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 6-2 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 x-axis. Then u and v are related by the equation.
Using equations (6.3-1) through (6.3-4) in equations (6.2-15) and (6.2-16) we can derive the following differential equations of bullet motion in terms of the new independent variable u :
The initial conditions for the solution of these equations are:
u o = v m
x o = 0
y o = 0
The velocity components v x and v y at any point in the trajectory are given by [equations (6.3-1) and (6.3-3)]:
V y = v x tan q = u cos q o tan q (6.3-10)
and the total velocity is
We now have a set of first-order 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.