Variable-mass system

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.^[1][2] 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

${\displaystyle \mathbf {F} _{\mathrm {ext} }+\mathbf {v} _{\mathrm {rel} }{\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}}$

where F_ext is the net external force on the body, v_rel 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.^[3] In astrodynamics, which deals with the mechanics of rockets, the term v_rel is often called the effective exhaust velocity and denoted v_e.^[4]