41-44] UNIFORMLY ACCELERATED MOTION. 43 



Again one function of t having the function u+ft for its 

 differential coefficient is ut + ^ftf, hence x must be of the form 

 C' + ut + ft 2 , where C' is an arbitrary constant. 



Putting t = 0, we find # = C', so that the constant is deter- 

 mined. 



Hence x = x Q -\-ut + \ftf- 



If s is the distance described in the interval t, s is x a? , so 

 that 



By elimination of t between this equation and the equation 

 v = u +ft, we find 



v*-i<; 2 = 2fs. 



In particular the velocity acquired in moving from rest over a 

 distance s is \/2/s. This is described as the "velocity due to 

 falling through s with an acceleration /. " 



43. Examples. 



1. Prove that, when the acceleration is uniform, the average velocity in 

 any interval of time is the velocity at the middle of the interval. 



2. Obtain the formula v 2 -u 2 =2fs by multiplying both sides of the 

 equation x=f by xdt and integrating. 



3. Suppose the distance s divided into a great number of equal segments, 

 and the sum of the velocities after describing those segments divided by their 

 number, a velocity will be obtained which will have a limit when the number 

 of segments is increased indefinitely, and this limit may be called the average 

 velocity in the distance. Prove that, when the initial velocity is zero, this 

 average velocity is equal to f of the final velocity. 



44. Acceleration due to gravity. The importance of the 

 case just discussed arises from the fact that it is very nearly 

 realised in nature. Suppose the frame of reference consists of 

 lines fixed with reference to the Earth's surface at a place, one of 

 them being the vertical at the place, i.e. the line in which a 

 loaded flexible string or chain at the place would hang so as to be 

 at rest relative to the Earth. Then it is very nearly true that, 

 if a body were let fall near this place, it would drop with a 

 constant acceleration in the vertical direction, or all its points 

 would move downwards with this acceleration. When the body 

 is let fall in the exhausted receiver of an air-pump, then it is 

 found that, the more perfect the vacuum, the more nearly is the 

 acceleration independent of the shape, size and material of the 



