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Mass measures a body's resistance to a change of velocity and its coupling to gravity. In Newton's second law for a body of fixed mass, $\mathbf{F}=m\mathbf{a}$, the inertial mass relates a known net force to the resulting acceleration. Applying the same force to two carts and comparing their accelerations measures their mass ratio.

A balance instead compares gravitational forces in a common gravitational field, probing gravitational mass. The proportionality of inertial mass and gravitational mass explains why freely falling bodies have the same acceleration when other forces can be neglected. Gravitational weight changes with the local gravitational field, while a body's mass need not change. In special relativity, mass-energy equivalence relates rest energy to mass through $E_0=m{c}^2$.

Put $\beta =v/c$ and $\gamma =(1-{\beta }^2{)}^{-1/2}$, the common Lorentz factor. The initial total relativistic energy and momentum areAssume that coalescence emits no additional radiation. By four-momentum conservation, the final rest mass $m_f$ is the rest mass of a coalescing relativistic collision:The final velocity is $V=c^2{P}_x/E=v/3$, the centre-of-momentum frame velocity.

For $v\ll c$,The leading term is the classical sum of masses. The first correction is the internal energy retained after sticking, divided by $c^2$: the initial classical kinetic energy is $3m{v}^2/4$, while the common-motion kinetic energy is $\frac{1}{2}(3m/2)(v/3{)}^2=m{v}^2/12$. Their difference is $2m{v}^2/3$, consistent with mass-energy equivalence.

The Stefan–Boltzmann law gives the radiated luminosityFor radiative mass loss at fixed luminosity, using mass-energy equivalence, the mass-loss magnitude and one-percent time areThe numerical constants are the external CODATA 2014 $\sigma =5.670367\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ and $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$; years here mean 365.25 days.^[ref] This estimates radiative mass loss with the supplied radius and temperature held constant. It does not predict a real stellar lifetime or include solar-wind mass loss.

For luminosity $L$, mass-energy equivalence gives $|\dot{M}|=L/c^2$. Holding $L$ constant gives a time $t=fM{c}^2/L$ to radiate a mass fraction $f$. A blackbody estimate uses the Stefan–Boltzmann law, $L=4\pi R^2\sigma T^4$. This is a radiative-energy estimate, not a model of a star's evolving nuclear fuel or mass loss in a wind.