Editing Spacecraft flight dynamics (section)

===Translunar injection===
{{Main|Translunar injection}}
This must be timed so that the Moon will be in position to capture the vehicle, and might be modeled to a first approximation as a Hohmann transfer. However, the rocket burn duration is usually long enough, and occurs during a sufficient change in flight path angle, that this is not very accurate. It must be modeled as a [[orbital maneuver#Low thrust for a long time|non-impulsive maneuver]], requiring [[numerical integration|integration]] by [[finite element analysis]] of the accelerations due to propulsive thrust and gravity to obtain velocity and flight path angle:{{sfnp|Kromis|1967| p=11:154}}
<math display="block">\begin{align}
\dot{v} &= \frac{F\cos\alpha}m - g\cos\theta\\
\dot{\theta} &= \frac{F\sin\alpha}{mv} + \left(\frac g v - \frac v r\right) \sin\theta, \\
v &= \int_{t_0}^t \dot{v}\, dt \\
\theta &= \int_{t_0}^t \dot{\theta}\, dt
\end{align}</math>
where:
*''F'' is the engine thrust;
*''α'' is the angle of attack;
*''m'' is the vehicle's mass;
*''r'' is the radial distance to the planet's center; and
*''g'' is the [[gravitational acceleration]], which varies with the inverse square of the radial distance:{{sfnp|Kromis|1967| p=11:154}} <math display="block">g = g_0\left(\frac{r_0}r\right)^2</math>

Altitude <math>h</math>, downrange distance <math>s</math>, and radial distance <math>r</math> from the center of the Earth are then computed as:{{sfnp|Kromis|1967| p=11:154}}
<math display="block">\begin{align}
h &= \int_{t_0}^t v \cos \theta\, dt \\
r &= r_0+h \\
s &= r_0 \int_{t_0}^t \frac v r \sin \theta\, dt
\end{align}</math>