decelerated Motion. 16? 



Small. Then, u being infinitely small with respect to T, which is 

 the velocity acquired during the infinite number of instants de- 

 noted by t) u is to be neglected in the equation s = (u -f v) X 1 1 

 i . r Cal. 6. 



which gives 



266. This being established, let us suppose that at the end 

 of the time *, the accelerating force ceases to act ; the body will 

 continue its motion with the velocity i>, that it had acquired ; 

 that is, in each unit of time, it will describe a space equal to V ; 22 

 accordingly, if it were to continue with this velocity during the 

 time <, it would describe a space equal to v X /, that is, double 

 the space s or v t, described in the same time by the successive 266. 

 action of the accelerating force. Therefore, in motion uniformly 

 and continually accelerated, the space described during a certain time 

 is half of that which the body would describe in an equal time with 

 the last acquired velocity continued uniformly. 



267. Since the acquired velocity increases with the times 

 elapsed, if we call g the velocity acquired at the end of a second, 

 the velocity acquired after a number t of seconds, will be g t ; 

 that is, we have, 



and accordingly the equation s = \ v /, found above, becomes 



If, therefore, we represent by 5' another space described in the 

 same manner during a time /', we shall have, according to the 

 above reasoning, 



f.j* ig*' 2 , 

 from which we deduce the proportion, 



s : s' :: %gt 2 : %gt' 2 : : t* : t? 2 ; 



we hence learn that with respect to motion uniformly accelerated. 

 the spaces described are as the squares of the times. 



268. Moreover, since the velocities are as the times, we con- 264 

 elude also, that the spaces described are a* the. squares of the -odor.i- 



