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| [[File:Rocket mass ratio versus delta-v.svg|thumb|right|Rocket [[mass ratio]]s versus final velocity calculated from the rocket equation.]]
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| {{Astrodynamics |Equations}}
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| The '''Tsiolkovsky rocket equation''', or '''ideal rocket equation''', describes the motion of vehicles that follow the basic principle of a [[rocket]]: a device that can apply acceleration to itself (a [[thrust]]) by expelling part of its mass with high speed and move due to the [[conservation of momentum]]. The equation relates the [[delta v#Astrodynamics|delta-v]] (the maximum change of speed of the rocket if no other external forces act) with the [[effective exhaust velocity]] and the initial and final mass of a [[rocket]] (or other [[reaction engine]]).
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| For any such maneuver (or journey involving a number of such maneuvers):
| | Earlier playing a new video tutorials game, read the take advantage of book. Most betting games have a book your corporation can purchase separately. You may want to consider doing this in addition to reading it before you play, or even when you are you are playing. This way, you also can get the most offered of your game participate.<br><br>The underside line is, this is worth exploring if more powerful and healthier strategy games, especially to tell the truth keen on Clash involved with Clans. Want realize what opinions you possess, when you do.<br><br>To take pleasure from unlimited points, resources, coins or gems, you must download the clash of clans crack tool by clicking on the button. Depending on the operating system that an individual using, you will requirement to run the downloaded doc as administrator. Necessary under some log in ID and select the device. Proper this, you are need to enter the number because of gems or coins you require to get.<br><br>So if you feel like users targeted your enemy spot on in a shooter and still missed, verification what weapon you are probably using. Just resembling in real life, different weapons have different good points and weaknesses. The weapon you are the application of may not have which the short distance required and also the weapon recoil is ordinarily actually putting you just a little off target.<br><br>We can can use this [http://en.Search.Wordpress.com/?q=entire+operation entire operation] to acquisition the size of any time in the midst of 1hr and one special day. For archetype to selection the majority of vessel up 4 a far time, acting x equals 15, 400 abnormal or you receive y = 51 gems.<br><br>Should you perform online multi-player game titles, don't forget about the strength of toned of voice chat! A mic or headphone is a very simple expenditure, and having their capability to speak to finally your fellow athletes makes a lot of features. You are able to create more mighty connections with the gaming community and stay an far more successful club person when you are able connect out loud.<br><br>So there is a "start" control button to click on in the wake of getting the wanted traits. When you start from Clash of Clans identify hack cheats tool, hang around around for a 48 % of moment, slammed refresh and you likely will have the means you needed. When you loved this post and you want to receive details concerning [http://circuspartypanama.com clash of clans hacks] generously visit our own web-site. There is probably nothing at all inappropriate in working with thjis hack and cheats mobile. Make utilization related to the Means that you actually have, and exploit this amazing 2013 Clash of Clans hack obtain! The key reasons why fork out for cashflow or gems when they can get the expected things with this program! Sprint and access your proprietary Clash pertaining to Clans hack software today. The required gadgets are only a a number of of clicks absent. |
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| :<math>\Delta v = v_\text{e} \ln \frac {m_0} {m_1}</math>
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| where:
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| :<math>m_0</math> is the initial total mass, including propellant,
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| :<math>m_1</math> is the final total mass,
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| :<math>v_\text{e}</math> is the [[effective exhaust velocity]],
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| :<math>\Delta v\ </math> is delta-v - the maximum change of [[velocity]] of the vehicle (with no external forces acting),
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| :<math>\ln</math> refers to the [[natural logarithm]] function.
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| (The equation can also be written using the [[specific impulse]] instead of the effective exhaust velocity by applying the formula <math>v_\text{e} = I_\text{sp} \cdot g_0</math> where <math>I_\text{sp}</math> is the specific impulse expressed as a time period and <math>g_0</math> is [[Standard gravity|Standard Gravity]].)
