.MOTION OF A BODY IN A RESISTING MEDIUM 887 



, m fkv + m\ f m \ m* kv + m 



Then h=> -j- { . - } { 1 - -. - } - -r-z- log. - - 

 kg\ k / V kv + m) k*g m 



m f1tv Q + m} f kv ^ m z kv + m 

 kg { k J \kv + m) ~~ g g * m~ 



mv 



kg 



- m 1 

 ~~^~ S 



(2) When the air resistance is proportional to the square of the 

 velocity of the body. 



If R Ib. is the resistance of the air, then R = kv z where k is a 

 constant, and if m Ib. is the mass of the body, 



then mg Tig = m x acceleration 



and - mg gkv 2 = m -j 



dv 



= - g(l + <xV), where a 2 = - 



m 



Hence I v = - g\dt + Const 



Jl + aV fe j 



Integrating - tan -1 <w = gt+ Const 



Since the initial velocity of projection is v Q , then when t = 0, 



v = v , and - tan" 1 au == Const. 

 a 



Hence gt = -{tan- 1 oo> tan -1 <xz>} 



oc 



= tan 



a ay 

 and tan arf = " 



(1 + a 2 u) tan a.gt = OLV O cuv 

 V(CL + a 2 y tan o.gt) = au tan ctgt 



cuv tan agt 



a(l + ar tan v.gt) 



- tan (9 vgt) , where tan 6 <xo 



