Is there some way to derive $E=mc^{2}$? I can understand that energy in something is proportional to its mass, but the $c^{2}$ part. I have no idea. It seems like the way the units are going it would end up as $kg*(m/s)^{2}$, which I think is the unit for Newtons, not energy. And why is there no constant of proportionality $k$? Did Einstein just set it up perfectly so that there wouldn't be one?
Answer
A simple derivation that is accessible to lay people who can only do primary school level math, starts from the fact that a pulse of electromagnetic radiation with energy $\mathbf{E}$ has a momentum of $\dfrac{\mathbf{E}}{\mathit{c}}$. In addition, one assumes conservation of momentum. We do a thought experiment involving a closed box containing two objects of mass $\mathtt{M}$. No external forces act on the box, therefore the momentum of the box is conserved. This means that whatever happens inside the box, if the box was at rest initially, it will stay at rest. In particular, it is impossible for the center of mass of the box to move.
We then consider the following process happening inside the box. One object emits a pulse of light of energy $\mathbf{E}$ that is completely absorbed by the other object. Due to the recoil of the light, the masses will move, but the velocities will obviously be very small (for macroscopic objects). We can thus safely make the assumption that classical mechanics will be valid after the pulse of light has been absorbed. If you want to be very rigorous, you must take the limit of $\mathtt{M}$ to infinity so that classical mechanics will be exactly correct.
Due to conservation of momentum, in the original rest frame of the box, the center of mass of the box will not change. Now classical mechanics doesn't give a correct description of relativistic phenomena like light, but it can be used to describe the situation in the box both before the pulse of light was emitted and after it was absorbed.
What is clear is that the object that emits the pulse of light will move away from the other object with the velocity of $$v = \dfrac{\mathbf{E}}{\mathtt{M} \cdot \mathit{c}}$$. If the other object is a distance of $\mathit{L}$ away, then it will take a time of $T = \dfrac{\mathit{L}}{\mathit{c}}$ before the other object absorbs that pulse of light. So, it seems that the center of mass will shift by $\frac{1}{2} v\cdot T = \frac{1}{2} \dfrac{\mathbf{E} \cdot \mathit{L}}{\mathtt{M} \cdot {\mathit{c}}^2}$ because of the time lag between when the object emitting the puls of light starts to move and when the object absorbing the pulse of light starts to move.
But, of course, a closed box upon which no external forces act cannot move all by itself. The masses must have changed due to the transfer of energy. Since classical mechanics is assumed to be valid we also have conservation of mass, so the sum of the masses of the objects will not have changed. If the emitting object has lost a mass of $d\mathtt{M}$, the receiving object will have gained a mass of $d\mathtt{M}$. A transfer of a mass of $d\mathtt{M}$ from the emitting object to the receiving object will have shifted the center of mass by an amount of $\mathit{L} \dfrac{d\mathtt{M}}{2\mathtt{M}}$ in the opposite direction as the shift due to the motion. For the two effects to cancel each other out, requires that $$\mathit{L} \dfrac{d\mathtt{M}}{2\mathtt{M}} = \frac{1}{2} \dfrac{\mathbf{E} \cdot \mathit{L}}{\mathtt{M} \cdot {\mathit{c}}^2} \implies \mathbf{E} = d\mathtt{M} {\mathit{c}}^2$$.
So, transferring the energy via the pulse of light led to mass being transferred according to $\mathbf{E} = \mathtt{M} {\mathit{c}}^2$. Due to conservation of energy and the ability of energy to transform from one form to another (e.g. the pulse of light will become heat, chemical energy or whatever), one can then argue that any transfer of energy implies a transfer of the equivalent mass according to $\mathbf{E} = \mathtt{M} {\mathit{c}}^2$. This in turn then leads to the conclusion that the mass of an object is simply the total energy content of the object divided by ${\mathit{c}}^2$.
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