ACCELERATION. 



101. If v be given as function of /, say ^ = (/>(/), we find from 

 (2) ds = vdt, and hence by inte- 

 gration 



j-j =j[W/, (3) 



where S Q is the space de- 

 scribed during the time / . 

 The equation v = <f>(t} furnishes 

 a graphical representation of Fig 2 9. 



the velocity, and formula (3) 



shows that the space s S Q described during the time tt Q is 

 represented by the area included between the curve v = <f>(t), the 

 axis Ot t and the ordinates ^ and v corresponding to / and /, 

 respectively (Fig. 29). 



102. Similarly, if v be given as a function of s, say v = \lr(s), 

 we have from (2) dt=ds/v, and hence 



(4) 



The two velocity curves v = <f>(t} and v ^r(s) are of course in 

 general different, and must not be confounded with the path of 

 the moving point, which is here supposed rectilinear. 



103. We have seen (Art. 91, equation (i")) that in the case of 

 uniform motion the velocity v=(s s^/t, i.e. the rate of change 

 of space with time, is constant. The simplest case of variable 

 motion is that in which the velocity varies uniformly. The rate 

 at which the velocity varies ivith the time is called the accelera- 

 tion ; we shall denote it by/. 



If the velocity vary uniformly, the acceleration is constant, and 

 we have j=(v v^/t t where / is the time during which the 

 velocity changes from V Q to v. 



By reasoning analogous to that employed in Art. 99, we find 

 for the acceleration of any rectilinear motion at the time t 



% = %; (s) 



