RATES 133 



mobile would be separated by a distance s from the 

 reference point in space. A plot may, therefore, be con- 

 structed as in Fig. 11. The rate of change of position 

 with respect to a change in time, that is, the speed of 

 the car, is given at each instant of time by the slope of 

 a tangent to the corresponding point of the curve. It 

 is therefore evident that for the concrete case of the 

 figure the car is originally at rest and so remains until 

 a time ti has elapsed. At a time ^ the maximum 

 speed is attained. This speed is maintained until 

 3 , following which the car is brought gradually to 

 rest at t 4 . It remains at rest for the balance of the 

 time indicated by the plot. 



This slope, involving unlike magnitudes, may be 

 termed a physical rate as distinct from the geometrical 

 rate, illustrated by the slope of a road. The only 

 difference is that of units, for geometrical rates are 

 pure numbers but a physical rate is expressed by a 

 numeric and a compound unit. Thus the unit of speed 

 is the quotient of unit length and unit time, e.g. 

 1 cm./l sec. 



The most familiar rates of everyday life are those 

 of speed and interest, both time-rates, if we name 

 them for their independent variables. For the scien- 

 tist, of course, there are as many different kinds of 

 rates as there are physical magnitudes which may be 

 considered independent variables. For example, he 

 may be concerned with a rate of expansion with respect 

 to temperature, with a rate of increase of electric 

 current with electromotive force, or with a rate of 

 change of energy with respect to space. In the following 

 chapter we shall see how force is a space rate of change 



