211, 212] RESISTED MOTION OF PROJECTILE. 199 



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



= const., 



u n g J cos n+1 

 an equation giving u, and therefore also v, in terms of &amp;lt;/&amp;gt;. 



Now the equation 



Cv 



gives t = I - sec &amp;lt;t&amp;gt;d(j) + const., 



J 9 



so that t is found in terms of &amp;lt;/&amp;gt;. Also the equations 



dx dy ds 



-j- = cos (/&amp;gt;, -f- sin 6, -T- = v, 

 cfc ds dt 



f v 2 C v 2 



give us x I d&amp;lt;t&amp;gt; + const., y = I tan &amp;lt;f)d$ + 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 being measured upwards. Now 



hence ^- ( Jy 2 ) 4- icy* g. 



Multiplying by e ZKy and integrating, we have 

 \f& Ky = -j- e 2 * y + const., 



giving if=Ce-**y-g\K. 



Again when the particle is descending we have, measuring 

 downwards, 



