211, 212] RESISTED MOTION OF PROJECTILE. 199 



This equation can be integrated when f(v) = KV H , and we have 



J,. , = const., 

 u" 1 g J cos n+1 <p 



an equation giving u, and therefore also v, in terms of <. 



Now the equation 



d<f> 



gives t = I - sec </> d(j> + const., 



so that t is found in terms of </>. Also the equations 



dx dy . ds 



5 = cos*, ^ = sm<, Jt = V , 



r v 2 [v 2 



give us x = I d(f> + const., y = I tan <f> deft + const. 



and thus the time and the position of the particle are determined 

 in terms of a single parameter <. 



It is not generally possible to integrate the equation for vertical 

 rectilinear motion even for the case here described where f(v)=v n . 

 In the special case, however, where the resistance is proportional 

 to the square of the velocity the velocity can be found in any 

 position. We have, when the particle is ascending, 



y = - 9 - K y 2 > 



y being measured upwards. Now 



. d d 



hence + 



Multiplying by e 2 " 27 and integrating, we have 



\&* = -j- &W + const., 

 ' 



gvng y 



Again when the particle is descending we have, measuring y 

 downwards, 



