Derivation of mass and energy?

In order to compensate for the apparent mass increase due to very high speeds we have to build it into our equations. We know that the mass increase can be accounted for by using the equation:

[Apparent mass increase due to speed.]

From this equation we know that mass (m) and the speed of light (c) are related in some way. What happens if we set the speed (v) to be very low? Einstein realised that if this is done we can account for the mass increase by using the term mc2(the exact arguments and mathematics required to derive this are quite advanced, but an example can be foundhere). Using this term we now have an equation that takes into account both the kinetic energy and the mass increase due to motion, at least for low speeds:

[Total energy at low speeds.]

This equation seems to solve the problem. We can now predict the energy of a moving bodyandtake into account the mass increase. What's more, we can rearrange the equation to show that:

[Relativistic and Newtonian energy.]

This result is fine for low speeds, but what about speeds closer to the speed of light? We know that mass increases at high speeds, but according to the Newtonian part of the equation that isn't the case. Therefore, we need to replace the Newtonian part of the formula in order to make the equation correct atallspeeds. How can we do this? We know that E - mc2is approximately equal to the Newtonian kinetic energy when v is small, so we can use E - mc2asthe definition of relativistic kinetic energy:

[A pure relativistic equation.]

We have now removed the Newtonian part of the equation. Note that we haven’t given a formula for relativistic kinetic energy. The reason for this will become apparent in a moment. Rearranging the result shows that:

[Total relativistic energy]

It can now be seen that relativistic energy consists of two parts. The first part is kinetic and depends on the speed of the moving body. The second part is due to the mass increase and does not depend on the speed of the body. However, both parts must be a form of energy, but what form? We can simplify the equation by setting the speed (i.e. the relativistic kinetic energy) of the particle to be zero, thereby removing it from the equation:

[Reduction to E = mc^2]

We now have the famous equation in the form it is most often seen, but what does it mean?

We have seen that a moving body increases in mass and has energy by virtue of its speed (the kinetic energy). Looking at the problem another way we can say that as the speed of a body gets lower there will be less and less kinetic energy until at rest the body will have no kinetic energy at all. So far so good, but what about the mass due to the speed of the body? Again, as the body slows down the mass will become progressively smaller but itcan'treach zero. The graph earlier showed that the lowest the mass can be is unity (one) and we can't just make the body disappear into nothing. The lowest possible mass the body can have is its "rest mass", i.e. the mass the body has when it is at rest. But the equation we have derived (E = mc2) isn't for mass, it's forenergy. The energy must somehow be locked up in the mass of the body.

Einstein therefore concluded that mass and energy are really the same thing, i.e. that any mass is really tightly packed energy. At the time he saw no mechanism for releasing the energy from the mass, and was in fact sceptical of the idea that it could ever be done. This didn't really matter to Einstein, however. As a theoretical physicist he was happy that his equations were consistent and he had a model to predict what happens to a body moving at very high speeds.