one second is 16.0916 feet. 



233. Theorem. The times are as the 

 square roots of the spaces directly, and of the 



forces inversely; they are also as the spaces 

 directly, and the final velocities inversely: 

 the final velocities are also as the spaces 

 directly, and the times inversely. 



Since x= — , f=v^(—), and if a=:i, (=1'=:»/(2t} ; 

 2 \ " •^ 



fv 2t 2X 



and since xzz-, t— — , and «:= — . 

 2 V t 



234. Theorem. The forces are as the 

 spaces directly, and the squares of the times 

 inversely, beginning from the state of rest. 



2.T 



For 2i:=a/<, and a^.—. 

 tt 



235. Theorem. The fluxions of the 

 squares of the velocities are as the fluxions of 

 the spaces, and as the forces, conjointly', 

 whether the forces be uniform or variable. 



In the evanescent time t', the variation of the force va- - 

 nishes in comparison with the whole, and v'zzat' (23o), 

 whence izzat; butiziut (23i), therefore aJi=:t)h'', ai=: 

 vi-=i\{vvy (49). 



236. Theorem. In considering the ef^ 

 fects of a retarding force, the body may be 

 supposed to be at rest in a moveable plane, 

 and the motion generated by the force may 

 be deducted from that of the plane. 



For X beecomes — i', and, ifWm, — xzi\tt, that is, the 

 diminution of the space which would be described varies 

 as the square of the time. It is also obvious, that the de- 

 grees by which an ascending body is retarded, being the 

 same as those by which it is accelerated in descending, the 

 velocities will be the same at the same heights. 



237. Theorem. If two forces act in the 

 same right line on a moveable body, varying 

 inversely as the square of its distance from 

 two given points, of which the distance is a, 

 their magnitudes being expressed by b and e 

 at the distance d, the square of the velocity 

 generated in the passage of the body from 

 any two points of which tlie distances from 

 the first centre are successive values of x, is- 

 the difference of the corresponding values of 



