Suppose we have a Rocket with initial mass $M_o$ and we want to sent it into space. The equation of motion is (i think... please tell me if there is something i forgot)

$$\frac {d(mu)}{dt}=-mg-\frac{1}{2}CρAu^2$$ where $m=m(t)$ is the mass of the rocket with $\dot{m}=-aM_o$ ($a$ is constant). $\qquad \qquad \qquad \qquad$ $g=g(r)$ is the Gravitational field $$ g(r)=\frac{G M_E}{r^2}$$ where $G$ is the gravitational constant and $M_E$ is the mass o the Earth and r is the distance of the rocket from the center of the Earth.$\qquad \qquad \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ $C$ is the drag coefficient, $A$ is the cross-sectional of the object and $ρ=ρ(r)$ is the density of the air changing with hight.$ \qquad \qquad \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad$ If we know the Final mass $M_f$ of the Rocket can we calculate the maximum hight it can get? (I forgot to include Coriolis and centrifugal force. Did Nasa include them in their missions or these forces are too small. And if yes what is the solution then). Is there any paper which solve this?


The thrust a rocket can generate is proportional to the mass flow (thus $\frac{dm}{dt}$) and the velocity at which this mass leaves the rocket, often called the effective exhaust velocity, $v_e$. So the sum of all forces acting on the rocket, when also neglecting the fictitious Coriolis and centrifugal forces and assuming a pure vertical ascent, will be equal to the mass of the rocket times it acceleration,

$$ m\frac{d^2r}{dt^2} = \frac{dm}{dt} v_e - \frac{GM_{\oplus}m}{r^2} - \frac{1}{2}C_D\rho A\left(\frac{dr}{dt}\right)^2 \tag{1} $$

This is a non-linear differential equation, which I believe does not have an analytical solution, but this still is a simplification, namely the density of the air, $\rho$, will change with altitude, since it is roughly proportional to the air pressure, the drag coefficient also increase significantly when your velocity is near the speed of sound and the effective exhaust velocity will also change during flight. All these extra variables which change during flight will make equation $(1)$ even more non-linear, so even less likely to be solved analytically, so you could resort to numerical calculations in order to approximate a flight.

When trying to achieve the greatest height, then you also might want to look at optimal control. So controlling the throttle/mass flow in such a way that you achieve the greatest height. For a vertical ascent this will be at terminal velocity (however this might not be optimal when you get close to Mach 1, due to the increase of $C_D$ near it).


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