30 LECTURE IV. 



The oblique motion of a projectile may be the most easily understood by 

 resolving its velocity into two parts the one in a horizontal, the other in a 

 vertical direction. It appears from the doctrine of the composition of 

 motion, that the horizontal velocity will not be affected by the force of 

 gravitation acting in a direction perpendicular to it, and that it will, there- 

 fore, continue uniform ; and that the vertical motion will also be the same 

 as if the body had no horizontal motion. Thus, if we let fall from the 

 head of the mast of a ship a weight which partakes of its progressive 

 motion, the weight will descend by the side of the mast in the same manner 

 and with the same relative velocity as if neither the ship nor the weight 

 had any horizontal motion. 



We may, therefore, always determine the greatest height to which a pro- 

 jectile will rise, by finding the height from which a body must fall in order 

 to gain a velocity equal to its vertical velocity, or its velocity of ascent ; 

 that is, by squaring one eighth of the number of feet that it would rise in the 

 first second if it were not retarded. For example, suppose a musket to be so 

 elevated that the muzzle is higher than the but-end by half of the length, 

 that is, at an angle of 30 ; and let the ball be discharged with a velocity 

 of 1000 feet in a second ; then its vertical velocity will be half as great, or 

 500 feet in a second ; now the square of one eighth of 500 is 3906, conse- 

 quently 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 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. 



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

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

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

 divide by 16 only we shall have the time of ascent and descent ; and then 

 the horizontal range may be found, by calculating the distance described 

 in this time with the uniform horizontal velocity. Thus, in the example 

 that we have assumed, dividing 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 hori- 

 zontal range is 31 times 886, that is, nearly 28,000 feet, or above 5 miles. 

 But the resistance of the air will reduce this distance also to less than one 

 mile. 



It may be demonstrated that the horizontal range of a body, projected 

 with a given velocity is always proportional to the sine of twice the angle 

 of elevation : that is, to the sine of the angle of] 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 ;* 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 elevation is somewhat less than 45, and some- 

 * Galileo, Dial. IV. Prop. 7, cor. 



