MOTION UNDER CONSTANT FORCES 189 



This gives the distance described in time t, when we know the 

 velocity v with which the particle arrives at the end of its 

 journey. 



Combining equations (45) and (46), we have 



s = $(u + v)t, (47) 



showing that the space described is the arithmetic mean of ut and 

 vt : the former is the space that would be described if the particle 

 maintained its original velocity u through the whole time t ; the 

 latter is that which would be described if the particle had its final 

 velocity throughout the whole time. 



Combining equation (47) with equation (44), which can be 



written in the form 



ft = (v- u), 



we obtain, on eliminating t, 



2fs = v 2 - u 2 , (48) 



an equation connecting the space described with the initial and 

 final velocities. 



This last equation may also be deduced from the equation of 

 energy. Since the work done on the particle is equal to the change 

 in its kinetic energy, we have 



9 



Ps = 1- mv 2 |- mu*, 

 and since P = mf, equation (48) follows at once. 



BODY FALLING UNDER GRAVITY 



155. The simplest application of these equations is to the 

 motion of a body which is allowed to fall freely under the influ- 

 ence of gravity, so that the acceleration is g. 



If the body starts from rest, we put u = 0, and measure s 

 vertically downwards. We find from equation (45) that after 

 time t the body has fallen a distance \gt 2 , while its velocity 