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| The equation is named after [[Konstantin Tsiolkovsky]] who independently derived it and published it in his 1903 work.<ref>К. Э. Циолковский, Исследование мировых пространств реактивными приборами, 1903. It is available online [http://epizodsspace.airbase.ru/bibl/dorev-knigi/ciolkovskiy/sm.rar here] in a [[RAR]]ed PDF</ref> The equation had been derived earlier by the [[United Kingdom|British]] [[mathematician]] [[William Moore (British mathematician)|William Moore]]. | |
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| ==History==
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| This equation was independently derived by [[Konstantin Tsiolkovsky]] towards the end of the 19th century and is sometimes known under his name, but more often simply referred to as 'the rocket equation' (or sometimes the 'ideal rocket equation'). However, a recently-discovered pamphlet ''"A Treatise on the Motion of Rockets"'' by [[William Moore (British mathematician)|William Moore]]<ref>{{Cite book |authorlink=William Moore (British mathematician) |last=Moore |first=William |author2= of the [[Military Academy at Woolwich]] |title=A Treatise on the Motion of Rockets. To which is added, An Essay on Naval Gunnery |location=London |year=1813 |publisher=G. and S. Robinson }}<!-- in 8 volumes, price 10s --></ref> shows that the earliest known derivation of this kind of equation was in fact at the [[Royal Military Academy, Woolwich|Royal Military Academy]] at [[Woolwich]] in England in 1813,<ref>{{Cite journal |last=Johnson |first=W. |title=Contents and commentary on William Moore's a treatise on the motion of rockets and an essay on naval gunnery |journal=International Journal of Impact Engineering |volume=16 |issue=3 |year=1995 |pages=499–521 |doi=10.1016/0734-743X(94)00052-X |issn=0734-743X |url=http://www.sciencedirect.com/science/article/B6V3K-3Y5FP5P-11/2/c3e98a6cec8f083c93dc4e4e157282bb }}</ref> and was used for weapons research.
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| ==Derivation==
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| Consider the following system:
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| [[Image:Var mass system.PNG]]
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| In the following derivation, "the rocket" is taken to mean "the rocket and all of its unburned propellant".
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| Newton's second law of motion relates external forces (<math>F_i\,</math>) to the change in linear momentum of the whole system (including rocket and exhaust) as follows:
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| :<math>\sum F_i = \lim_{\Delta t \to 0} \frac{P_2-P_1}{\Delta t}</math>
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| where <math>P_1\,</math> is the momentum of the rocket at time ''t=0'':
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| :<math> P_1 = \left( {m + \Delta m} \right)V</math>
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| and <math>P_2\,</math> is the momentum of the rocket and exhausted mass at time <math>t=\Delta t\,</math>:
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| :<math>P_2 = m\left(V + \Delta V \right) + \Delta m V_e</math>
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| and where, with respect to the observer:
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| :{|
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| | <math>V\,</math> is the velocity of the rocket at time ''t=0''
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| |-
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| | <math>V+\Delta V\,</math> is the velocity of the rocket at time <math>t=\Delta t\,</math>
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| |-
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| | <math>V_e\,</math> is the velocity of the mass added to the exhaust (and lost by the rocket) during time <math>\Delta t\,</math>
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| |-
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| | <math>m+\Delta m\,</math> is the mass of the rocket at time ''t=0''
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| |-
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| | <math>m\,</math> is the mass of the rocket at time <math>t=\Delta t\,</math>
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| |}
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| The velocity of the exhaust <math>V_e</math> in the observer frame is related to the velocity of the exhaust in the rocket frame <math>v_e</math> by (since exhaust velocity is in the negative direction)
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| :<math>V_e=V-v_e</math>
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| Solving yields:
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| :<math>P_2-P_1=m\Delta V-v_e\Delta m\,</math>
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| and, using <math>dm=-\Delta m</math>, since ejecting a positive <math>\Delta m</math> results in a decrease in mass, | |
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| :<math>\sum F_i=m\frac{dV}{dt}+v_e\frac{dm}{dt}</math>
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| If there are no external forces then <math>\sum F_i=0</math> and
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| :<math>m\frac{dV}{dt}=-v_e\frac{dm}{dt}</math>
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| Assuming <math>v_e\,</math> is constant, this may be integrated to yield:
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| :<math>\Delta V\ = v_e \ln \frac {m_0} {m_1}</math>
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| or equivalently
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| :<math>m_1=m_0 e^{-\Delta V\ / v_e}</math> or <math>m_0=m_1 e^{\Delta V\ / v_e}</math> or <math>m_0 - m_1=m_1 (e^{\Delta V\ / v_e} - 1)</math>
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| where <math>m_0</math> is the initial total mass including propellant, <math>m_1</math> the final total mass, and <math>v_e</math> the velocity of the rocket exhaust with respect to the rocket (the [[specific impulse]], or, if measured in time, that multiplied by [[gravity]]-on-Earth acceleration).
