OF ACCELERATING FORCES. 83 



enters the space, will receive equal additions during the 

 passage through it. 



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. 



In this case a being negative, we have vzzb—at, and 

 dxzzvdtzubdt^atdt, whence x=.bt—^at^, ht being the 

 space described by the initial velocity, and ^at- being 

 deducted from it by the effect of the retarding force. 



Scholium. The degrees, by which an ascending 

 body loses its motion, are the same as those by which it is 

 again accelerated at the same points, when it has acquired 

 its greatest height and again descends. We may thus 

 calculate to what height a body will rise, when projected 

 upwards with a given velocity, and retarded by the force 

 of gravitation. Since the force of gravitation produces or 

 destroys a velocity of 32 feet in every second, an initial 

 velocity of 320 feet, for instance, will be destroyed in 10 

 seconds ; and in 10 seconds a body would fall through 100 

 times 16 feet, or 1600 feet, which is therefore the height, 

 to which a velocity of 320 feet in a second will carry a 

 body, moving without resistance in a vertical direction. 

 We may also obtain the same result by squaring one 

 eighth of the velocity ; thus one eighth of 320 is 40, of which 

 the square is 1600, the height corresponding to the given 

 velocity ; and this velocity is sometimes called the velocity 

 due to the height, being found by multiplying its square 

 root by 8 ; thus v'lCOO x 8 =320. 



G 2 



