Motion of Projectiles. 1 93 



or, 



o* v z a; sin a 



cos a 



or, putting for ^ its value 4 ft, and multiplying both members 



2" o 



by cos a 3 , 



4 Tiy cos a s = 4 hx sin a cos a a 2 , 

 which will furnish us with the following properties. 



305. As the velocity communicated to the projectile is sup- 

 posed to be limited to a certain measure, its effect in a vertical 

 direction must be exhausted at the end of a certain time by the 

 action of gravity, so that at a certain point the body will cease 

 to ascend, and thence will commence a downward motion ; but, 

 as its horizontal velocity does not change when it has reached 

 its highest point, as jB, it will describe the second branch BC of 

 the same curve, and will again meet the .horizontal line AC in 

 another point C. Now in order to determine the distance AC, 

 called the horizontal range* of the projectile, we have only to sup- 

 pose y = 0. We have, accordingly, 



4 h x sin a cos a x 2 or x (4 h sin a cos a x) = ; 



which gives x = 0, and x = 4 h sin a cos a. The first value of 

 x indicates the point A ; the second is that of AC, which may be 

 determined by producing XA till AK is equal to 4 ft, and letting 

 fall from the point K upon AZ the perpendicular KL, and from 

 the point L upon AC the perpendicular LC; since we have 



R = 1 : sin K = sin a : : AK = 4 h : AL 4 h sin a, and 

 #=1 : sin #LC=cos a : : AL=4 h sin a : AC 4 h siq a cos . 



306. If with the same velocity of projection we would know 

 what angle would give the greatest horizontal range, we take the 

 differential of the value of AC, by regarding a as variable, and 

 put this differential equal to zero ; thus c . , 



4 h d a cos a 2 4 h d a sin a 2 = ; 



* Sometimes called also random and amplitude. 

 Mech. 25 



