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Concept
Variable-mass system
Summary
In mechanics, a variable-mass system is a collection of matter whose mass varies with time. It can be confusing to try to apply Newton's second law 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 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). 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 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
Noting that u - v is the velocity of dm relative to m, symbolized as vrel, this final equation can be arranged as
In a system where mass is being ejected or ablated from a main body, the derivation is slightly different.
Official source
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