Motion of Projectiles. 191 



advanced by a quantity equal to AP, it will not have risen to 

 a height PJV, equal to that at which it would have arrived, 

 uninfluenced by gravity, but to some lower point M in the same 

 vertical PN -, because its velocity in a vertical direction, being 

 directly opposed to that of gravity, the space which it would 

 have described in virtue of this vertical velocity, must be dimin- 

 ished by the space which the action of gravity would have 

 caused the body to describe in the same time. 



Accordingly let v denote the velocity communicated in the 

 direction AZ, or the number of feet that the projectile would 

 describe uniformly each second, in virtue of this velocity, and 

 / the time, or number of seconds or parts of a second, employed 

 in passing from A to some point JV, we shall have 



AN = v t. 



Let g be the velocity communicated by gravity in a second, 

 | g t 2 will be the space that a heavy body would describe in a 

 number t of seconds. If therefore M be the point where the 

 body will arrive at the expiration of the time J, we shall have 



Through the point A, draw the vertical AX, and through the 

 point M the straight line MQ parallel to the tangent AZ. Call- 

 ing AQ, x', and QM, which is equal to ^JV, */', we shall have 



x' = i g t 2 , and y' = v t. 

 If from this last equation, we deduce the value of /, namely, 



and substitute it in the first, we shall obtain 



or 



D 

 But - expresses the height from which a heavy body must 



o 



fall to acquire the velocity v ; hence, if we call this height h, we 277. 



