How do I calculate where a ballistic missile is going to land?

Question

I've been trying to design a homemade rocket, and I'm stumped on how I will calculate where the rocket will land. I can't figure out how to calculate the velocity (in m/s) a rocket will be at when its motor burns out. Is there an equation (or equations) I can use to determine a rockets velocity at burnout? I would prefer if the equations ignored air drag.a

Answer 1

If you ignore drag, the equations are simple but the error will be large. You need the dry mass of the rocket, the mass of the propellant, the thrust/time curve and the launch angle. If we define $m_0$ as the dry mass of the rocket, $m_p$ as the mass of the propellant, $F(t)$ as the thrust as a function of time, $\theta$ as the angle off the vertical of the rocket, and $g$ the acceleration of gravity our dynamic variables are $x(t),y(t), \theta(t), m(t)$ First compute the specific impulse as $I_{sp}=\frac 1{m_p}\int F(t)dt$ Then $$\dot m=-\frac {F(t)}{I_sp}\\ \ddot x=\frac{F(t)}{m(t)}\sin \theta(t)\\ \ddot y=\frac{F(t)}{m(t)}\cos\theta(t)-g\\ \theta(t)=\operatorname{arccot} \frac {\dot y}{\dot x}$$ and you can integrate these numerically.