PROJECTILES 207 



There is no horizontal force acting on the particle, so that its 

 horizontal velocity remains unaltered throughout its flight. Thus 



u = v cos a. 



The latus rectum of the parabola is accordingly 

 2u 2 2 v* cos 2 a 



9 9 



The velocity v is that due to a fall from the directrix to A, so 

 that if NX is the directrix in fig. 118, 



The time of flight from A to is the tune required for gravity 



to destroy a vertical velocity vsina; it is therefore -- In 



9 

 this time the horizontal distance AM is described with a uniform 



horizontal velocity u, so that 



* 



v sin a v 2 sin a; cos a 

 AM - u = - 



The vertical distance described, OM, is by equation (47) equal 

 to half of the time multiplied by the initial vertical velocity. 



2 9 



The total range on a horizontal plane, AA f , is twice A M, so that 



2 v 2 sin a cos a ^ 2 sin 2 a 



A A' = = = 



9 9 



169. If the value of v is fixed (as, for instance, it would be if 

 we were firing a shot with a given charge of powder), while the 



angle a can be varied, then the range AA 1 can never exceed > for 



the factor sin 2 a can never exceed unity. Thus the greatest range 

 attainable on a horizontal plane with a given velocity of projection 



