Probably dumb question... and one I probably knew the answer to at some stage... why does the equation showing the conversion from mass to energy involve the constant of the speed of light?
For historical reasons we measure time and space using different units (seconds and meters in SI) but phisically space and time ought to be measured with the same unit. It is possible to construct such a measurement system, indeed many of them, known as natural units[1]. In natural units, c, i.e the ratio between the space and time unit, would be simply be set to dimensionless 1. In such a measurement system the E=mc^2 equation simply reduces to E=m i.e. each unit of max correspond to one unit of energy.
I can't think of a concise/non-technical explanation. This Feynman lecture is about as close as you'll get: http://www.feynmanlectures.caltech.edu/I_15.html
If you trade away all the time, you get a maximum velocity: c. So the things moving at c don't experience time, and have momentum but not mass.
http://www.math.ucr.edu/home/baez/physics/ParticleAndNuclear...
In the same direction, if you take a mass and accelerate it, it experiences less time. There you have relativistic time dilation, as beloved by science fiction stories.
https://en.wikipedia.org/wiki/Time_dilation
And you simultaneously get length contraction in the direction of each velocity:
https://en.wikipedia.org/wiki/Length_contraction
These are all aspects of the same thing.
Really not a dumb question at all.
E=mc^2 cosh(eta)
where eta is the velocity parameter. https://en.wikipedia.org/wiki/Proper_velocity
In terms of the coordinate velocity v,
E=mc^2 cosh(atanh(v/c))=mc^2 / sqrt(1-(v/c)^2) = mc^2 (1 + 1/2 (v/c)^2 + 3/8 (v/c)^4 + ...
So mc^2 is the irreducible energy due to mass and isn't relevant at low speeds because it doesn't change; the first-order approximation of kinetic energy is just 1/2 mv^2 and higher order terms aren't significant at low speeds.
There is a relationship between mass and the speed of light that people may or may not know about. As you accelerate a particle towards the speed of light, the mass actually increases. Since the mass increases, it's harder to accelerate, which means that you need more energy to make it go faster. To get it to the speed of light you would need infinite energy and it would have infinite mass.
We can also rewrite the equation as c = sqrt(E/m). In other words, the speed of light is determined from the ratio of energy to mass in the system. When looked at from this perspective, it's not really that surprising that the ratio of energy to mass is a constant. What's kind of surprising is that this constant is the speed of light. Even just trying to figure out how the units might work is a bit mind bending -- m/s = sqrt(g/J)
I think this is where you start to understand that the concept of space, time, energy and mass are all intertwined. We kind of think of them as separate things because from our macro perspective, that's how they look. Under the hood, though, they are all intricately bound together. The consequences of special relativity (where this equation came from) is that space, time, and mass are functions of energy, which is both super cool and super confusing :-)