t 
PRINCIPLES OF GUNNERY. 
261 
In the same way, if gravity were not acting, 
the vertical distance traversed in time t would be 
V sin a X t ; 
nr OM = Vt sin a. 
This distance is diminished by the distance that would be traversed 
by the projectile falling freely under gravity in time t. 
This latter distance, MP ', = \gV. (Vide Elementary Treatises on 
Dynamics). Hence, 
PN=P'0= OM—MP*; 
or y = Vt sin a — ^gfi. ..... 
,( 2 ) 
By eliminating t, as before, the equation of the trajectory of an 
unresisted projectile is 
y = x tan a 
2F' 3 cos 2 a.. 
The trajectory is a parabola, with the axis vertical, as in the figure. 
(3) 
To find the range on a horizontal plane passing through the muzzle The range 
of the gun, let y — 0 in equation (3), then SpSieto' 
gx = 2 V 2 sin a cos a; 
or if X denote the range, 
terms of 
muzzle 
. velocity 
and angle 
of depar¬ 
ture. 
X — 
2 V 2 sin a cos a F 3 sin 2a 
(D 
It is evident that for a given muzzle velocity, V 3 the range will be 
the greatest when sin 2a = 1, i.e. when 2a = 90° or a — 45°. 
Again, if T be the time of flight for range X , since horizontal velo- The time 
city, V cos a, is constant, 
X = VT cos a* 
X 2 V 2, sin a cos a 2 V sin a 
or Ts— - 
V cos a 
Vg cos a 
of flight on 
a horizontal 
, plane in 
terms of 
muzzle 
velocity 
and angle of 
(5) departure. 
Since the horizontal velocity, V cos a, is constant, it is evident that The projec. 
£ yi tile reaches 
the projectile will traverse the horizontal distance - c y in time ^ • 
0 u a 
Bisect OB in C, then 
00 = CB = 
Draw CII perpendicular to OB , cutting the trajectory in IP. 
its highest 
point at 
one-half of 
its range on 
the hori¬ 
zontal 
yl^ne, §nd 
ijni ^lie-half' 
pfits total 
time of 
flight. 
