200 NEWTON S PRINCIPIA. 



have always the same sign. Hence the velocity is always 



greater or always less than +/ m * 



x 



* x 



/3. Since e ~ m * continually decreases as x increases, 



v continually becomes more and more equal to \/ Be. 



tt 



y. As t increases and finally becomes infinite, the 

 equation connecting v and t shows that v continually 



approaches and finally becomes equal to A/ ~ } a fact 



* /v 



which we could not infer from the first equation. This 

 velocity is called the &quot; terminal velocity.&quot; Let us repre 

 sent it by the letter u. The equations may then be put 

 under the simple forms 



-, U + V , U + V si f * 



log _L --- log - ! = 2 J - t. 

 ^ u v u V u 



Suppose the particle to begin to fall from rest, then we 

 have V=0 and/ =g. 



v = u . A/ 1 



U U , * 8 j. 



x ut log 2 H log (1 4- e ) 



y y 



(2.) Let the velocity of projection be in a direction 

 opposite to that in which f acts. Here f is negative, 

 write f for it, we learn that v 0, or the particle will 

 come to rest after having described a space given by the 

 equation 



