# On Four New Methods of Analytical Calculation of Rocket Trajectories

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## Abstract

**:**

## 1. Introduction

## 2. Dynamical Equations of the Rocket Trajectory

#### 2.1. Orthogonal Frame Moving with the Velocity

#### 2.2. Influence of the Lift-to-Drag Ratio

#### 2.3. Fixed Cartesian Frame with Axis Along Altitude

## 3. Trajectory Calculation via a Power Series of Time

#### 3.1. Dimensionless Gravity, Thrust and Drag Parameters

#### 3.2. Atmospheric Mass Density as a Function of the Mass of Burned Propellant

#### 3.3. Leading Coefficients of Power Series Expansion

## 4. Alternative Methods for Short Times or Long Times

#### 4.1. Residual Mass Fractions as Time Variable

#### 4.2. Atmospheric Mass Density as a Function of Residual Mass

#### 4.3. Determination of Exponent and Coefficients of the Series Solution

## 5. Three Distinct Methods of Trajectory Calculation

#### 5.1. Comparison of Accuracy for Short Times and Long Times

#### 5.2. Direct Calculation of the Taylor Series Expansion

#### 5.3. Peak Altitude in Post Burnout Flight

## 6. Computation of a Powered Followed by a Ballistic Ascent

#### 6.1. Input Data for Trajectory Calculation

#### 6.2. Altitude and Ascent Velocity in Powered Flight

#### 6.3. Peak Altitude in Unpowered Ballistic Flight

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Approximation to the trajectory of a rocket by tangents with a constant flight path angle.

No | Effect | Present Work | [1] | [2] | [3] | [4] | [5] |
---|---|---|---|---|---|---|---|

1 | rocket mass as function of time | √ | √ | √ | √ | √ | |

2 | drag included | √ | √ | √ | √ | √ | √ |

3 | lift included | √ | |||||

4 | aerodynamic forces depend on velocity | √ | √ | √ | √ | ||

5 | aerodynamic forces depend on altitude | √ | |||||

6 | flight path angle variable | √ | √ | √ | √ | √ | |

7 | non-zero angle-of-attack | √ | |||||

8 | thrust vector angle to flight path | √ | |||||

9 | multistage | √ | √ | ||||

10 | wind | √ |

**Table 2.**Input data for calculations of rocket trajectories by methods I in Section 3, method II in Section 4 and method III in Section 5.1 and Section 5.2.

Symbol | Meaning | Value | Unit |
---|---|---|---|

Environment | |||

g | acceleration of gravity | 9.81 | $\mathrm{m}{\mathrm{s}}^{-1}$ |

${\rho}_{0}$ | sea level mass density | 1.225 | $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ |

ℓ | atmospheric scale height | $2.6\times {10}^{4}$ | $\mathrm{m}$ |

Propulsion | |||

T | thrust | $1.555\times {10}^{7}$ | $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-2}$ |

c | propellant flow rate | $2.029\times {10}^{3}$ | $\mathrm{k}\mathrm{g}{\mathrm{s}}^{-1}$ |

${m}_{0}-{m}_{1}$ | propellant mass | $2.841\times {10}^{5}$ | $\mathrm{k}\mathrm{g}$ |

${t}_{1}-{t}_{0}$ | burn time | $1.4\times {10}^{2}$ | $\mathrm{s}$ |

Aerodynamics | |||

${m}_{0}$ | initial mass | $7.77\times {10}^{5}$ | $\mathrm{k}\mathrm{g}$ |

S | cross-sectional area | 3.76 × 10 | ${\mathrm{m}}^{2}$ |

${C}_{\mathrm{D}}$ | drag coefficient | 0.15 | ^{*} |

Calculated parameters | |||

${m}_{*}$ | reference mass | $1.198\times {10}^{6}$ | $\mathrm{k}\mathrm{g}$ |

a | weight parameter | 5.533 × 10 | ^{*} |

b | thrust parameter | $1.129\times {10}^{2}$ | ^{*} |

f | drag parameter | $1.156\times {10}^{-1}$ | ^{*} |

${t}_{*}$ | reference time | $3.829\times {10}^{2}$ | $\mathrm{s}$ |

${v}_{*}$ | reference velocity | 6.790 × 10 | $\mathrm{m}{\mathrm{s}}^{-1}$ |

**Table 3.**Calculation of powered trajectories with four discretizations for method I in Section 6.1 and method III in Section 6.2.

