208-211] RESISTANCE VARYING AS VELOCITY. 197 



We have the equation 



mx mn 2 x KX, 

 or x + \x + ri*x = 0, 



where X is written for /c/m. The complete primitive of this 

 equation takes different forms according as n 2 > or < JX 2 . In the 

 former case, which is practically the more important, it is 



x = e-*^ [A cos {t V(^ 2 - JX 2 )} + B sin [t VO 2 - JX 2 )}]. 



The motion represented may be roughly described as simple 

 harmonic motion with period 2?r/\/(^ 2 JX 2 ), and with amplitude 

 diminishing according to the exponential function e~$ M . It will 

 be observed that the period is lengthened by the resistance, and 

 that the amplitude falls off in geometric progression as the time 

 increases in arithmetic progression. Thus the motion rapidly dies 

 away. 



211. Examples. 



1. A particle is projected vertically upwards with velocity v in a medium 

 in which the resistance is proportional to the velocity. It rises to a height h 

 and returns to the point of projection with velocity w. Prove that 



where V is the terminal velocity in the medium. 



2. A particle moves under gravity in a medium whose resistance varies 

 as the velocity, starting with horizontal and vertical component velocities u ot 

 y , and returning to the horizontal plane through the point of projection 

 with component velocities M 15 ^ ; show that the range R and time of flight t 

 are given by the equations 



*>o -Vi=fft, &=* K ~ w i)/(log u - log Uj). 



Prove also that R=u Q Vt/(V+v \ where Fis the terminal velocity in the 

 medium. 



3. A body performs rectilinear vibrations under an attractive force to a 

 fixed centre proportional to the distance in a medium whose resistance is 

 proportional to the velocity. Prove that, if T is the period, and a, 6, c are 

 the coordinates of the extremities of three consecutive semi- vibrations, then 

 the coordinate of the position of equilibrium and the time of vibration if 

 there were no resistance are respectively 



a+c- 



