MOTION OF A BODY IN A RESISTING MEDIUM 885 



But when t - 0, s - 0. Hence Const = 

 and * = -5- log,, cosh v.gt 



* o 



m i k , 



= TT lo g cosh W-t 

 Kg 



Note. To integrate tanh <xgt, put v.gt = x. 



Then I tanh a/rf d< = I tanh x dx 



J . a ^J 



1 fsinh x . 

 dx 



= log, cosh x 

 <* 



Since the numerator is the differential coefficient of the de- 

 nominator, 



hence I tanh ogJ dt = log e cosh y.gt 



j V-Q 



170. To investigate the motion of a body projected vertically 

 upwards with an initial velocity v , the resistance of the air being 

 taken into account. 



(1) When the air resistance is proportional to the velocity of 

 the body. 



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

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



then mg Rg = mass x acceleration 



dv 

 ~dt 

 dv kg 



Now S 



m 



and the integrating factor is therefore e m 



Jcot r tot 



Hence ve m = g\e m dt+ Const 



m & 



= -r-e m + Const 

 k 



but when t = 0, v = v , since v is the velocity of projection 

 then V Q = -- 77 + Const 



K 



rn ~dt = -g- 