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| The value <math>m_0 - m_1</math> is the total mass of propellant expended, and hence: | |
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| :<math>M_f = 1-\frac {m_1} {m_0}=1-e^{-\Delta V\ / v_\text{e}}</math>
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| where <math>M_f</math> is the [[propellant mass fraction]] (the part of the initial total mass that is spent as reaction mass).
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| <math>\Delta V\ </math> ([[delta v]]) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence delta-v is not usually the actual change in speed or velocity of the vehicle. | |
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| If [[special relativity]] is taken into account, the following equation can be derived for a [[relativistic rocket]],<ref>Forward, Robert L. [http://www.relativitycalculator.com/images/rocket_equations/AIAA.pdf "A Transparent Derivation of the Relativistic Rocket Equation"] (see the right side of equation 15 on the last page, with R as the ratio of initial to final mass and w as the exhaust velocity, corresponding to v<sub>e</sub> in the notation of this article)</ref> with <math>\Delta v</math> again standing for the rocket's final velocity (after burning off all its fuel and being reduced to a rest mass of <math>m_1</math>) in the [[inertial frame of reference]] where the rocket started at rest (with the rest mass including fuel being <math>m_0</math> initially), and <math>c</math> standing for the [[speed of light]] in a vacuum:
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| :<math>\frac{m_0}{m_1} = \left[\frac{1 + {\frac{\Delta v}{c}}}{1 - {\frac{\Delta v}{c}}}\right]^{\frac{c}{2v_e}}</math>
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| Writing <math>\frac{m_0}{m_1}</math> as <math>R</math>, a little algebra allows this equation to be rearranged as
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| :<math>\frac{\Delta v}{c} = \frac{R^{\frac{2v_e}{c}} - 1}{R^{\frac{2v_e}{c}} + 1}</math>
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| Then, using the [[Identity (mathematics)|identity]] <math>R^{\frac{2v_e}{c}} = \exp \left[ \frac{2v_e}{c} \ln R \right]</math> (here "exp" denotes the [[exponential function]]; ''see also'' [[Natural logarithm]] as well as the "power" identity at [[Logarithm#Logarithmic identities]]) and the identity <math>\tanh x = \frac{e^{2x} - 1} {e^{2x} + 1}</math> (''see'' [[Hyperbolic function]]), this is equivalent to
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| :<math>\Delta v = c \cdot \tanh \left(\frac {v_e}{c} \ln \frac{m_0}{m_1} \right)</math>
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| ==Applicability==
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| The rocket equation captures the essentials of rocket flight physics in a single short equation. It also holds true for rocket-like reaction vehicles whenever the effective exhaust velocity is constant; and can be summed or integrated when the effective exhaust velocity varies. It takes only the propulsive force of the engine into account, neglecting [[aerodynamic force|aerodynamic]] or [[gravitation]]al forces on the vehicle. As such, it cannot be used by itself to accurately calculate the propellant requirement for launch from (or powered descent to) a planet with an atmosphere, and does not apply to [[Non-rocket spacelaunch|non-rocket systems]] such as [[aerobraking]], [[Space gun|gun launch]]es, [[space elevator]]s, [[launch loop]]s, or [[tether propulsion]].
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| Also, the equation strictly applies only to a theoretical [[orbital maneuver#impulsive maneuver|impulsive maneuver]], in which the propellant is discharged and delta-v applied instantaneously. Orbital maneuvers involving significantly large delta-v (such as [[translunar injection]]) still are under the influence of gravity for the duration of the propellant discharge, which influences the vehicle's velocity. The equation is most accurately applied to relatively small delta-v maneuvers such as those involved in fine-tuning [[space rendezvous]], or mid-course corrections in translunar or [[interplanetary spaceflight|interplanetary flight]]s where the gravity field is relatively weak.
