262 
PRINCIPLES OF GUNNERY. 
The height 
of the tra¬ 
jectory at 
any given 
time. 
The maxi¬ 
mum heigh 
of trajec¬ 
tory. 
Since vertical velocity at any time l 
— V sin a — gt, 
T 
therefore vertical velocity at time ^ 
. (jT 
— V sm a — J . 
2 
But at highest point this must be zero; so that 
F sin a — ^ = 0, 
2 
T V sin a 
or - = -; 
i.e.j the projectile has traversed, when it reaches its highest point, one- 
half of its range on the horizontal plane in one-half of the total time of 
flight. 
The height of the trajectory at any time may be found when the 
whole time of flight is known, thus : 
Suppose the whole time of flight; then from equation (5), 
2 sin a 
Substituting this value of V in equation (2), 
y = Vt sin a - § =%Tt- \g &; 
or y = ff fT-t) .(6) 
Kef erring to the figure (p. 260), if 
time over OB — T\ 
• OP = t, 
u PB = (T-t)=f, 
then PN= ^ (T—t) = f 
2 2 
or PN = | x (time over OP) X (time over PB). 
2 
T 
But the height of the trajectory will be a maximum when t = -r; so 
.t U 
that substituting in equation (6), it follows that 
the maximum height of trajectory = 
_^ 3 
