54 KINEMATICS. [108. 



This equation gives the space or distance passed over in 

 terms of the time. 



108. Eliminating/ between (6) and (7), we obtain the relation 



which shows that in uniformly accelerated motion the space 

 can be found as if it were described uniformly with the mean 

 velocity J ( 



109. To obtain the velocity in terms of the space, we have 

 only to eliminate t between (6) and (7) ; we find 



V) = /('-'<)) (8) 



This relation can also be derived by eliminating dt between the 

 differential equations v = ds/dt, dv/dt=j, which gives vdv = jds, 

 and integrating. The same equation (8) is also obtained 

 directly from the fundamental equation of motion d 2 s/dt 2 =jby 

 a process very frequently used in mechanics, viz. by multiplying 

 both members of the equation by dsjdt. This makes the left- 

 hand member the exact derivative of \(ds/dt}^ or |V, and the 

 integration can therefore be performed. 



110. The three equations (6), (7), (8) contain the complete 

 solution of the problem of uniformly accelerated motion. For 

 uniformly retarded motion, taking the direction of motion as 

 positive, we have only to write / for +/. 



If the spaces be counted from the position of the moving 

 point at the time t=o, we have ^ =o, and the,equations become 



