Why do we relate the square of the speed of light in energy?

You’re squaring the speed at which light travels, not light itself.

@dxs Why do we do that?

We don’t “do” that. It is the relation between energy and mass, and defines their transmutability. c is a constant, not a variable.

@zenvelo Why was C a constant? Why square it?

The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source.

Maximum speed is finite: No physical object, message or field line can travel faster than the speed of light in a vacuum.

Einstein found that the speed of light in a vacuum is a constant and that the proportion of matter to energy was equal to the square of that constant.

@zenvelo How did he do that? Was it guess and check? I took high school physics and they never covered how he came up with his equation e=mc (squared)? What was his proof?

Eistein was working on his theory based on a number of well known experiments that were done between 1881 and 1906. And his work was an attempt to demonstrate how light and time are connected and can be bent by gravity. His E=mc^2 came out of that and experiments that he used to support his theory.

It is a bit beyond a high school level physics class.

Your questions here read as though you believe that physical phenomena are dependent on our explanations for them. “We” are the lucky recipients of Einstein’s great insights, and insights they were because he largely visuslized these concepts, then ginned up the appropriate mathematics.
The thing I find ironic in your questions is that Einstein himself became irritated at the silly notion that our description of things might actually be responsible for the way they behave. But along came quantum mechanics to torment us all.

One way to test the validity of a formula is to look at the units. I believe the definitions for energy and force still hold in relativity. I am having trouble formatting the square operation for Fluther. In the equations below, interpret a 2 by a symbol as being the square.

Energy = force x distance = (mass x acceleration) x distance.
Lett m stand for mass, d for distance and t for time. Acceleration = velocity/t = (d/t)/t) = d/t^2.

Substituting, we get energy = m d/ t^{2 x d = m d}2/t^2.

Now let’s check the units in E = mc^{2 = m(d/t)}2 = m d^2/t2. The units match, so the formula passes this test. The formula would definitely be invalid without squaring c.

Thank you @LostInParadise ! I had an inkling it was somewhere in there.

Thanks @zenvelo
A quicker way would be take the equation for kinetic energy, ½ m v^{2 and see that it has the same form as mc}2, since c is a velocity.

It has to do with the derivation of the Lorentz Transformation. Go to Wikipedia to read about it.