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



energy than kinetic energy of visible motion, potential energy of 

 strain, and potential energy of the parts of the system in the field 

 of force. 



209. Resistance proportional to the Velocity. Since 

 the velocity of a particle is a vector whose direction and sense 

 are determined by the resolved parts x, y, z, the resistance has 

 resolved parts KX, icy, KZ, where K is a constant. 



Suppose the motion takes place under gravity parallel to the 

 negative direction of the axis y, and first suppose the particle to 

 move vertically. The equation of motion is 



my = -mg- icy, 

 or 



where X is written for /c/m. Multiplying by e u and integrating, we 

 have 



, 



A, 



where C is a constant of integration. Hence 

 y = Ce~ Kt/m mg/K. 



If the particle continues to fall for a sufficiently long time the 

 value of y will ultimately differ very little from gm//c, or the 

 particle falls with a practically constant velocity when it has been 

 falling for some seconds. 



The equation last written can easily be integrated again so as 

 to express y as a function of t. 



Again suppose that the particle is projected in any other than a 

 vertical direction, then the vertical motion is the same as before, 

 but for the horizontal motion we have an equation 



mx = KX, 

 giving x = Ae~ Ktlm , 



where A is a constant of integration. This equation can easily be 

 integrated again so as to express a? as a function of t. 



210. Resisted Simple Harmonic Motion. Consider the 

 case where, apart from the resistance, the motion would be simple 

 harmonic in period 27r/n, and the resistance is proportional to the 

 velocity. 



