(6.3-1)
Then
(6.3-2)
Also
(6.3-3) (6.3-4)
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
:
(6.3-5) (6.3-6) (6.3-7) (6.3-8)
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
(6.3-9)
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
x
= u cos
q
o
V
y
= v
x
tan
q
= u cos
q
o
tan
q
(6.3-10)
and the total velocity is
(6.3-11)
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.
