# Analytic Derivation of Ascent Trajectories and Performance of Launch Vehicles
## 4 Analytic Derivation of Launcher Trajectories
By these solutions, the velocity, altitude and range profiles can be computed using Equations (13)-(15) in the ranges \([0,z_{20}],\ldots[z_{N\ 0},z_{f}]\). For instance, for a three-stage launcher, one has  
\[t_{b1}=\frac{A_{1}}{g}\left(\frac{z_{20}^{n_{1}-1}}{n_{1}-1}+\frac{z_{20}^{n_{1 }+1}}{n_{1}+1}\right) \tag{24}\]  
\[t_{b2}=\frac{A_{2}}{g}\Bigg{[}\Bigg{(}\frac{z_{30}^{n_{2}-1}}{n_{2}-1}+\frac{ z_{30}^{n_{2}+1}}{n_{2}+1}\Bigg{)}-\left(\frac{z_{20}^{n_{2}-1}}{n_{2}-1}+\frac{z_{20}^{ n_{2}+1}}{n_{2}+1}\right)\Bigg{]} \tag{25}\]  
\[t_{b3}=\frac{A_{3}}{g}\left[\left(\frac{z_{3}^{n_{3}-1}}{n_{3}-1}+\frac{z_{3}^ {n_{3}+1}}{n_{3}+1}\right)\right]_{z_{3_{0}}}^{z_{3f}} \tag{26}\]  
\[A_{1}=A_{2}z_{20}^{\overline{n}_{2}-\overline{n}_{1}} \tag{27}\]  
\[A_{2}=A_{3}\ z_{30}^{\overline{n}_{3}-\overline{n}_{2}} \tag{28}\]  
(these five Equations (24)-(28) with the five unknowns \(A1\), \(A_{2}\), \(A_{3}\), \(z_{20}\), \(z_{30}\)). One solution is to write the variables \(A_{1}\) and \(A_{2}\) as function of \(A_{3}\) using Equations (27) and (28). Then, the ratios \(\frac{t_{b_{2}}}{t_{b_{1}}}\), \(\frac{t_{b_{2}}}{t_{b_{1}}}\) are obtained dividing (25) and (26) by the Equation (24). The two ratios give two equations on the two unknown \(z_{20}\), \(z_{30}\). The solutions can be found numerically within the interval [0,1], and they determine the values of the flight-path angles at the end of the first and second stages. The values \(A_{1}\), \(A_{2}\), \(A_{3}\) are derived by Equations (24) and (26) and and the velocities at the end of each stage are obtained. For a four-stage launcher, there are seven equations in the seven unknowns \(A1\), \(A_{2}\), \(A_{3}\), \(A_{4}\), \(z_{20}\), \(z_{30}\), \(z_{40}\)  
\[A_{1}=A_{2}z_{20}^{\overline{n}_{2}-\overline{n}_{1}} \tag{29}\]  
\[A_{2}=A_{3}\;z_{30}^{\overline{n}_{3}-\overline{n}_{2}} \tag{30}\]  
\[A_{3}=A_{4}\;z_{40}^{\overline{n}_{4}-\overline{n}_{3}} \tag{31}\]  
\[t_{b1}=\frac{A_{1}}{g}\left(\frac{z_{20}^{n_{1}-1}}{n_{1}-1}+\frac{z_{20}^{n_ {1}+1}}{n_{1}+1}\right) \tag{32}\]  
\[t_{b2}=\frac{A_{2}}{g}\Bigg{[}\left(\frac{z_{30}^{n_{2}-1}}{n_{2}-1}+\frac{z_ {30}^{n_{2}+1}}{n_{2}+1}\right)-\left(\frac{z_{20}^{n_{2}-1}}{n_{2}-1}+\frac{ z_{20}^{n_{2}+1}}{n_{2}+1}\right)\Bigg{]} \tag{33}\]  
\[t_{b2}=\frac{A_{2}}{g}\Bigg{[}\Bigg{(}\frac{z_{30}^{n_{2}-1}}{n_{2}-1}+\frac{ z_{30}^{n_{2}+1}}{n_{2}+1}\Bigg{)}-\left(\frac{z_{20}^{n_{2}-1}}{n_{2}-1}+\frac{z_{20}^{ n_{2}+1}}{n_{2}+1}\right)\Bigg{]} \tag{25}\]0  
\[t_{b2}=\frac{A_{2}}{g}\Bigg{[}\Bigg{(}\frac{z_{30}^{n_{2}-1}}{n_{2}-1}+\frac{ z_{30}^{n_{2}+1}}{n_{2}+1}\Bigg{)}-\left(\frac{z_{20}^{n_{2}-1}}{n_{2}-1}+\frac{z_{20}^{ n_{2}+1}}{n_{2}+1}\right)\Bigg{]} \tag{25}\]1  
The first three equations allow us to write \(A_{1}\), \(A_{2}\), \(A_{3}\) as function of \(A_{4}\) only. Then, the three ratios \(\frac{t_{b_{2}}}{t_{b_{1}}}\), \(\frac{t_{b_{3}}}{t_{b_{1}}}\), \(\frac{t_{b_{4}}}{t_{b_{1}}}\) are functions of the three unknowns \(z_{20}\), \(z_{30}\), \(z_{40}\). The solution is used then to derive \(A_{1}-A_{4}\). Note that the above formulas for three- or four-stage launchers provide the values of the flight-path angles and the velocities at the end of each stage for the launcher trajectory with a fixed value of the final flight-path angle.