When I Google the units for $E=mc^2$ it says m is mass in kg speed of light is in m/s and energy is in joules. Why does there not need to be some sort of constant? Why does the total energy in joules relate to the speed of light in meters per second. Isn't the second based of how long the earth takes to spin round. If a second was slightly longer would the joule be slightly smaller? If I had 1kg of water, would it have 300 million joules of energy?
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With our current definitions of meter and second, $$c = 299, 792 ,458\frac{m}{s}.$$ Let's say we were to use some other unit of second, let's call it the "zepond" and denote it with $z$ instead of $s$. Let's say one zepond is $2$ seconds, so $$ z = 2 s. $$ Then the speed of light, measured in meters per zepond (instead of meters per second) would be $$ c = 599, 584, 916\frac{m}{z}. $$ Note that this isn't really saying anything. In fact, it just trivially follows from $z = 2s$ as \begin{align} 299, 792 ,458\frac{m}{s} &= 299, 792 ,458\frac{m}{ \tfrac{1}{2} z } \\ &= 2 \times 299, 792 ,458\frac{m}{ z } \\ &= 599, 584, 916\frac{m}{ z }. \end{align}
This is just how units work. $c$ is a physical quantity, and has a meaning no matter what units you use. For instance, we could write it in miles per hour if we want. $$ 299, 792 ,458\frac{m}{ s } = c = 670,616,629 \frac{\rm miles}{\rm hour}. $$ They're still equal to each other. The number is only different because the units are different. So really, your question doesn't have anything to do with $E=mc^2$ in particular, it's just a question about how units work.
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