6.3
Siacci’s Method
Siacci’s
appr simplified their solutions.
6.3.1
The Change of Independent Variables
Siacci
introduced the “pseudovelocity”
u
shown in
Figure 62
as a new independent variable replacing time
t
in our previous equations. The pseudovelocity
u
is a velocity in the direction of the extended bore centerline which
would give the correct component
v
x
resolved along the xaxis. Then
u
and
v
are related by the equation.
(6.31)
Then
(6.32)
Also
(6.33) (6.34)
Using equations (6.31) through (6.34) in equations (6.215) and (6.216) we can derive the following differential equations of bullet motion in terms of the new independent variable
u
:
(6.35) (6.36) (6.37) (6.38)
The initial conditions for the solution of these equations are:
t
o
= 0
u
o
= v
m
x
o
= 0
y
o
= 0
(tan
q
)
o
= tan
q
o
(6.39)
The velocity components
v
x
and
v
y
at any point in the trajectory are given by [equations (6.31) and (6.33)]:
v
x
= u cos
q
o
V
y
= v
x
tan
q
= u cos
q
o
tan
q
(6.310)
and the total velocity is
(6.311)
We now have a set of firstorder differential equations for the time of flight
t
, the range
x
, the vertical coordinate
y
, and the trajectory slope
tan
X. But these simplified equations of the bullet motion are still nonlinear coupled.
