CH. VII ACCELERATION 151 



than the final speed by an amount given by the 

 equation. 



Equation 69 may be regarded still more generally. 

 We have seen that the speed of a motor is given by 



. , and that an alteration in any one of the quantities 



making up this expression will cause an alteration in the 

 speed. The effect of an alteration in the tension of 

 the line has already been considered. If the load is 

 increased c will increase, and the speed of the motor will 

 fall to that given by inserting the new value of c in the 

 expression for n. Similarly, if the load is reduced, the 

 motor will speed up according to the same law. During 

 these changes the value of r will remain unaltered. 



If in Equation 70 we put = r, we see that the speed 



is e nf. or 0-632w,, since e = 2-718. We find then that 



e 



when the time from the start is given by =r the speed is 

 a constant fraction of the final speed. The constant r 

 might then be defined as being the time from the start 

 when the speed is O632 of the final speed. We shall call 

 this constant the Time Factor. We only know at 

 present that it varies directly as R, inversely as M, 

 inversely also as the constant 1\, the value of which we have 

 not yet determined. 



By differentiating Equation 70 we obtain 



This gives us the value of the tangent to the curve at 

 any moment, that is to say the acceleration. We see that 

 the acceleration at the moment of starting up from rest, 



