![Soyuz TMA-18 launching](https://upload.wikimedia.org/wikipedia/commons/e/e9/Soyuz_TMA-18_launching.jpg)
In
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, a variable-mass system is a collection of
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
whose
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
varies with
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
. It can be confusing to try to apply
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
of motion directly to such a system.
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
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
carried by mass entering or leaving the system. The general equation of variable-mass motion is written as
:
where F
ext is the
net external force on the body, v
rel is the
relative velocity
The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.
Classical mechanics
In one dimension (non-relativistic)
We begin with relative motion in the classi ...
of the escaping or incoming mass with respect to the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the body, and v is the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of the body.
In
astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
, which deals with the mechanics of
rocket
A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entirely fr ...
s, the term ''v''
rel is often called the
effective exhaust velocity
Specific impulse (usually abbreviated ) is a measure of how efficiently a reaction mass engine (a rocket using propellant or a jet engine using fuel) creates thrust. For engines whose reaction mass is only the fuel they carry, specific impulse is ...
and denoted ''v''
''e''.
Derivation
There are different derivations for the variable-mass system motion equation, depending on whether the mass is entering or leaving a body (in other words, whether the moving body's mass is increasing or decreasing, respectively). To simplify calculations, all bodies are considered as
particle
In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
s. It is also assumed that the mass is unable to apply external forces on the body outside of accretion/ablation events.
Mass accretion
![Variable-mass system derivation](https://upload.wikimedia.org/wikipedia/commons/8/86/Variable-mass_system_derivation.svg)
The following derivation is for a body that is gaining mass (
accretion
Accretion may refer to:
Science
* Accretion (astrophysics), the formation of planets and other bodies by collection of material through gravity
* Accretion (meteorology), the process by which water vapor in clouds forms water droplets around nucl ...
). A body of time-varying mass ''m'' moves at a velocity v at an initial time ''t''. In the same instant, a particle of mass dm moves with velocity u with respect to ground. The initial
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
can be written as
:
Now at a time ''t'' + d''t'', let both the main body and the particle accrete into a body of velocity v + dv. Thus the new momentum of the system can be written as
:
Since d''m''dv is the product of two small values, it can be ignored, meaning during d''t'' the momentum of the system varies for
:
Therefore, by
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
:
Noting that u - v is the velocity of d''m''
relative
Relative may refer to:
General use
*Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives''
Philosophy
*Relativism, the concept that ...
to ''m'', symbolized as v
rel, this final equation can be arranged as
:
Mass ablation/ejection
In a system where mass is being ejected or
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
:
Assuming u to be the velocity of the ablated mass dm with respect to the ground, at a time ''t'' + d''t'' the momentum of the system becomes
:
where u is the velocity of the ejected mass with respect to ground, and is negative because the ablated mass moves in opposite direction to the mass. Thus during d''t'' the momentum of the system varies for
:
Relative velocity v
rel of the ablated mass with respect to the mass ''m'' is written as
:
Therefore, change in momentum can be written as
:
Therefore, by
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
:
Therefore, the final equation can be arranged as
:
Forms
![Inflated rocket balloon](https://upload.wikimedia.org/wikipedia/commons/0/0e/Inflated_rocket_balloon.jpg)
By the definition of
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
, a = dv/d''t'', so the variable-mass system motion equation can be written as
:
In bodies that are not treated as particles a must be replaced by a
cm, the acceleration of the
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the system, meaning
:
Often the force due to
thrust
Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that syst ...
is defined as
so that
:
This form shows that a body can have acceleration due to thrust even if no external forces act on it (F
ext = 0). Note finally that if one lets F
net be the sum of F
ext and F
thrust then the equation regains the usual form of Newton's second law:
:
Ideal rocket equation
![Tsiolkovsky rocket equation](https://upload.wikimedia.org/wikipedia/commons/f/f6/Tsiolkovsky_rocket_equation.svg)
The
ideal rocket equation, or the
Tsiolkovsky rocket equation, can be used to study the motion of vehicles that behave like a
rocket
A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entirely fr ...
(where a body accelerates itself by ejecting part of its mass, a
propellant
A propellant (or propellent) is a mass that is expelled or expanded in such a way as to create a thrust or other motive force in accordance with Newton's third law of motion, and "propel" a vehicle, projectile, or fluid payload. In vehicles, the e ...
, 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
ext = 0) the variable-mass system motion equation reduces to
:
If the velocity of the ejected propellant, v
rel, is assumed have the opposite direction as the rocket's acceleration, dv/d''t'', the
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
equivalent of this equation can be written as
:
from which d''t'' can be canceled out to give
:
Integration by
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
gives
:
:
By rearranging and letting Δ''v'' = ''v''
1 - ''v''
0, one arrives at the standard form of the ideal rocket equation:
:
where ''m''
0 is the initial total mass, including propellant, ''m''
1 is the final total mass, ''v''
rel is the
effective exhaust velocity
Specific impulse (usually abbreviated ) is a measure of how efficiently a reaction mass engine (a rocket using propellant or a jet engine using fuel) creates thrust. For engines whose reaction mass is only the fuel they carry, specific impulse is ...
(often denoted as ''v''
''e''), and Δ''v'' is the maximum change of speed of the vehicle (when no external forces are acting).
References
{{reflist, colwidth=33em
Classical mechanics
Mechanics