# Mass Energy Equivalence and Kinetic Energy

If a particle is moving through free space (no forces acting upon it) should its kinetic energy equal its mass energy equivalence or am I getting confused. In other words is an object's kinetic energy absorbed within its mass?

Is the following true?

$$E_{kinetic} = \frac{p^{2}}{2m} = mc^{2} = E_{mass}$$

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Doc Al
Mentor
No.
(1) For one thing, mc², where m is the invariant mass, is the rest energy of the particle.
(2) KE = p²/2m is only approximately valid when speeds are low enough.
(3) The total energy of a free particle is given by

$$E = (\frac{1}{\sqrt{1 - v^2/c^2}}) mc^2$$

The KE energy is the total energy minus the rest energy:

$$KE = (\frac{1}{\sqrt{1 - v^2/c^2}} - 1) mc^2$$

Thanks!

Yes, I realised I asked my question badly. So to clarify the total mass of the moving particle is equal to the initial mass-energy plus the kinetic energy so the kinetic energy could be thought of as being absorbed into the total mass of the moving particle.

Correct?

Doc Al
Mentor
So to clarify the total mass of the moving particle is equal to the initial mass-energy plus the kinetic energy so the kinetic energy could be thought of as being absorbed into the total mass of the moving particle.

Correct?
If by 'total mass' you mean the so-called relativistic mass, then yes: The relativistic mass reflects the total energy of the particle.

But I think you're better off sticking with invariant mass and thinking in terms of rest mass energy plus kinetic energy.

So has science determined if there is a concrete difference between mass and energy? If the "relativistic" mass were increased by kinetic energy on an atom rather than a particle then would that affect the internal balance of forces of the atom? Could it alter the orbital radius &c?

Doc Al
Mentor
So has science determined if there is a concrete difference between mass and energy?
While there is certainly an equivalency between mass and energy, I wouldn't think of them as being the same thing.
If the "relativistic" mass were increased by kinetic energy on an atom rather than a particle then would that affect the internal balance of forces of the atom? Could it alter the orbital radius &c?
If I understand your question, it's equivalent to asking: If an atom is moving, will it have different characteristics than an atom at rest. As far as the moving atom is concerned, from its frame there is no difference.

Granted, but if I observe the atom's state when it was stationary and again when it was moving, both times from my own stationary frame, would there be any difference?

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Granted, but if I observe the atom's state when it was stationary and again when it was moving, both times from my own stationary frame, would there be any difference?
You would observe that it's moving, and it has higher energy (kinetic). for low speeds you can approximate this extra energy as 2*E(k)=mv²

Explanation of mass with Tensors

10:00 onwards, it shows how the above os derived

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So has science determined if there is a concrete difference between mass and energy?

While there is certainly an equivalency between mass and energy, I wouldn't think of them as being the same thing.
I definitely would however think of them as connected... or related....we strongly suspect the fource forces are all related; similarly relativity shows us energy and mass are related as are space and time...you can read more about all these related to "phase transition" associated with the big bang...or "grand unification epoch"...

Doc Al
Mentor
Granted, but if I observe the atom's state when it was stationary and again when it was moving, both times from my own stationary frame, would there be any difference?
Sure. Viewed semi-classically, the electrons would seem to be moving slower about the nucleus. The energy levels would be different.

All of this would conspire to support the relativistic time dilation and Doppler effects.