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I was trying to reverse engineer Einstein's formula for energy, E=γmc^2 by re-engineering Newton's Law of motion, F=ma. I was talking with my physics prof about deriving energy from this because I got two different answers but it gets weird because the incorrectly derived formula works.

F = ma = dp/dt -> F dx = mv dv -> E = ∫ F dx = ∫ mv dv = .5mv^2 + C

Then I did this

F = dp/dt = v dp (dx/dx) -> F dx = v dp -> E = ∫ v dp = vp + C

My prof told me that my last integral, ∫ v dp, is an illegal operation and that v must be converted into p/m which makes sense because it then follows that E = .5mv^2 = p^2/2m.

I did some fiddling around though because I was curious and I was able to derive E = γmc^2 and the formula always works. What I derived from the above was:

E = vp + C = vp + mc^2/γ, p=γmv

I'm just curious if anyone can point out why it works.

Also, I know that energy for a photon is equal to |p|c. When m=0 then v=c and I find it interesting that the rest mass, m/γ, is introduced above given energy equivalence. So, bad math or is there something to this?

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# Is this a legal derivation?

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