I have a problem creating a model for a horizontal rocket flight.
I want to model a rocket with constant force, drag constant and gravity. I also have to account for a changing mass and drag. I know I could calculate movement of this rocket by something like Runge-Kutta algorithm but I want to know if it is possible to create a equation that would give me a position (height) for a given time.
Basically those are my functions:
$$F_G (t) = - g \, m (t)$$
$$F_D (t) = - c \, v^2 (t)$$
$$F (t) = F_T (t) - F_G (t) + F_D (t)$$
where $F_T$ is the (constant) thrust, $F_G$ is gravity, and $F_D$ is drag.
I'm kinda stuck here so any help would be appreciated.
Assuming that you rocket is still not fast enough. By Newton's second law (like gt6989 said):
$m\ddot{x}+\dot{m}\dot{x}=F(t)-gm-c\dot{x}^2$ or $$m\ddot{x}+\dot{m}\dot{x}+c\dot{x}^2=F(t)-gm$$ You still need to give models for the thrust and also for the mass which is lost. An explicit solution will not be that easy.
You could also introduce the substitution $\dot{x}=z$
$$m\dot{z}+\dot{m}z+cz^2=F(t)-gm$$
This is a riccati differnetial equation. You might be lucky if there is a general explicit solution.
$$\dot{z}=\frac{m_d}{m}z-\frac{c}{m}z^2+\frac{F(t)}{m}-g$$
$$\dot{z}=\frac{m_d}{m_0 - m_d t}z-\frac{c}{m_0 - m_d t}z^2+\frac{F_0}{m_0 - m_d t}-g$$
Maple gives an explicit solution with Bessel functions.