What is the energy of each photon produced in the decay of the pion?

[cashmoney805]

Homework Statement

The neutral pion is an unstable particle that decays very quickly after its creation into two photons (“particles” of light: v = c, mo = 0). The pion has a rest-mass energy of 135 MeV. Consider a pion that has a kinetic energy of 90 MeV
1) Determine the v of this pion
2) Determine the momentum of the pion
3) Determine the sum of the energies of the photons

Homework Equations

1) m_oc²\gamma = KE + Rest E
2) m_ov\gamma = p
3) E_photons= pc

The Attempt at a Solution

I'm pretty sure I got 1 and 2. For 1 I added the KE + RE, converted to proper units, and finally got v = .80c
For 2 I got p = 9.6 x 10^-20 kg m/s
Now I'm not so sure about #3. I think E_photons = pc because momentum is conserved, and all of the pion's momentum gets turned into the photons' momentum. However, what about adding up the original rest mass energy + KE of the pion? If I do that I get a bigger energy then if I do pc. Thanks for all the help!

 

[Nabeshin]

Homework Statement

The neutral pion is an unstable particle that decays very quickly after its creation into two photons (“particles” of light: v = c, mo = 0). The pion has a rest-mass energy of 135 MeV. Consider a pion that has a kinetic energy of 90 MeV
1) Determine the v of this pion
2) Determine the momentum of the pion
3) Determine the sum of the energies of the photons

Homework Equations

1) m_oc²\gamma = KE + Rest E
2) m_ov\gamma = p
3) E_photons= pc

The Attempt at a Solution

I'm pretty sure I got 1 and 2. For 1 I added the KE + RE, converted to proper units, and finally got v = .80c
For 2 I got p = 9.6 x 10^-20 kg m/s
Now I'm not so sure about #3. I think E_photons = pc because momentum is conserved, and all of the pion's momentum gets turned into the photons' momentum. However, what about adding up the original rest mass energy + KE of the pion? If I do that I get a bigger energy then if I do pc. Thanks for all the help!

Just using pure energy conservation, how much energy is there before and after the decay?

 

[cashmoney805]

Just using pure energy conservation, how much energy is there before and after the decay?

According to my calculations, E before = (90 + 135) MeV = 225 MeV
After E = pc = 181.25 MeV

 

[cashmoney805]

I calculated p a different way this time, p = sqrt(2mKE) where m is the relativistic mass. When I do this then multiply p by c to get E, I get E = 201 MeV. Oh boy...

edit: actually I'm not sure if that equation works for speeds close to c...

 

[Nabeshin]

According to my calculations, E before = (90 + 135) MeV = 225 MeV
After E = pc = 181.25 MeV

Hrm, you're letting the calculations get bogged down too much. The change in energy is zero, right? You know initial energy, so final energy must be.. the same.

 

[cashmoney805]

I get what you're saying, but I don't understand why the equations don't work here. It seems to me that momentum isn't conserved.

 

[cashmoney805]

and there is one more part to this problem which I thought I could get myself, but I can't. Here is a pic of the question/diagram

http://i44.tinypic.com/k12kja.gif