127.] RECTILINEAR MOTION. 65 



integrating, we find 



, = 2_log^^, (24) 



2fJ, g-pV 



the constant of integration being o if the initial velocity be o. 

 Solving for z/, we have 



- ~* 



. (25) 



As the numerator, apart from a constant factor, is the deriva- 

 tive of the denominator, the second integration can at once be 

 performed, giving 



J=j5 log (** +*-'*) + 



For /=o, we have s = o; hence o = ^log2 + 7. Hence 



t* 



*~ M ')- (26) 



To find s in terms of v, we may eliminate dt between the 

 fferential equati 

 resulting equation 



-differential equations dsvdt and dv = -(g* i&v*)dt. The 



o 



is readily integrated ; as v = o when s = o, we find : 



log 



/LA 



127. Exercises. 



2 /LA 2 



(1) Show that as / increases, the motion considered in Art. 126 

 approaches more and more a state of uniform motion without ever 

 reaching it. 



(2) Show that when ^, and hence K, becomes o, the equations of 

 Art 126 reduce to those for bodies falling in vacuo. 



(3) Investigate the motion of a particle thrown vertically upwards in 

 the air with a given initial velocity, the resistance of the air being pro- 

 portional to the square of the velocity. 



PART i 5 



