
233Trajectories
But because we are in an inertial frame, i
= j
= k
= 0 and therefore
ijk
VVV
xyz
=++
(8.50)
If we break Equation 8.47 into its components, we get three coupled, ordinary, nonlinear
differential equations
=
(8.51)
VCVV g
D
(8.52)
=
(8.53)
The equation that couples Equations 8.51 through 8.53 is
V
xyz
=++
(8.54)
We can linearize these equations by making a few assumptions. First, let us assume that
there is no crosswind, so V
z
= 0, and if we further constrain the ratio of the vertical velocity
to the horizontal velocity to |V
y
/V
x
| = tan ϕ < 0.1, then V and V
x
are within 0.5% of each
other, and w