# Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles
## 3 A Second Step (Gravity Losses)
\tag{9}\]  
Dividing the first by the second equation of the system (9), one has  
\[\frac{dV}{dz}=\bar{n}\frac{V}{z}-\frac{V}{z}\frac{1-z^{2}}{1+z^{2}}\]  
that is  
\[\frac{dV}{V}=\left(\frac{\bar{n}}{z}-\frac{1}{z}\,\frac{1-z^{2}}{1+z^{2}} \right)dz\]By integration  
\[V=A\,z^{n-1}(1+z^{2}) \tag{10}\]  
The velocity in (10) is a function of the variable \(z\). The time dependence can be obtained deriving the relation between time and \(z\). Substitution of (10) in the second equation of the system (9) gives  
\[\dot{z}=\frac{dz}{dt}=-\frac{g}{A}\,\frac{1}{z^{n-2}(1+z^{2})}\]  
by separation of variables and integrating from \(t_{0}=0\) to \(t\)  
\[t=\frac{A}{g}\left(\frac{z^{n-1}}{\bar{n}-1}+\frac{z^{n+1}}{\bar{n}+1}\right) \tag{11}\]  
The launch initial conditions are in the given hypothesis  
\[\left\{ \begin{array}{l}V_{0}=0\\ \gamma_{0}=\frac{\pi}{2}\\ \chi_{0}=0\Rightarrow z_{0}=0\end{array} \right.\]  
Imposing these initial conditions in the solution (10), one has  
\[V_{0}=A\,z_{0}^{n-1}(1+z_{0}^{2})=0\quad\forall A\]  
Then, the initial conditions are verified for any value of the constant \(A\). That is, there are infinite (gravity-turn) solutions for the velocity, parameterized by A, corresponding to the same initial condition. This violation of the [PERSON] theorem on uniqueness of the solution of a differential equation is due to the singularity of \(\dot{z}\) for \(V=0\) in the system (9). One can take advantage of this multiplicity of solutions to select for instance the solution that achieves a chosen flight-path angle \(\gamma_{f}\) at the burn-out time \(t_{b}\). The chosen \(\gamma_{f}\) defines a value \(z_{f}=\tan(\frac{\pi/2-\gamma_{f}}{2})\). By Equation (11), one obtains  
\[A=g\,t_{b}\left(\frac{z_{f}^{n-1}}{\bar{n}-1}+\frac{z_{f}^{n+1}}{\bar{n}+1} \right)^{-1} \tag{12}\]  
and the related velocity is obtained inserting this value of \(A\) into Equation (10). In particular, the velocity at burn out is  
\[V_{f}=A\,z_{f}^{n-1}(1+z_{f}^{2}) \tag{13}\]  
In addition, the altitude \(h\) and range \(s\) can be easily derived [16]. For the altitude  
\[\dot{h}=V\,\sin\gamma=V\,\cos\chi=V\,\frac{1-z^{2}}{1+z^{2}}\]  
Then  
\[\frac{dh}{dz}=\frac{\dot{h}}{z}=\frac{V^{2}}{g\,z}\,\frac{1-z^{2}}{1+z^{2}}\]  
By Equation (10), one has  
\[\frac{dh}{dz}=\frac{A^{2}}{g}\left(z^{2n-3}-z^{2n+1}\right)\]The elementary integration of the above equation between current \(z\) and initial \(z_{0}\) gives the formula for the altitude  
\[h(z)=h(z_{0})+\frac{A^{2}}{g}\left[\frac{z^{2n-2}}{2\pi-2}-\frac{z^{2n+2}}{\bar {n}+2}\right]_{z_{0}}^{z_{f}} \tag{14}\]  
For the range \(s\)  
\[s=V\,\cos\gamma=V\,\sin\chi=2\,V\frac{z}{1+z^{2}}\]  
then  
\[\frac{ds}{dz}=\frac{s}{z}=2\frac{A^{2}}{g}\left(z^{2n-1}+z^{2n+1}\right)\]  
The integration gives the formula for the range  
\[s(z)=s(z_{0})+2\frac{A^{2}}{g}\left[\frac{z^{2n}}{2\bar{n}}+\frac{z^{2n+2}}{2 \bar{n}+2}\right]_{z_{0}}^{z_{f}} \tag{15}\]  
Equations (10), (12), (14) and (15), together with (11), provide the state variables of the gravity-turn trajectory of a single-stage rocket with flight-path angle \(\gamma_{f}\) at burn-out time \(t_{b}\). For instance, let us consider initial lift off condition \((V_{0}=0,\chi_{0}=0,h_{0}=0,s_{0}=0)\) and choose the final condition \(\gamma_{f}=0\) this condition corresponds to \(z_{f}=1\). Then Equation (12) gives  
\[A=g\,t_{b}\left(\frac{z_{f}^{n-1}}{\bar{n}-1}+\frac{z_{f}^{n+1}}{\bar{n}+1} \right)^{-1} \tag{12}\]0  
By Equations (10), (14) and (15), the velocity, altitude and range at burn out are equal to  
\[A=g\,t_{b}\left(\frac{z_{f}^{n-1}}{\bar{n}-1}+\frac{z_{f}^{n+1}}{\bar{n}+1} \right)^{-1} \tag{12}\]1  
\[A=g\,t_{b}\left(\frac{z_{f}^{n-1}}{\bar{n}-1}+\frac{z_{f}^{n+1}}{\bar{n}+1} \right)^{-1} \tag{12}\]2  
\[A=g\,t_{b}\left(\frac{z_{f}^{n-1}}{\bar{n}-1}+\frac{z_{f}^{n+1}}{\bar{n}+1} \right)^{-1} \tag{12}\]3  
Note that the final velocity formula of a rocket is generally obtained by the Tziolkowski formula that takes into account only the thrust acceleration.