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. 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
where Fext is the net external force on the body, vrel 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. In astrodynamics, which deals with the mechanics of rockets, the term vrel is often called the effective exhaust velocity and denoted ve.
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 particles. It is also assumed that the mass is unable to apply external forces on the body outside of accretion/ablation events.
The following derivation is for a body that is gaining mass (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
Now at a time t + dt, 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 dmdv is the product of two small values, it can be ignored, meaning during dt the momentum of the system varies for
Therefore, by Newton's second law
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
Since the main body will be losing mass, dm will be negative, meaning that at a time t + dt the momentum of the system becomes
where u is the velocity of the ejected mass. Thus during dt the momentum of the system varies for
This is the same dp as found in the mass accretion case above, meaning that the same conclusion holds.
By the definition of acceleration, a = dv/dt, so the variable-mass system motion equation can be written as
In bodies that are not treated as particles a must be replaced by acm, the acceleration of the center of mass of the system, meaning
Often the force due to thrust 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 (Fext = 0). Note finally that if one lets Fnet be the sum of Fext and Fthrust then the equation regains the usual form of Newton's second law:
Ideal rocket equation
The ideal rocket equation, or the 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 (Fext = 0) the variable-mass system motion equation reduces to
If the velocity of the ejected propellant, vrel, is assumed have the opposite direction as the rocket's acceleration, dv/dt, the scalar equivalent of this equation can be written as
from which dt can be canceled out to give
Integration by separation of variables gives
By rearranging and letting Δv = v1 - v0, one arrives at the standard form of the ideal rocket equation:
where m0 is the initial total mass, including propellant, m1 is the final total mass, vrel is the effective exhaust velocity (often denoted as ve), and Δv is the maximum change of speed of the vehicle (when no external forces are acting).
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