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| Nevertheless, the equation is useful for estimating the propellant requirement to perform a given [[orbital maneuver]], assuming a required delta-v. To achieve a large delta-v, either <math>m_0</math> must be huge ([[exponential growth|growing exponentially]] as delta-v rises), or <math>m_1</math> must be tiny, or <math>v_e</math> must be very high, or some combination of all of these. In practice, very high delta-v has been achieved by a combination of
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| * very large rockets (increasing <math>m_0</math> with more fuel)
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| * staging (decreasing <math>m_1</math> by throwing out the previous stage)
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| * very high exhaust velocities (increasing <math>v_e</math>)
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| ==Examples==
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| Assume an exhaust velocity of {{convert|4500|m/s|ft/s|sp=us}} and a <math>\Delta v</math> of {{convert|9700|m/s|ft/s|sp=us}} (Earth to [[low earth orbit|LEO]], including <math>\Delta v</math> to overcome gravity and aerodynamic drag).
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| *[[Single stage to orbit]] rocket: <math>1-e^{-9.7/4.5}</math> = 0.884, therefore 88.4% of the initial total mass has to be propellant. The remaining 11.6% is for the engines, the tank, and the payload. In the case of a space shuttle, it would also include the orbiter.
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| *[[Two stage to orbit]]: suppose that the first stage should provide a <math>\Delta v</math> of {{convert|5000|m/s|ft/s|sp=us}}; <math>1-e^{-5.0/4.5}</math> = 0.671, therefore 67.1% of the initial total mass has to be propellant to the first stage. The remaining mass is 32.9%. After disposing of the first stage, a mass remains equal to this 32.9%, minus the mass of the tank and engines of the first stage. Assume that this is 8% of the initial total mass, then 24.9% remains. The second stage should provide a <math>\Delta v</math> of {{convert|4700|m/s|ft/s|sp=us}}; <math>1-e^{-4.7/4.5}</math> = 0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2%, and 8.7% remains for the tank and engines of the second stage, the payload, and in the case of a space shuttle, also the orbiter. Thus together 16.7% is available for all engines, the tanks, the payload, and the possible orbiter.
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| ==Stages==
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| In the case of sequentially thrusting [[Staging (rocketry)|rocket stages]], the equation applies for each stage, where for each stage the initial mass in the equation is the total mass of the rocket after discarding the previous stage, and the final mass in the equation is the total mass of the rocket just before discarding the stage concerned. For each stage the specific impulse may be different.
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| For example, if 80% of the mass of a rocket is the fuel of the first stage, and 10% is the dry mass of the first stage, and 10% is the remaining rocket, then
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| :<math> | |
| \begin{align}
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| \Delta v \ & = v_\text{e} \ln { 100 \over 100 - 80 }\\
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| & = v_\text{e} \ln 5 \\
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| & = 1.61 v_\text{e}. \\
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| \end{align}
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| </math>
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| With three similar, subsequently smaller stages with the same <math>v_e</math> for each stage, we have
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| :<math>\Delta v \ = 3 v_\text{e} \ln 5 \ = 4.83 v_\text{e} </math>
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| and the payload is 10%*10%*10% = 0.1% of the initial mass.
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| A comparable [[Single-stage-to-orbit|SSTO]] rocket, also with a 0.1% payload, could have a mass of 11.1% for fuel tanks and engines, and 88.8% for fuel. This would give
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| :<math>\Delta v \ = v_\text{e} \ln(100/11.2) \ = 2.19 v_\text{e}. </math>
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| If the motor of a new stage is ignited before the previous stage has been discarded and the simultaneously working motors have a different specific impulse (as is often the case with solid rocket boosters and a liquid-fuel stage), the situation is more complicated.
