200 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



or jfy (&) + *#* = 9, 



giving 2/ 2 = Ce~^ y . 



iC 



As in the case of resistance proportional to the velocity, there 

 is a limiting velocity, i/(g/ie), which is practically attained when 

 the particle has fallen through a considerable height. 



*213. Examples. 



1. A particle is projected vertically upwards in a medium whose resist- 

 ance varies as the square of the velocity. Prove that the interval that 

 elapses before it returns to the point of projection is less than it would be if 

 there were no resistance. 



Prove also that, if the particle is let fall from rest, then in time t it 

 acquires a velocity ftanh (fft/U) and falls a distance U 2 g " l log cosh (gtlU\ 

 where U is the terminal velocity in the medium. 



2. A particle of weight W moves in a medium whose resistance varies 

 as the nth power of the velocity. Prove that, if F is the resistance when 

 the direction of motion makes an angle <f> with the horizon, then 



W f 



= ncos n (f) l 



3. A particle of unit mass moves in a straight line under an attraction 

 /u (distance) to a point in the line, and a resistance K (velocity) 2 . Prove that 

 if it starts from rest at a distance a from the centre of force it will first 

 come to rest at a distance 6, where 



4. The bob of a simple pendulum moves under gravity in a medium of 

 which the resistance per unit of mass is < (velocity) 2 , and starts from the 

 lowest point with such velocity that if it were unresisted the angle of oscilla- 

 tion would be a. Prove that it comes to rest after describing an angle 6 which 

 satisfies the equation 



(1 + 4* 2 Z 2 ) cos a = 4* 2 Z 2 - 2*Z sin Qe*** + cos Be* 16 , 

 where I is the length of the pendulum. 



