334 PRACTICAL MATHEMATICS 



(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* where k is a 

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

 mg Rg = m x acceleration 



dv 



and mg gkv 2 = m -r 



dv 



k 



= g (1 <x 2 w 2 } where a 2 = 



Hence g\ dt + Const = I 5-3 



If dv If dt 



gt +Const = - + -I- 



2jl+au 2J1- 



-log a (1 - 



dv 



ay 



_l 



" 2a ge I - au 



But when t = 0, v = 0. Hence Const = 

 1 1 + au 



and log <^ 



1 + OP 



1 au 



+ 1) = e 2 ^ - 1 

 _ 1 g 2agf - 1 

 = 



= - tanh agf 

 a 



giving the velocity of the body in terms of the time. 



Now v = -r = - tanh tx.gt 



dt v. 



Integrating s = log e cosh agt + Const 



& g 



