mass energy equivalance

Mass–velocity relationship

In developing special relativity, Einstein found that the kinetic energy of a moving body is

with the velocity, the rest mass, and γ the Lorentz factor.
He included the second term on the right to make sure that for small velocities, the energy would be the same as in classical mechanics:

Without this second term, there would be an additional contribution in the energy when the particle is not moving.
Einstein found that the total momentum of a moving particle is:

and it is this quantity which is conserved in collisions. The ratio of the momentum to the velocity is the relativistic mass, m.

And the relativistic mass and the relativistic kinetic energy are related by the formula:

Einstein wanted to omit the unnatural second term on the right-hand side, whose only purpose is to make the energy at rest zero, and to declare that the particle has a total energy which obeys:

which is a sum of the rest energy m₀c² and the kinetic energy. This total energy is mathematically more elegant, and fits better with the momentum in relativity. But to come to this conclusion, Einstein needed to think carefully about collisions. This expression for the energy implied that matter at rest has a huge amount of energy, and it is not clear whether this energy is physically real, or just a mathematical artifact with no physical meaning.