ON DEFLECTIVE FORCES. S^ 



Wc may easily find the time of the body's ascent from its initial velocity ; 

 for the time of ascent is directly proportional to the velofcity, and may be found 

 in seconds by dividing the vertical velocity in feet by 32; or if we divide by 

 16 only, we sllall have the time of ascent and descent; and then the horizontal 

 rano-e mavl>e: found, by calculating the distance described in this time, with the 

 imiform horizontal velocity. Thus, in the example that wc have assumed, di- 

 vidino- 500 by 16, we have 31 seconds for the whole time of the range ; but the 

 hypotenuse of our triangle being 1000, and the perpendicular 500, the base 

 will be 886 feet; consequently the horizontal range is 31 times 886, that is, 

 nearly 28000 feet, or above 5 miles. Biit the resistance of the air will reduce 

 this distance also to less than one mile. 



It may be demonstrated that the horizontal rat>ge of a body, projected with a 

 given velocity, is always proportional to the sine of twice the angle of elevation r 

 that is, to the elevation of the muzzle of the piece in a situation twice as remote 

 from a horizontal position as its actual situation. Hence it follows, that the 

 greatest horizontal range will be when the elevation is half a right angle; sup- 

 posing thebody 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 obseive in the track of a 

 bomb. ' For both these reasons, the best elevation is somewhat less than 45°, 

 andsometijnes, when the velocity is very great, as little as 30°. But it usually 

 happens in the operations of natural causes, that neAr the point where any 

 quantity is greatest or least, its 'variation is slower than elsewhere*! a small 

 difference, therefore, in the angle of elevation, is of little consequence to the ex- 

 tent of the range, provided that it contiftu'e between the limits of 45" and 35°; 

 and for theisanve reason^ the angular adjtistmettt requires, less accuracy in this- 

 position than in any other, "Which besides the economy of powder, makes it 

 the best elevation for practice. (PlatC' 11. Fig. I7, 18.) 



Tlie path of a projectile, supposetl to move without resistance, is always a: 

 parabola. This interesting proposition was first discovered by Galileo; it fol- 

 lows very readily from the doctrine of the composition of motion, combined 

 with the' laws which that philosopher established concerning the fall of heavy 

 bodies. If from any points of a given right liTie, as many lines be drawn, 

 parallel to each other, and proportional to the squares of tlie corresponding 



