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Name: Pat
Status: student
Grade: 12+
Location: Outside U.S.
Country: Australia
Date: Spring 2013


Question:
I understand the principle of e=mc2, but the other day, I found out that it is actually e2=(mc2)2+ (pc)2. This apparently applies for objects in motion. Could you please explain the meaning of this rule to me?

Replies:
Pat, Potential energy is within the force between objects, not within an object itself. E^2=(mc^2)^2+(pc)^2 represents the relation between total energy within an object and the momentum of that same object. For very small speeds, speeds much less than the speed of light, this reduces to E=mc^2 +(p^2)/2m: mass energy plus kinetic energy. At very high speeds, according to relativity theory, the more complex equation must be used. At rest, the energy within an object is E=mc^2. For an object without mass, such as a photon of light, E=pc (momentum times the speed of light, p=mv does not apply). When an object has both mass and momentum, they relate to the energy within the object according to E^2=(mc^2)^2+(pc)^2. M=mass of the particle at rest. Both p=momentum and E=energy must be measured in the same frame of reference.

Dr. Ken Mellendorf Physics Instructor Illinois Central College


Pat From this URL: http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence “In inertial reference frames other than the rest frame or center of mass frame, the equation E = mc² remains true if the energy is the relativistic energy and the mass the relativistic mass. It is also correct if the energy is the rest or invariant energy (also the minimum energy), and the mass is the rest mass, or the invariant mass. However, connection of the total or relativistic energy (Er) with the rest or invariant mass (m0) requires consideration of the system total momentum, in systems and reference frames where the total momentum has a non-zero value. The formula then required to connect the two different kinds of mass and energy, is the extended version of Einstein's equation, called the relativistic energy–momentum relation:

(Equarions are not presented)

or

(Equarions are not presented)

Here the (pc)2 term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation reduces to E = mc² when the momentum term is zero. For photons where m0 = 0, the equation reduces to Er = pc.

View a video explanation of the full equation at Minute Physics on Youtube”

Sincere regards, Mike Stewart


Hi Pat,

Thanks for the question. Yes, you are correct that the full equation is E^2 = (mc^2)^2 + (pc)^2. We use the full equation to account for the total energy, E, of an object with both rest energy (i.e., potential energy) and kinetic energy. The kinetic energy is included by the p in the equation. p in the equation represents the momentum of the object. If the object is at rest and p = 0, the equation simplifies to E = mc^2.

I hope this helps. Please let me know if you have more questions. Thanks Jeff Grell


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