# Why c^2 in e=mc^2 ?

1. Aug 14, 2010

### MarsWTF

Why c^2 in "e=mc^2"?

Ok, i'v got that e=mc^2 meens some mass at max possible speed will equal max possible energy.
But max possible speed is c, not c^2. Nothing can have speed greater than speed of light, isnt it?

2. Aug 14, 2010

### cjl

Re: Why c^2 in "e=mc^2"?

You're kind of misinterpreting the equation. It doesn't mean that the mass has the maximum possible energy at maximum speed. It means that the mass itself has an inherent energy, even when it isn't moving, that is equal to m*c2. The full equation is actually E2 = (m*c2)2 + (p*c)2, which accounts for any motion that the object has in addition to the rest energy.

Oh, and part of the reason that the c needs to be squared is simply for the units to work out - mass times velocity gives units of kg*m/s, which is momentum. Energy has units of kg*m2/s2 (J).

Last edited: Aug 14, 2010
3. Aug 14, 2010

### MarsWTF

Re: Why c^2 in "e=mc^2"?

Exactly. In my view this is like do some precise mechanism with duct tape.
There is no objectivity reasons for c^2 in this formula, only for the units to work :/
Imho, the formula must be
E=mc
what do you think? :)

Last edited: Aug 14, 2010
4. Aug 14, 2010

### cjl

Re: Why c^2 in "e=mc^2"?

You'll notice I said that was part of the reason, not the whole reason. In order for the equation to be valid, the units must work out, but that alone isn't sufficient.

I remember that there's a fairly good explanation for why E=mc2 is correct, but I don't remember it off the top of my head at the moment. I might have to glance through one of my textbooks to refresh my memory.

5. Aug 14, 2010

Re: Why c^2 in "e=mc^2"?

I think your units don't work.

You could give mass the units of energy and have E=m.

Last edited: Aug 14, 2010
6. Aug 14, 2010

### MarsWTF

Re: Why c^2 in "e=mc^2"?

Yes it would not work. But squaring с isnt correct cause light cannot exist on the speed of c^2.

7. Aug 14, 2010

### Theorem.

Re: Why c^2 in "e=mc^2"?

The equation is not suggesting light can reach speeds of c^2 at all.

8. Aug 14, 2010

### maxverywell

Re: Why c^2 in "e=mc^2"?

The rest mass of a photon is zero, so its energy is E=pc.

Last edited: Aug 14, 2010
9. Aug 14, 2010

### MarsWTF

Re: Why c^2 in "e=mc^2"?

how? when? where? 0_o

10. Aug 14, 2010

### maxverywell

Re: Why c^2 in "e=mc^2"?

Take for example the classical kinetic energy which is K=1/2 m u^2. This doesn't mean that the speed of particle is u^2.

11. Aug 14, 2010

### Staff: Mentor

Re: Why c^2 in "e=mc^2"?

Again, C^2 is not a speed so what you are saying is meaningless. Check the units!

....and consider that this equation is very similar to the kinetic energy equation: e=.5mv^2

12. Aug 14, 2010

### Staff: Mentor

I agree with Russ, c² isn't a speed. The units are different. I think you need to learn to be more careful with units. That should have been the first thing you were taught in physics.

Let's say that we begin with the supposition that mass has some intrinsic amount of energy, even when it is at rest, and we want to find the formula that determines how much energy is in a given amount of mass. First, we know that the units of mass and energy differ by a speed² so we can immediately write E=kmv².

So, what v can we use? The mass is at rest (v=0). So we cannot use its speed; it must be some sort of universal constant with units of speed. The only such constant is the speed of light, so we know E=kmc² just from consideration of units.

Determining that k=1 requires some actual derivation, so if anything you should ask why k=1 instead of k=1/2. But c² should be obvious: what else could it possibly be?

Last edited: Aug 14, 2010
13. Aug 14, 2010

### Jimmy Snyder

Re: Why c^2 in "e=mc^2"?