Method | I | III |
---|---|---|

Coefficients of series | ||

first | ${A}_{0}=1.00$ | ${D}_{0}=0$ |

second | ${A}_{1}=0.00$ | ${D}_{1}=0$ |

third | ${A}_{2}=28.77$ | ${D}_{2}=5.10\mathrm{m}{\mathrm{s}}^{-2}$ |

fourth | ${A}_{3}=18.81$ | ${D}_{3}=8.71\times {10}^{-3}\mathrm{m}{\mathrm{s}}^{-3}$ |

fifth | ${A}_{4}=391.4$ | N/A |

Altitude versus time ($\mathrm{m}$) | ||

$t=35\mathrm{s}$ | $0.646\times {10}^{4}$ | $0.662\times {10}^{4}$ |

$t=70\mathrm{s}$ | $2.396\times {10}^{4}$ | $2.798\times {10}^{4}$ |

$t=105\mathrm{s}$ | $4.554\times {10}^{4}$ | $6.633\times {10}^{4}$ |

$t=140\mathrm{s}$ | $6.620\times {10}^{4}$ | $1.239\times {10}^{5}$ |

Velocity versus time ($\mathrm{m}{\mathrm{s}}^{-1}$) | ||

$t=35\mathrm{s}$ | $3.668\times {10}^{2}$ | $3.891\times {10}^{2}$ |

$t=70\mathrm{s}$ | $5.935\times {10}^{2}$ | $8.422\times {10}^{2}$ |

$t=105\mathrm{s}$ | $6.161\times {10}^{2}$ | $1.359\times {10}^{3}$ |

$t=140\mathrm{s}$ | $5.593\times {10}^{2}$ | $1.941\times {10}^{3}$ |

**Table 4.**Calculation of ballistic trajectory by method IV in Section 5.3, using initial data from methods I or II in Section 6.3.

Method | I + IV | III + IV | Units |
---|---|---|---|

Burnout data (${m}_{1}=4.929\times {10}^{5}\mathrm{k}\mathrm{g}$) | |||

${z}_{1}$ | $6.610\times {10}^{4}$ | $1.239\times {10}^{5}$ | $\mathrm{m}$ |

${\dot{z}}_{1}$ | $5.593\times {10}^{2}$ | $1.941\times {10}^{3}$ | $\mathrm{m}{\mathrm{s}}^{-1}$ |

${\rho}_{1}={\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{-z/\ell}$ | $9.602\times {10}^{-2}$ | $1.044\times {10}^{-2}$ | $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ |

Calculated values | |||

${m}_{**}={\rho}_{1}S\ell $ | $9.390\times {10}^{4}$ | $1.021\times {10}^{4}$ | $\mathrm{k}\mathrm{g}$ |

$\vartheta ={C}_{\mathrm{D}}{m}_{**}/{m}_{1}$ | $2.857\times {10}^{-2}$ | $3.107\times {10}^{-3}$ | * |

${\overline{E}}_{0}={\dot{z}}_{1}^{2}/(2g\ell )$ | $6.132\times {10}^{-1}$ | 7.382 | * |

${E}_{0}={\overline{E}}_{0}\phantom{\rule{0.166667em}{0ex}}{\mathrm{e}}^{-\vartheta}$ | $5.959\times {10}^{-1}$ | 7.359 | * |

Apogee of trajectory (${m}_{1}=4.929\times {10}^{5}\mathrm{k}\mathrm{g}$) | |||

$X={E}_{0}/(1-\vartheta )$ | $6.134\times {10}^{-1}$ | 7.382 | * |

${z}_{2}-{z}_{1}=X\ell $ | $1.595\times {10}^{4}$ | $1.919\times {10}^{5}$ | $\mathrm{m}$ |

${z}_{2}$ | $8.215\times {10}^{4}$ | $3.158\times {10}^{5}$ | $\mathrm{m}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Campos, L.M.B.C.; Gil, P.J.S.
On Four New Methods of Analytical Calculation of Rocket Trajectories. *Aerospace* **2018**, *5*, 88.
https://doi.org/10.3390/aerospace5030088

**AMA Style**

Campos LMBC, Gil PJS.
On Four New Methods of Analytical Calculation of Rocket Trajectories. *Aerospace*. 2018; 5(3):88.
https://doi.org/10.3390/aerospace5030088

**Chicago/Turabian Style**

Campos, Luís M. B. C., and Paulo J. S. Gil.
2018. "On Four New Methods of Analytical Calculation of Rocket Trajectories" *Aerospace* 5, no. 3: 88.
https://doi.org/10.3390/aerospace5030088