RATES 135 



the middle of the interval, t, and also that for each 

 instantaneous velocity below this average there is a 

 corresponding velocity equally above the average. 

 So far as concerns the total distance traversed in the 

 time t the result is the same as if the body moved with 

 a uniform velocity, v, equal to the average value. Con- 

 sider then the case of a body uniformly accelerated 

 from rest. The average velocity is half the sum of the 

 initial and final velocities, that is v = at/2. The dis- 

 tance traversed is s = vt and hence s = at 2 /2. (For 

 example, in the case of a body falling freely from rest 

 the acceleration is of value "g" and s = gt 2 /2.) The 

 distance is therefore proportional to the square of the 

 time. 



For our later purposes, however, we wish to express 

 the relation between the total space traversed and the 

 final velocity which the body has acquired. This 

 velocity is v = at and hence t = v/a. Substituting 

 v 2 /a 2 for t 2 in s = at 2 /2 gives s = v 2 /2a or v 2 =2as, as 

 the desired relation. The distance, therefore, in- 

 creases as the square of the velocity. 



The three relations for free fall from rest, namely, 

 v = gt, s = gt 2 /2 and v 2 =2gs were derived by Galileo, 

 whose experiments from the leaning tower of Pisa 

 were more spectacular but less fruitful in scientific 

 development than those of his other methods. His 

 study followed the line of assumption and experimen- 

 tation. Several assumptions he disproved himself by 

 further analysis before he hit upon the correct one, for 

 in his day acceleration was unknown and its concept 

 is due to him. Starting, finally, with the assumption 

 that if two bodies are allowed to fall from rest, one, 



