Expressions

In relativistic dynamics,

$$\mathbf p=\gamma_v m\mathbf v,\, E=\gamma mc^2$$

where $m$ is the rest mass. These quantities are conserved in all collisions. These have been empirically proven to be true. A rationalisation can be made through 4-vectors.

This also implies a relationship between momentum, energy, and velocity -

$$\frac{\mathbf{p}}{E}=\frac{\mathbf{v}}{c^2}$$

implying any two gives us the other.

Furthermore, for collisions,

$$E^2-|\mathbf p|^2c^2=\gamma^2m^2c^4\left( 1-\frac{v^2}{c^2} \right)=(mc^2)^2$$

usually expressed as $E^2=p^2c^2+m^2c^4$. Notably for photons ($m=0$) this reduces to $E=pc$.

When $v\ll c$ and $\gamma \to 1$, we have that

$$\mathbf p \approx m\mathbf v, \, E\approx mc^2+\frac{1}{2}mv^2+\cdots$$

where we neglect higher-order terms in $\frac{v}{c}$ for energy. These match Newtonian expectation; for energy when the collision is elastic, there is no change to mass - hence

$$E_1=E_2\implies \frac{1}{2}mv_1^2=\frac{1}{2}mv_2^2$$

For non elastic collisions, heat is converted into a small amount of mass. Hence $mc^2$ terms on each side differ, and conservation of ‘kinetic’ energy (the $\frac{1}{2}mv^2$ term) does not hold.

Mass