Editing Tsiolkovsky rocket equation (section)

{{Short description|Mathematical equation describing the motion of a rocket}}
{{Astrodynamics |Preflight engineering}}
[[File:Tsiolkovsky rocket equation.svg|thumb|right|A rocket's required [[mass ratio]] as a function of [[effective exhaust velocity]] ratio]]

The '''classical rocket equation''', or '''ideal rocket equation''' is a mathematical equation that describes the motion of vehicles that follow the basic principle of a [[rocket]]: a device that can apply acceleration to itself using [[thrust]] by expelling part of its mass with high [[velocity]] and can thereby move due to the [[conservation of momentum]].
It is credited to [[Konstantin Tsiolkovsky]], who independently derived it and published it in 1903,<ref name="here">К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online [http://epizodsspace.airbase.ru/bibl/dorev-knigi/ciolkovskiy/sm.rar here] {{webarchive|url=https://web.archive.org/web/20110815183920/http://epizodsspace.airbase.ru/bibl/dorev-knigi/ciolkovskiy/sm.rar |date=2011-08-15 }} in a [[RAR (file format)|RARed]] PDF)</ref><ref name="ReactiveFlyingMachines">{{Cite web|last=Tsiolkovsky|first=K.|title=Reactive Flying Machines|url=http://epizodsspace.airbase.ru/bibl/inostr-yazyki/tsiolkovskii/tsiolkovskii-nhedy-t2-1954.pdf}}</ref> although it had been independently derived and published by [[William Moore (British mathematician)|William Moore]] in 1810,<ref name="moore1810">{{Cite journal|last=Moore|first=William|date=1810|title=On the Motion of Rockets both in Nonresisting and Resisting Mediums|url=https://www.biodiversitylibrary.org/page/36067032|journal=Journal of Natural Philosophy, Chemistry & the Arts|volume=27|pages=276–285}}</ref> and later published in a separate book in 1813.<ref name="moore">{{Cite book|last=Moore|first=William|url=https://books.google.com/books?id=nrVgAAAAcAAJ|title=A Treatise on the Motion of Rockets: to which is added, an Essay on Naval Gunnery, in theory and practice, etc|date=1813|publisher=G. & S. Robinson|language=en}}</ref> [[Robert Goddard]] also developed it independently in 1912, and [[Hermann Oberth]] derived it independently about 1920.

The maximum change of [[velocity]] of the vehicle, <math>\Delta v</math> (with no external forces acting) is:

<math display="block">\Delta v = v_\text{e} \ln \frac{m_0}{m_f} = I_\text{sp} g_0 \ln \frac{m_0}{m_f},</math>
where:
* <math>v_\text{e}</math> is the [[effective exhaust velocity]];
**<math>I_\text{sp}</math> is the [[specific impulse]] in dimension of time;
**<math>g_0</math> is [[standard gravity]];
* <math>\ln</math> is the [[natural logarithm]] function;
* <math>m_0</math> is the initial total mass, including [[propellant]], a.k.a. wet mass;
* <math>m_f</math> is the final total mass without propellant, a.k.a. dry mass.

Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g., [[orbital speed]] or [[escape velocity]]), and a given dry mass <math>m_f</math>, the equation can be solved for the required wet mass <math>m_0</math>:
<math display="block">m_0 = m_f e^{\Delta v / v_\text{e}}.</math> The required propellant mass is then <math display="block">m_0 - m_f = m_f (e^{\Delta v / v_\text{e}} - 1)</math>

The necessary wet mass grows exponentially with the desired delta-v.