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Demosthenes 2010-03-30 12:19 AM

Deriving e=mc^2
 
We begin with the definition of relativistic mass:

.


We can now define relativistic momentum as:

.


Since force is defined as



we can define force in the relativistic context analogously:

.


In classical physics work is defined as

.


We can reduce this to a problem in a single dimension for simplicity, and recall that the kinetic energy added to a body is equal to the work done on the body by an external force:

.


From here we can say:

.


Applying the definition of relativistic mass, the last expression becomes:

[
.


Integrating through, we have



The term on the left,
, is the total energy of the body after an external force is applied, the term on the right,
is the rest energy, and the final expression is the kinetic energy of the body.

The kinetic energy can be expressed more clearly, in my opinion, as

.


And there you have it. Mass is energy!

Demosthenes 2010-03-30 12:20 AM

Latex isn't working entirely properly. But you should get the gist of it.

Demosthenes 2010-03-30 12:24 AM

I have fixed it for now, but certain things aren't rendering correctly.

!King_Amazon! 2010-03-30 08:23 AM

I'm glad I know this now. Or rather that I've read it.

Skurai 2010-03-30 08:28 AM

Well, at least now I know HOW e = mc^2
I'd heard it used to many times in cartoons, it was in my head before teachers ever got around too it (infact, I've probably learned more from cartoons than text books), but at least now I know how it fits together.

Demosthenes 2010-03-30 10:24 AM

Well, I'm glad it helped someone, and was not an exercise in LaTeX. If there is anything I can clear up, let me know.


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