206 



MOTION UNDER CONSTANT FORCES 



167. At the velocity of the particle is u, which is the velocity 



u 



X 



due to a height This is equal to a quarter of the latus rectum, 



*9 



and is therefore equal to the depth of 0, the vertex of the parabola, 

 below the directrix XM. Thus the total energy of the projectile 



when at is equal to that of the same 

 projectile at rest at X y or, of course, at 

 any other point of the directrix, since 

 this is horizontal. 



Since the total energy remains con- 

 stant, we see that when the particle is 

 at any point P of its path, its kinetic 

 energy is that due to a fall through PM, 

 the distance from P to the directrix. 



FIG. 117 



This is expressed by saying that 



The velocity of a projectile at any point is that due to a fall 

 from the directrix. 



168. Instead of supposing that the particle is projected horizon- 

 tally at 0, the vertex of the parabola, we can suppose that it has 

 arrived at in its flight through the air, having been previously 

 projected from some point A. The same reasoning which shows 

 'that the part of the path described 

 after passing is parabolic, will 

 show that the path described be- 

 fore reaching is parabolic also. 

 Thus the path of a particle pro- 

 jected from any point in any 

 manner is a parabola. 



Suppose that a particle is pro- 

 jected from A with velocity v, in 



FIG. 118 



a direction which makes an angle 



a with the horizontal. Let be the vertex of its path, and let us 

 suppose that when the projectile passes through 0, its velocity is 

 u, this velocity being of course horizontal 