In Einstein's original paper, it said E = mc with a reference to footnote 2. He intended to write in the footnote: E = ma and E = mb are rejected for orthographical reasons. However, he forgot to add the footnote and thus his fame was assured.

14. Aug 14, 2010

### DrGreg

Re: Why c^2 in "e=mc^2"?

Nice joke, but of course totally untrue.

15. Aug 14, 2010

### Theorem.

Re: Why c^2 in "e=mc^2"?

You ignored the rest of the sentence entirely...

16. Aug 14, 2010

### MarsWTF

Re: Why c^2 in "e=mc^2"?

i like kinetic energy equation. because some speed v' may be grater than v (if only v<<<c), so its allowed to be squared

17. Aug 14, 2010

### Staff: Mentor

Re: Why c^2 in "e=mc^2"?

Huh?

18. Aug 14, 2010

### MarsWTF

Re: Why c^2 in "e=mc^2"?

my English is bad, i dont know how to say my thought more clearly, sorry :(

19. Aug 14, 2010

### Mute

Re: Why c^2 in "e=mc^2"?

You are seriously confused about what these formulas mean. As has already been said, $E = mc^2$ is the amount of energy in an object of mass m in its rest frame, it is NOT "the energy of a mass m moving at maximum speed". i.e., it is the object's internal energy due to the fact that it has mass. The $c^2$ is the conversion factor between mass and rest energy. It's no different that how the Boltzmann constant is a conversion factor between energy and temperature, e.g. $E = k_B T/2$.

20. Aug 14, 2010

### Staff: Mentor

Re: Why c^2 in "e=mc^2"?

There is no v' in any of the equations we've discussed so far...

....you're not trying to say that v'=v^2, are you? If you are, you need to reread the entire thread (actually read it this time!) and then repeat the following over and over again until it sinks in:

v^2 is not a speed.
v^2 is not a speed.
v^2 is not a speed.
v^2 is not a speed.
v^2 is not a speed.
v^2 is not a speed.

Last edited: Aug 14, 2010
21. Aug 14, 2010

### Jimmy Snyder

Re: Why c^2 in "e=mc^2"?

Neither is v for that matter.

22. Aug 14, 2010

### brainstorm

Re: Why c^2 in "e=mc^2"?

So why does this squaring occur in the energy equation but not in the momentum equation? I always think of kinetic energy as equivalent to particle momentum in qualitative terms.

23. Aug 14, 2010

### novop

Re: Why c^2 in "e=mc^2"?

Why should it? Momentum is defined as the product of mass and velocity. What other units would you expect it to have?

Of course, the units for energy are a direct consequence of it's definition as well.

Since $$E = \int{F dx}$$ then $$E = \int{v dp} = \frac{mv^2}{2}$$

which means that energy has units of $$\frac{kg*m^2}{s^2}$$

24. Aug 14, 2010

### brainstorm

Re: Why c^2 in "e=mc^2"?

I googled "energy" to find out why it is defined in this way mathematically, but there were too many different types of energy and corresponding equations for me to figure out why this one you mention is general and what it refers to, specifically - let alone why the units are squared in this way.

25. Aug 15, 2010

### novop

Re: Why c^2 in "e=mc^2"?

My choice of deriving kinetic potential was arbritary, but you can be assured that the units of energy are the same for all types. In all cases, energy is a quantity with units of force times distance. Why?

The definition of energy is $$E = \int{F dx}$$

Since the force "F" has units of Newtons, and $$1~N = 1~\frac{kg\cdot m}{s^2}}$$, and $$dx$$ has units of metres, then $$F dx$$ = $$\frac{kg*m^2}{s^2}$$.

The "squared units" are a direct consequence of how energy is defined. We could have called the quantity $$\int{F dx}$$ "spaghetti tuesday", but then you'd be asking why the units of "spaghetti tuesday" are what they are.

Last edited: Aug 15, 2010