332 PRACTICAL MATHEMATICS 



, n R . 



and CgL = sm * " * cos 



IH 



R2 + 



(R sin pt pL cos pt) + _,, 

 \ t- t- ^ ' R 2 + 



, 







' 



sin (p< - 0) 

 2 



y 

 VR 2 



where = tan" 1 ^5- 

 ri, 



_Ef 



It should be noticed that as t becomes large e L becomes 



small, and therefore C tends to the form . = sin (pt - 0), 



VR 2 



a periodic function of the same frequency as v, but of amplitude 



/ R 2 T 

 (b) Let the initial condition be C = C when t = 0. 



m 



R< v e L 



Then Ce L = (R sin pt pL cos pi) + Const 



and c - - ^ + Const 



g 



- C = 2 22 (R sin pt - pL cos pt) + 





71 gL / t^Lf 



222 (R Sin ^ - ^ L C S ^ 



R 



and C - ^ (R Sin ^ " 



sin (p ' ~ 0) + c 



R 

 'L 



169. To investigate the motion of a body falling from rest under 

 the action of gravity, the resistance of the air being taken into 

 account. 



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

 the body. 



