Impossible Sp. Relativity question?

[Zorba]

Homework Statement

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Homework Equations

Below

The Attempt at a Solution

I think this question is impossible, i keep getting an answer that says that the particle recoils with a velocity > c_0.

Cons. Of Energy
$E_1 = E_2 + E_\gamma$
$m_0c^2 = \gamma m_1 c^2 + E_\gamma$

Cons. Of Mom.
$0 = P_1 - P_\gamma$
$\gamma m_1 = \frac{E_\gamma}{vc}$

$\Rightarrow m_0 c^2 = E_\gamma \cdot \frac{c}{v} + E_\gamma$

Some manipulation yields:
$v = \frac{cE_\gamma}{m_0c^2 - E_\gamma}$

And I plug the numbers in and I get 1.46c. ... !

 

[kuruman]

The correct equation for the total energy is

E = γm₀c², where m₀ is the rest mass.

 

[Zorba]

But does the rest mass not change, ie. the restmass before the photon is emitted, is different from after the rest mass is emitted? How else would the photon come about then..?

 

[kuruman]

The rest mass is always the same. In this case the given 6x10^-30 kg is the rest mass m₀. It is a number less than the initial mass because the particle is in an excited state before it emits the photon. You need to find the kinetic energy, which is the difference between the total energy and the rest mass energy. Note that you can write the total energy as

$$E=\sqrt{p^2c^2+m_0^2c^4}$$

Use the momentum conservation equation to eliminate p, subtract the rest mass energy and you are done.

 

[Zorba]

The rest mass is always the same.

Even if I have a particle that pumps out 100 photons, its rest mass is still invariant? That makes no sense to me, where is the energy coming from?

The particle initially is at rest, so it can't come from kinetic energy, thus it must come from rest mass energy, thus its rest mass energy must decrease after emitting the photon, thus its mass must decrease? Is this line of reasoning all wrong?

 

[kuruman]

Even if I have a particle that pumps out 100 photons, its rest mass is still invariant? That makes no sense to me, where is the energy coming from?

The particle initially is at rest, so it can't come from kinetic energy, thus it must come from rest mass energy, thus its rest mass energy must decrease after emitting the photon, thus its mass must decrease? Is this line of reasoning all wrong?

The particle is at rest, but in an excited state. If it were in the ground (lowest energy) state, it would not be able to emit a photon. It got in that excited state perhaps because it absorbed a photon or it underwent a collision with another particle and picked some energy, or whatever. That's where the energy is coming from.

 

[Zorba]

The particle is at rest, but in an excited state. If it were in the ground (lowest energy) state, it would not be able to emit a photon. It got in that excited state perhaps because it absorbed a photon or it underwent a collision with another particle and picked some energy, or whatever. That's where the energy is coming from.

But, but, but... where in the question does it say that? I have to assume that?!

 

[kuruman]

You don't have to assume anything. You know that a particle that is in the ground state and at rest cannot possibly emit a photon. That's because, by definition, the lowest possible energy state is the ground state and furthermore does not any have kinetic energy in that frame of reference that it can trade for photon energy. Like you say, if the particle emitted a photon, where would the photon energy come from? Therefore the particle that is given in the problem must be in an excited state. Either that or Energy Conservation does not hold. Take your pick.

 

[Zorba]

Hmm interesting, it seems to imply that a particle cannot "degrade" so to speak without some energy input, although I thought mass-energy equivalence would allow this - just have some of that mass converted into energy to emit the photon, but that seems to break postulate wrt to invariance of c...I think.

Would I be correct in saying that rest mass is an entirely invariant quantity?

(thanks a lot for answering all these qustions)

 

[kuruman]

Would I be correct in saying that rest mass is an entirely invariant quantity?

That's how I understand rest mass. See

http://en.wikipedia.org/wiki/Mass_in_special_relativity