c is not the square root of the speed of the light; c is the speed of light.
[Well, except in the sense that the speed of light, if considered to be 1, is indeed its own square root…]
Well, I understand that, and I understand more than I can put in words. But I’ll try this one, why is c squared used in Einstein’s formula instead of c?
All physical equations need to be dimensionally consistent, meaning that the quantity measured on the left hand side of the equation needs to be in the same units as the quantity measured on the right hand side. You can’t say that distance = time, or that force = acceleration, since these are different units.
Energy is measured in joules:
1 J = 1 N * m
= 1 (kg * m / s ^ 2) * m
= 1 kg * m^2 / s^2
Speed (or more strictly velocity) is measured in units of distance / time, or m / s. If the formula was just E = mc, the units on the right hand side of the equation would be kg * m / s, which is not equal to the left hand side.
Having E = mc^2 gives us kg * m^2 / s^2 on both sides, so the equation is dimensionally consistent. I’m not suggesting that the square is inserted merely as a trick to gain dimensional consistency, but it’s necessary for an equation to be valid. The actual equation derives from more fundamental equations of Special Relativity. I’m using dimensional analysis somewhat loosely here, but it’s trying to give you a flavour of what it’s about.
Thank you. Unfortunately, this isn’t going to help clear up the general misimpressions among the populace. Usually I tell them it just looked cooler in the Twilight Zone opening. While I lack too many details to perform the calculations, I understand the more general constraints. I should have gone to class more often, but you see it was the 70’s, and, well, getting high seemed like a better idea at the time. I suppose I can tell those guys, (some of them still think its the 70’s), that it’s necessary to form a valid equation. Sometimes the best way to fight ignorance is with a smack to the head. Would have done wonders for me 40 years ago.
c is the absolute constant which happens to be equal to the speed of light (or any electromagnetic radiation) in a vacuum. And it’s an absolute value regardless of the units used to measure it: 186,282 miles per second, or 300.000,000 meters per second, or 3,304,791,705,600 fathoms per fortnight. It’s not the number, it’s the value it represents in the measurement system chosen, that is constant.
And it is the square of that number, whatever it is, that also, constantly, represents tghe ratio between matter, measured as X weight-units, and its equivalent in energy, measured in distance/weight-units. If you are looking at ten pounds of matter and seeking how much ‘stored’ energy it represents, measured in foot-pounds, then the value of c is 186282 mps * 5280 ft/mi = 983,568,960 ft/sec, c[sup]2[/sup] = 967,407,899,075,481,600 ft/sec[sup]2[/sup]; multiply by ten to get the equivalent of ten pounds mass.
As stated above, the reason the speed is squared is that, well, it has to be in order for the units to match up.
I remember the first time I realised this (not with reference to E=mc[sup]2[/sup], but with equations in general) while studying physics at school. It was a real “a-ha!” moment. If you plug the units into the equation, you’ll always get appropriate units out. (You might need to expand out the units into the base SI units to do this.)
e.g. if you apply a force of 10 newtons to a stationary 5kg mass for 20 seconds, what will its speed be (ignoring friction)?
a = F/m = 10 N / 5 kg = 10 kg m s[sup]-2[/sup] / 5 kg = 2 m s[sup]-2[/sup]
v = at = 2 m s[sup]-2[/sup] x 20 s = 40 m s[sup]-1[/sup].
You divide a force by a mass and multiply by a time and you get a speed. Magic! 
Nitpick: While 3E8 meters per second is a good enough approximation to use for most purposes, the exact value is 299,792,458 m / s . The meter is, in fact, currently defined to be equal to 1/299792458 of the distance light travels in a second. They could have instead, when they made the new definition, said that it was 1/300000000 of the distance light travels in a second, but a tenth of a percent change in the definition of the meter would throw off a lot of other things in science and engineering, so they made the new definition as close to the old one as they could.
Talking to myself, patching my miswanderings up:
Although I suppose no one actually denotes relativistic addition of speeds simply by +, and everyone does use + to denote the operation corresponding to pointwise addition of the corresponding linear functions from lengths of time to distance instead. E.g., no one writes “c + c = c”, so far as I know; people simply write out “(v + w)/(1 + v/c * w/c)” to denote relativistic composition of speeds, instead of redefining “v + w” to mean this. So I guess there isn’t really any problem here.
“C” is a constant and refers to the speed of light in the equation:
E = MC^2
“M” stands for mass and “E” stands for energy. Basically it states that there is a direct relationship between the mass of an object and how much energy is has. You can almost think of mass as a form of stored energy.
There is a difference between mass and weight. Mass is a property of the object itself and doesn’t change. Weight is that mass under the acceleration (usually due to gravity) and will change from place to place. If you have a mass of 70Kg, you’ll have the same 70Kg mass whether you’re on the moon or on Earth. If you have a weight of 150 pounds on Earth, you’ll have a weight of about 25 pounds on the moon.
Although the idea of speed squared seems strange, the units work out and are equal on both sides. Energy is measured in Joules which is Newtons * Meters. A Newton is a unit of weight which is mass (in Kilograms) * acceleration (in m/s^2). C^2 is meters squared over second squared (m^2/s^2).
Thus the equation in units:
E in Joules = M in kilograms * (Speed of light in meters/seconds)^2
J = kg * m^2/s^2
N * m = kg * m^2/s^2
kg * m/s^2 * m = kg * m^2/s^2
kg * m^2/s^2 = kg * m^2/s^2
Yes, squaring a speed doesn’t make much sense. After all, what does meters meters per second per second represent in the real world? But, the amazing thing is that the units balanced.
But that’s not a very satisfying answer, is it?
Try this - let’s say you’re throwing a baseball. How much energy does it take to throw it (or, how much energy does the moving ball have, or how much energy is available to break stuff it hits)?
Well, you can figure that it depends - on the mass of the ball and the speed you throw it. Experiments bear this out. But more specifically, a ball that’s twice as heavy will have twice the energy. That’s a nice little relationship, and not really surprising. But if you throw it twice as fast, the ball actually has four times the energy. The kinetic energy in the baseball is in proportion to the ball’s mass, and it’s in proportion to the square of the speed. Experiments and basic math of physics show the relationship to be:
E = m * v[sup]2[/sup] / 2
That baby is going to have units of mass times length squared over distance squared. No matter what energy you calculate, energy always has these units, whether you’re talking about the energy in a compressed spring or the stored electrical energy in a capacitor. Neat, huh?
What Einstein did was to figure out that the energy contained in the mass of an object is its mass times the speed of light squared. This of course gives the same units as above - there would be something wrong if his answer didn’t come out that way.