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No equation is more famous than E = mc², and few are simpler. Indeed, the immortal equation’s fame rests largely on that utter simplicity: the energy E of a system is equal to its mass m multiplied by c², the speed of light squared. The equation’s message is that the mass of a system measures its energy content. Yet E = mc² tells us something even more fundamental. If we think of c, the speed of light, as one light year per year, the conversion factor c²equals 1. That leaves us with E = m. Energy and mass are the same.
“Why is the conversion factor c²?"
There are two parts to the question. First, why is the factor something squared? Second, why is that something c, the speed of light.
The answer to the first question is that the factor has to be some speed squared to get the units right. Most people don't value units enough. You can often solve a problem just knowing the units involved, and you can certainly tell if something is askew if the units aren't right. For example, if you ask someone, "what's the area of your living room?" and they answer, "Thirty feet," you know they misheard the question because an area has to be in square-feet.
The units of energy are mass·speed², for example, kg·m²/s². The kinetic energy of a mass m moving and velocity v is ½ m v².
So any formula writing energy in terms of mass has to involve something with units of speed squared.
But why the speed of light? Well, the theory of relativity is based on the idea that all observers, no matter how fast they are moving relative to you, must observe the same physical laws. And there is only one special speed which they all measure to be the same-the speed of light. In fact, you can show that the theory doesn't work if you try to add a second special speed. A short way of putting it is that in relativity there is only one constant to work with, the speed of light.
Now I could derive the formula for you, or show you it gives the correct kinetic energy in the low-speed limit, but a simple answer is that there is no other speed it could be!
So in summary, the factor has to be a speed squared to get the units right, and the only speed it could be is the speed of light.
To know more watch this entire video till the end.
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