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[[File:Soyuz TMA-18 launching.jpg|thumb|150px|[[Rockets]], which lose significant amounts of mass as fuel during flight, are an example of a variable-mass system.]]
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In [[mechanics]], a '''variable-mass system''' is a collection of [[matter]] whose [[mass]] varies with [[time]]. [[Newton's second law]] of motion cannot directly be applied to such a system because it 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|doi=10.1007/BF00052611|bibcode=1992CeMDA..53..227P|url=http://articles.adsabs.harvard.edu/full/seri/CeMDA/0053//0000227.000.html|accessdate=2011-12-30}}</ref><ref name="Basavaraju">{{cite book|last1=Basavaraju|first1=G|last2=Ghosh|first2=Dipin|title=Mechanics and Thermodynamics|date=1985-02-01|publisher=[[Tata McGraw-Hill]]|isbn=978-0-07-451537-2|pages=162–165}}</ref> Instead, the time dependence of the mass ''m'' can be calculated by rearranging Newton's second law and adding a term to account for the [[momentum]] carried by mass entering or leaving the system. The general equation of variable-mass motion is written as
 
:<math>\mathbf{F}_{\mathrm{ext}} + \mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d} m}{\mathrm{d}t} = m {\mathrm{d} \mathbf v \over \mathrm{d}t}</math>
 
where '''F'''<sub>ext</sub> is the [[net force|net external force]] on the body, '''v'''<sub>rel</sub> is the [[relative velocity]] of the escaping or incoming mass with respect to the [[center of mass]] of the body, and '''v''' is the [[velocity]] of the body.<ref name="Plastino" />  In [[astrodynamics]], which deals with the mechanics of [[rocket]]s, the term ''v''<sub>rel</sub> is often called the [[effective exhaust velocity]] and denoted ''v''<sub>''e''</sub>.<ref name="NASA">{{cite web|url=http://microgravity.grc.nasa.gov/education/rocket/rktpow.html|title=Ideal Rocket Equation|last=Benson|first=Tom|publisher=[[NASA]]|accessdate=30 December 2011}}</ref>
 
== Derivation ==
There are different derivations for the variable-mass system motion equation, depending on whether is mass is entering or leaving a body (in other words, whether the moving body's mass is increasing or decreasing).  To simplify calculations, all bodies are considered as [[particle]]s.
 
=== Mass accretion ===
[[File:Variable-mass system derivation.svg|thumb|300px|At instant 1, a mass d''m'' with relative velocity '''u''' is about to collide with the main body of mass ''m'' and velocity '''v'''. After a time d''t'', at instant 2, both particles move as one body with velocity '''v'''&nbsp;+&nbsp;d'''v'''.]]
The following derivation is for a body that is gaining mass ([[Accretion (astrophysics)|accretion]]). Let a body of time-varying mass ''m'' moves at a velocity '''v''' at an initial time ''t''. Meanwhile, at this instant, let a particle of mass dm moves with velocity '''u'''.  The initial [[momentum]] can be written as<ref name="Cveticanin">{{cite book|last=Cveticanin|first=L|title=Dynamics of Machines with Variable Mass|edition=1|date=1998-10-21|publisher=[[CRC Press]]|isbn=978-90-5699-096-1|pages=15–20|accessdate=2011-12-25}}</ref>
 
:<math>\mathbf{p}_{\mathrm{1}} = m\mathbf{v} + \mathbf{u}\mathrm{d}m</math>
 
Now at a time ''t'' + d''t'', let both the main body and the particle accrete into a body of velocity '''v''' + d'''v'''. Thus the new momentum of the system can be written as
 
:<math>\mathbf{p}_{\mathrm{2}} = (m + \mathrm{d}m)(\mathbf{v} + \mathrm{d}\mathbf{v}) = m\mathbf{v} + m\mathrm{d}\mathbf{v} + \mathbf{v}\mathrm{d}m + \mathrm{d}m\mathrm{d}\mathbf{v}</math>
 
Since d''m''d'''v''' is the product of two small values, it can be ignored, meaning during d''t'' the momentum of the system varies for
 
:<math>\mathrm{d}\mathbf{p} = \mathbf{p}_{\mathrm{2}} - \mathbf{p}_{\mathrm{1}} = (m\mathbf{v} + m\mathrm{d}\mathbf{v} + \mathbf{v}\mathrm{d}m) - (m\mathbf{v} + \mathbf{u}\mathrm{d}m) = m\mathrm{d}\mathbf{v} - (\mathbf{u} - \mathbf{v})\mathrm{d}m</math>
 
Therefore, by [[Newton's second law]]
 
:<math>\mathbf{F}_{\mathrm{net}} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \frac{m\mathrm{d}\mathbf{v} - (\mathbf{u} - \mathbf{v})\mathrm{d}m}{\mathrm{d}t} = m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} - (\mathbf{u} - \mathbf{v})\frac{\mathrm{d}m}{\mathrm{d}t}</math>
 
Noting that '''u''' - '''v''' is the velocity of d''m'' [[relative velocity|relative]] to ''m'', symbolized as '''v'''<sub>rel</sub>, this final equation can be arranged as<ref name="Giancoli">{{cite book|last=Giancoli|first=Douglas C.|title=Physics for Scientists & Engineers|edition=4, illustrated|volume=2|year=2008|publisher=Pearson Education|isbn=978-0-13-227359-6|pages=236–238}}</ref>
 
:<math>\mathbf{F}_{\mathrm{ext}} + \mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d} m}{\mathrm{d}t} = m {\mathrm{d} \mathbf v \over \mathrm{d}t}</math>
 
