[deepground]

When looking at the mass of a particle at "rest" what does at rest mean? Relative to what? The earth is moving and the solar system is moving etc etc, so how do we determine what the mass is at a rested state for a particle.. 

[pantone159]

At rest relative to the reference frame im which you are measuring the mass.

If you have a particle at rest wrt the Earth, and measure its mass, you get m0, the rest mass.
If you measure the mass of the same particle from a frame fixed to the sun (so now the particle is moving with the Earth wrt you), you get a different answer.

(Sort of.  Opinions vary as to whether one should really think of the mass as varying with velocity, or instead just adjust the momentum/energy equations to account for the changes.  I was personally taught that mass equals mass equals mass, i.e. it doesn't change, but on the other hand, momentum does not equal mass times velocity, rather it equals mass times velocity divided by sqrt (1 - v^2 / c^2).)

[deepground]

Thank you for your reply. I suppose this means that the mass numbers we have found for all of the elementary particles at rest are simply what they happen to be when traveling at the speed of our earth.

Do we have any idea of how to find out what mass they would be if they were not moving at all in the grand scheme of things? 

[pantone159]

If I have a particle that is at rest wrt the Earth, and I measure its mass, I get the true rest mass.
It doesn't matter that the Earth is moving wrt something else (i.e. the sun.)  There is no such thing as the 'true' velocity of something, it always has to be measured relative to an arbitrary frame.

If I measure the mass somehow when the particle is moving in my lab, I can calculate the rest mass from the equation:  apparent mass = rest mass / sqrt (1 - v^2/c^2)

[Yggdrasil]

As long as the object is moving in the same frame of reference as the observer, the observer will measure the true rest mass of the object.

The point of relativity is that all inertial frames of reference are the same.  If the particle is at rest wrt to the Earth and the observer is at rest wrt the Earth, the observer will measure the same mass of the particle as a observer at rest wrt the sun measuring a particle at rest wrt the sun.

[deepground]

Ok, I think I see what your saying, because the speed of light is a constant maximum speed.

So by using the speed of light that equation would give the same mass of a particle at rest no matter what planet your lab was located on