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\begin{equation} E^2 = (mc^2)^2 + (pc)^2 \end{equation}

If I am using this equation to figure out the energy of something, what units would I use? Would it be the metric system? I.e. kilograms for $m$, meters per second for $p$, kilometers per second for $c$? And what units of measurement are used for $E$?

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Use the SI units. Joules for E. –  Michael Luciuk Nov 28 '12 at 11:03
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"meters per second for p" Er...no. Kg m / s in SI, and likewise the velocity there should be in m/s so that you don't have to mess around with loose factors of $10^3$. Of course particle physicists would use $c=1$ units and measure energy, mass and momentum all in GeV. –  dmckee Nov 28 '12 at 16:43
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up vote 7 down vote accepted

Any consistent system will do. That's the entire point of systems of units--if you stick to one, you don't need to worry about the units too much. And it never happens that a certain equation only works in a certain system*.

In this case, you would use joules ($\:\mathrm{J}\equiv\:\mathrm{kg\:m^2\:s^{-2}}$), the metric unit of energy. If you were using the cgs system, $m$ would be in grams, $p$ would be in $\:\mathrm{g\:cm\:s^{-1}}$, $c$ would be in centimetres per second, and $E$ would be in ergs ($\:\mathrm{erg}\cong\:\mathrm{g\:cm^2\:s^{-2}}$),

Physical constants may change. Also, some equations have some constants set to one (eg Planck units, Gaussian units), so they may disappear entirely. For example, if $c=1$ (Planck units), the equation becomes $E^2=m^2+p^2$.

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keep in mind that constants of value $1$ are omitted, thus it is not necessarily true that all equations are equally valid in all unit systems - examples are Lorentz–Heaviside vs Gaussian vs SI units of elecromagnetism as well as various systems of natural units –  Christoph Nov 28 '12 at 11:26
    
@Christoph: True... I'll add that. –  Manishearth Nov 28 '12 at 11:27
    
Why are you saying that 1 J is only approximately equal to 1 kg m$^{2}$ s$^{-2}$? –  user12345 Nov 28 '12 at 12:07
    
@user16307: No, $\cong$ and $\equiv$ mean "congruent to" or "equivalent". $\sim$ and $\simeq$ are "similar to". –  Manishearth Nov 28 '12 at 12:50
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by (some) convention, $\equiv$ is used for equal by definition and $\cong$ for isomorphic (ie structurally equivalent but not necessarily equal); in your case, $\equiv$ would probably be more appropriate –  Christoph Nov 28 '12 at 18:55
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