=== Mass ablation/ejection ===
 
In a system where mass is being ejected or [[ablation|ablated]] from a main body, the derivation is slightly different.  At time ''t'', let a mass ''m'' travel at a velocity '''v''', meaning the initial momentum of the system is
 
:<math>\mathbf{p}_{\mathrm{1}} = m\mathbf{v}</math>
 
Since the main body will be losing mass, d''m'' will be negative, meaning that at a time ''t'' + d''t'' the momentum of the system becomes
 
:<math>\mathbf{p}_{\mathrm{2}} = (m + \mathrm{d}m)(\mathbf{v} + \mathrm{d}\mathbf{v}) + \mathbf{u}(-\mathrm{d}m) = m\mathbf{v} + m\mathrm{d}\mathbf{v} + \mathbf{v}\mathrm{d}m + \mathrm{d}m\mathrm{d}\mathbf{v} - \mathbf{u}\mathrm{d}m</math>
 
where '''u''' is the velocity of the ejected mass. Thus during d''t'' the momentum of the system varies for
 
:<math>\mathrm{d}\mathbf{p} = \mathbf{p}_{\mathrm{2}} - \mathbf{p}_{\mathrm{1}} = (m\mathbf{v} + m\mathrm{d}\mathbf{v} + \mathbf{v}\mathrm{d}m - \mathbf{u}\mathrm{d}m) - (m\mathbf{v}) = m\mathrm{d}\mathbf{v} - (\mathbf{u} - \mathbf{v})\mathrm{d}m</math>
 
This is the same d'''p''' as found in the mass accretion case above, meaning that the same conclusion holds.<ref name="Cveticanin" />
 
== Forms ==
[[File:Inflated rocket balloon.jpg|thumb|100px|When released, this rocket [[toy balloon|balloon]] ejects a significant amount of its mass as air, causing a large acceleration.]]
By the definition of [[acceleration]], '''a''' = d'''v'''/d''t'', so the variable-mass system motion equation can be written as
 
:<math>\mathbf{F}_{\mathrm{ext}} + \mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{d}t} = m\mathbf{a}</math>
 
In bodies that are not treated as particles '''a''' must be replaced by '''a'''<sub>cm</sub>, the acceleration of the [[center of mass]] of the system, meaning
 
:<math>\mathbf{F}_{\mathrm{ext}} + \mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{d}t} = m\mathbf{a}_{\mathrm{cm}}</math>
 
Often the force due to [[thrust]] is defined as <math>\mathbf{F}_{\mathrm{thrust}} = \mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{d}t}</math> so that
 
:<math>\mathbf{F}_{\mathrm{ext}} + \mathbf{F}_{\mathrm{thrust}} = m\mathbf{a}_{\mathrm{cm}}</math>
 
This form shows that a body can have acceleration due to thrust even if no external forces act on it ('''F'''<sub>ext</sub> = 0). Note finally that if one lets '''F'''<sub>net</sub> be the sum of '''F'''<sub>ext</sub> and '''F'''<sub>thrust</sub> then the equation regains the usual form of Newton's second law:
 
:<math>\mathbf{F}_{\mathrm{net}} = m\mathbf{a}_{\mathrm{cm}}</math>
 
=== Ideal rocket equation ===
[[File:Rocket mass ratio versus delta-v.svg|thumb|right|Rocket [[mass ratio]]s versus final velocity calculated from the rocket equation]]
{{main|Tsiolkovsky rocket equation}}
The [[Tsiolkovsky rocket equation|ideal rocket equation]], or the [[Konstantin Tsiolkovsky|Tsiolkovsky]] rocket equation, can be used to study the motion of vehicles that behave like a [[rocket]] (where a body accelerates itself by ejecting part of its mass, a [[propellant]], with high speed). It can be derived from the general equation of motion for variable-mass systems as follows: when no external forces act on a body ('''F'''<sub>ext</sub> = 0) the variable-mass system motion equation reduces to<ref name="Basavaraju" />
 
:<math>\mathbf{v}_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{d}t}= m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}</math>
 
If the velocity of the ejected propellant, '''v'''<sub>rel</sub>, is assumed have the opposite direction as the rocket's acceleration, d'''v'''/d''t'', the [[scalar (mathematics)|scalar]] equivalent of this equation can be written as
:<math>-v_{\mathrm{rel}}\frac{\mathrm{d}m}{\mathrm{d}t} = m{\mathrm{d} v \over \mathrm{d}t}</math>
 
from which d''t'' can be canceled out to give
 
:<math>-v_{\mathrm{rel}}\mathrm{d}m = m\mathrm{d}v \,</math>
 
Integration by [[separation of variables]] gives
 
:<math>-v_\mathrm{rel}\int_{m_0}^{m_1} \frac{\mathrm{d}m}{m} = \int_{v_0}^{v_1} \mathrm{d}v</math>
 
:<math>v_\mathrm{rel}\ln{\frac{m_0}{m_1}} = v_1 - v_0</math>
 
By rearranging and letting Δ''v'' = ''v''<sub>1</sub> - ''v''<sub>0</sub>, one arrives at the standard form of the ideal rocket equation:
 
:<math>\Delta v = v_\mathrm{rel} \ln \frac {m_0} {m_1}</math>
 
where ''m''<sub>0</sub> is the initial total mass, including propellant, ''m''<sub>1</sub> is the final total mass, ''v''<sub>rel</sub> is the [[effective exhaust velocity]] (often denoted as ''v''<sub>''e''</sub>), and Δ''v'' is the maximum change of speed of the vehicle (when no external forces are acting).
 
== References ==
{{reflist|colwidth=33em}}
 
[[Category:Classical mechanics]]
[[Category:Mechanics]]

Latest revision as of 21:10, 11 May 2014

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