246 
PRINCIPLES OP GUNNERY. 
Equation 
of motion 
connecting 
time and 
Telocity. 
It now becomes necessary to investigate the conditions op velo¬ 
city with respect to time . 
From equation (2), 
retardation = 
dv 
dt 
d 3 K# 
w ' (1000) 3 ’ 
supposing the projectile to move in a straight line unaffected by gravity; 
or, inverting and integrating, 
w /" 9 (1000 )*dv % 
d*J ZP ’ 
••( 6 ) 
the limits of this integral being the velocities at the beginning and 
end of the time, t. 
Approxi¬ 
mate calcu¬ 
lation of 
remaining 
velocity 
with regard 
to time. 
For the range of velocities in which the cubic law holds, K may be 
taken as constant for similarly shaped projectiles; so that equation (6) 
becomes 
+ _ w (1000)3 f*v dv 
~d*~~ir~J v 
which may be integrated thus 
d*, K _ 1 1 
w * (lOOO) 3 ” 2v 2 2 
cr ^ f —Y 1 l yiooo) 3 . 
•J \v 3 F*) 2 K ’ 
(?) 
Where V is the muzzle velocity and v the remaining velocity at a given 
time, t } from the muzzle. 
Prom (7), 
/ 2 K <P .m,, 
V mw'^ 
.( 8 ) 
V J IV 
which gives the remaining velocity at a given interval of time. 
Distance 
and Velo¬ 
city Table, 
Time and 
Velocity 
Table. 
For velocities where the cubic law holds, and the projectile moves 
approximately in a straight line, the remaining velocity at a given time 
may be fairly accurately determined by the above formula by sub¬ 
stituting the mean value of K= 108*5. 
But in order to facilitate the computation of these problems at all 
velocities where the cubic law does not hold and the value of K changes 
rapidly, Bashforth has calculated equation (3), s —J] ; 
(the value of K continually changing, so as to have the proper value 
corresponding to the velocity), and tabulated the results in what may 
be Called a Distance and Velocity Table [vide Table I., p. 250); also the 
c p W7oo (lOOOWft; 
value of equation (6), — t = / -—— , in what may be called a 
10 
Time and Velocity Table 
J 500 
[vide Table II., p. 252), 
Table I. connects distance and velocity, and is denoted by 
Table IL connects time and velocity, and is denoted by T v . 