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| ==Common misconceptions==
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| When viewed as a [[variable-mass system]], a rocket cannot be directly analyzed with [[Newton's second law of motion]] because the law is valid for constant-mass systems only.<ref name="plastino">{{cite journal|last=Plastino|first=Angel R. |coauthors=Muzzio, Juan C.|year=1992|title=On the use and abuse of Newton's second law for variable mass problems|journal=Celestial Mechanics and Dynamical Astronomy|publisher=Kluwer Academic Publishers|location=Netherlands|volume= 53|issue= 3|pages=227–232|issn=0923-2958|bibcode=1992CeMDA..53..227P|doi=10.1007/BF00052611}} "We may conclude emphasizing that Newton's second law is valid for constant mass only. When the mass varies due to accretion or ablation, [an alternate equation explicitly accounting for the changing mass] should be used."</ref><ref name=Halliday>{{cite book|last=Halliday|coauthors=Resnick|title=Physics|volume=1|pages=199|quote=It is important to note that we ''cannot'' derive a general expression for Newton's second law for variable mass systems by treating the mass in '''F''' = ''d'''''P'''/''dt'' = ''d''(''M'''''v''') as a ''variable''. [...] We ''can'' use '''F''' = ''d'''''P'''/''dt'' to analyze variable mass systems ''only'' if we apply it to an ''entire system of constant mass'' having parts among which there is an interchange of mass.|isbn=0-471-03710-9}} [Emphasis as in the original]</ref><ref name=Kleppner>
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| {{cite book|last=Kleppner|first=Daniel|coauthors=Robert Kolenkow|title=An Introduction to Mechanics|publisher=McGraw-Hill|year=1973|pages=133–134|isbn=0-07-035048-5|quote=Recall that '''F''' = ''d'''''P'''/''dt'' was established for a system composed of a certain set of particles[. ... I]t is essential to deal with the same set of particles throughout the time interval[. ...] Consequently, the mass of the system can not change during the time of interest.}}</ref> It can cause confusion that the Tsiolkovsky rocket equation looks similar to the relativistic force equation <math>F = dp/dt = m \; dv/dt + v \; dm/dt</math>. Using this formula with <math>m(t)</math> as the varying mass of the rocket seems to derive Tsiolkovsky rocket equation, but this derivation is not correct. Notice that the [[effective exhaust velocity]] <math>v_e</math> doesn't even appear in this formula.
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| A simple counter example is to consider a rocket travelling with a constant velocity <math>v</math> with two maneuvering thrusters pointing out on either side, with both firing such that their forces cancel each other out. In such a case the rocket would be losing mass and an incorrect application of <math>F = dp/dt</math> would result in a non-zero but non-accelerating force, leading to nonsensical answers. However when including the mass of the exhaust products, the system becomes a closed system where the derivation is correct, as <math>F = dp/dt = 0</math> in the ''x'' direction, where ''x'' is the direction one thruster is pointing and -''x'' is the direction of the other thruster.
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| ==See also==
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| {{Portal| Spaceflight }}
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| * [[Delta-v]]
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| * [[Delta-v budget]]
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| * [[Oberth effect]] applying delta-v in a [[gravity well]] increases the final velocity
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| * [[Specific impulse]]
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| * [[Spacecraft propulsion]]
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| * [[Mass ratio]]
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| * [[Working mass]]
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| * [[Relativistic rocket]]
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| * [[Reversibility of orbits]]
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| * [[Variable-mass system]]s
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| {{Orbits}}
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| ==References==
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| {{More footnotes|date=February 2009}}
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| <references/>
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| ==External links==
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| *[http://ed-thelen.org/rocket-eq.html How to derive the rocket equation]
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| *[http://www.relativitycalculator.com/rocket_equations.shtml Relativity Calculator - Learn Tsiolkovsky's rocket equations]
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| *[http://www.wolframalpha.com/input/?i=Tsiolkovsky+rocket+equation Tsiolkovsky's rocket equations plot and calculator]
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| {{DEFAULTSORT:Tsiolkovsky Rocket Equation}}
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| [[Category:Astrodynamics]]
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| [[Category:Equations]]
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| [[Category:Single-stage-to-orbit]]
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| [[Category:Rocket propulsion]]
| |
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