Relativistic Energy Equations: When to Use Each

[Kara386]

When would I use the equation $E = \gamma mc^2$ and when would I use $E^2 = (mc^2)^2 + (pc)^2$? I'm a little confused because my textbook calls them both total energy equations. I know that for a particle at rest it has energy $E=mc^2$. It can't be at rest for the equation $E = \gamma mc^2$ because $\gamma$ involves velocity, so I assume the object has to be moving. So when do I use that equation? And when do I use the $E^2$ one?

Thanks for any help! :)

 

[Dale]

When would I use the equation E=γmc^2 and when would I use E^2 = (mc^2)^2 + (pc)^2

I would recommend always using the second one. It reduces to the first whenever appropriate.

 

[PAllen]

You can use any of them for a particle at rest; gamma is just 1 for zero velocity. The formula with gamma is no good for light because it is undefined for speed c. The energy squared relation is good for all cases, including light. Obviously, m is 0 for light. For m not zero, you can demonstrate algebraically that it is the same as the gamma formula.

 

[Kara386]

Seems like the best thing then is to stick with the $E^2$ equation. Thanks! :)

 

[PeroK]

When would I use the equation $E = \gamma mc^2$ and when would I use $E^2 = (mc^2)^2 + (pc)^2$? I'm a little confused because my textbook calls them both total energy equations. I know that for a particle at rest it has energy $E=mc^2$. It can't be at rest for the equation $E = \gamma mc^2$ because $\gamma$ involves velocity, so I assume the object has to be moving. So when do I use that equation? And when do I use the $E^2$ one?

Thanks for any help! :)

I would use each of the equations when it appears useful. For example, if I knew the mass and gamma factor of a particle and wanted the energy, I would use the first equation.

If I knew the mass and the momentum, I would use the second.

 

[stevendaryl]

The drawback to using $E = \sqrt{p^2 c^2 + m^2 c^4}$ is that it's harder to connect it with the velocity. That extra information is provided by:

$v = \frac{pc^2}{E}$

That's valid whether the object is massless or not. Another relation that gives the same answer, but is interesting because it is true both classically and relativistically, is:

$v = \frac{dE}{dp}$