32 



THE HUMAN MOTOR 



But if the point already has 

 a uniform movement, there will 

 be a resultant movement. It is 

 what is found, for example, in 

 the fall of a body moving in a 

 horizontal direction, this being 

 the movement of projectiles. Let 

 a point, M, having a speed v, in the 

 direction XX', be attracted by 

 gravity in the direction YY' 

 (fig. 56) ; then it will describe a 

 parabola ( 5). In time t, the 

 space traversed will be vt on 

 XX', J gt* on YY'; the par- 

 allelogram of these movements 

 will give the required trajectory. 



Suppose a projectile to be 



thrown from a point O with a speed v and at an angle of inclination 

 a (fig. 57) ; it will eventually reach the horizontal ox (the ground) 

 by reason of gravity. But the speed v has a vertical component 

 v 1 v sin a, which would be added algebraically to the speed of 

 the fall in the direction ZZ', and a horizontal speed v 2 v cos a. 

 Therefore the vertical speed will be V = v sin a gt at the instant 

 /, and the space traversed will be 



Z = (v sin a) t \gt 2 . 

 And in the direction Ox the speed and the space traversed will be 



v z = v cos a, and x = (v cos a) t. 

 Thus 



F,O 57 



This is the equation of a parabola having a vertical axis AA'. 

 When the projectile is at the point A, its speed V = v sin a _ gt, 

 which is equal to zero, and therefore 



o 



Thus the duration / of the ascension is found ; it must be the 



same for the descent, so that the total duration is 



Sm g 



and 



corresponds to the range OO' of the projectile or to the amplitude 

 of the throw. The equation of the range OO' will be therefore : 



g 



