OF DEFLECTIVE FORCES. 131 



in a second : now the square of one eighth of 500 is 

 — — ^zi3906, consequently the height, to which the ball 



would rise, if unresisted by the air, is 3906 feet, or three 

 quarters of a mile. But in fact a musket ball, actually 

 shot directly upwards, with a velocity of 1670 feet in a 

 second, which would rise six or seven miles in a vacuum, 

 is so retarded by the air, that it does not attain the height 

 of a single mile. The time, in which the velocity of 500 

 feet would be destroyed, is found by dividing it by 32, or 

 twice the time if we divide by 16 : we have, therefore, 31 

 seconds for the time of the whole range ; and the horizon- 

 tal velocity, being 1000 x V (l-i) zi 886 feet, the ball 

 would describe in 31 seconds, with this velocity, nearly 

 28000 feet, or above five miles. But the resistance of the 

 air will reduce this distance also to less than one mile. 



276. Theorem. With a given velocity, 

 the horizontal range is proportional to the sine 

 of twice the angle of elevation. 



The time of ascent being as the vertical velocity, that is 

 as the sine of the angle of elevation, when the oblique 

 velocity is given, the range must be as the product of the 

 horizontal and vertical velocities, or as the product of the 

 sine and cosine ; that is, as the sine of twice the angle 

 (140). 



Scholium. Hence it follows, that the greatest hori- 

 zontal range will be when the elevation is half a right 

 angle; supposing the body to move in a vacuum. But the 

 resistance of the air increases with the length of the path, 

 and the same cause also makes the angle of descent much 

 greater than the angle of ascent, as we may observe in the 

 track of a bomb. For both these reasons, the best eleva- 



K 2 



