NEWTON S PRINCIPIA. 199 



integrating throughout the motion, 



i- 



r &quot; / = - 2- x 



Log - m 



1 - -V* 



which is true, whether the body is moving in the direction 

 in which / acts or the opposite, provided only we give / 

 its proper sign. 

 Again we have 



dv ,. x 2 

 _ = f v 



dt J m 



=fdt 



X. 



^7 1 



-v 1 -V .V 

 *. 1 lo g y-^ - - 



1 &quot; ? v 



provided /be positive, but if negative, say /, 



tan&quot; 1 */ v - tan- v/-^- V = - %/^ * 

 ??* / 77Z/ m 



the first or second equation being true, according as the 

 particle moves in the direction in which / acts or the 

 opposite. 



From these equations we can infer the nature of the 

 motion. 



(1.) Let the velocity of projection be in the direction 

 in which the force /acts. Then /is positive. 



- X 



a. Since e m can never change sign, the quantities 



m f , TTO m f 

 v 2 ^ and V 2 



o 4 



