Pi meson with velocity v decays into two photons

[jfbych]

Homework Statement

A ∏° meson moving with a velocity v in the z direction decays into two photons. One of the photons travels in the z direction and the other travels in the minus z direction.

a) If one photon has an energy that is nine times greater than the other photon, calculate the speed of the ∏° meson.

b)If the speed of the ∏° meson is c/2 determine the energies of the two photons.

First time user here, so sorry in advance for formatting problems!

Homework Equations

E'=E*sqrt(1+β/1-β)

Conservation of momentum:
P_∏°=P₁+P₂
Where P₁ and P₂ are the momentum of the photons.

The Attempt at a Solution

Not sure if this is along the right tracks, but I did:
E'=9*E=E*sqrt(1+β/1-β)
β=0.976
v=0.976*c
For part b I am not even sure where to start...

 

[mfb]

Homework Statement

A ∏° meson moving with a velocity v in the z direction decays into two photons. One of the photons travels in the z direction and the other travels in the minus z direction.

a) If one photon has an energy that is nine times greater than the other photon, calculate the speed of the ∏° meson.

b)If the speed of the ∏° meson is c/2 determine the energies of the two photons.

First time user here, so sorry in advance for formatting problems!

Homework Equations

E'=E*sqrt(1+β/1-β)

Conservation of momentum:
P_∏°=P₁+P₂
Where P₁ and P₂ are the momentum of the photons.

The Attempt at a Solution

Not sure if this is along the right tracks, but I did:
E'=9*E=E*sqrt(1+β/1-β)
β=0.976
v=0.976*c
For part b I am not even sure where to start...

For part b, you can just work your way back. You start with v=0.5 c, what is E'/E?
What do you know about the total energy of the photons?

Where does the formula for (a) come from? I get a different result with relativistic momenta.

 

[jfbych]

So we find E'/E=sqrt(3), and we know that the energy of the two photons have to add up to the energy of the ∏° meson. I am still confused on where to go from there. For part (a) I tried to solve for the velocity of the ∏° meson by using the 9:1 photon energy relationship using the doppler shift equation.

 

[mfb]

So we find E'/E=sqrt(3),

Again, I think you use a wrong formula, or the correct formula in a wrong way.

and we know that the energy of the two photons have to add up to the energy of the ∏° meson.

Right

I am still confused on where to go from there. For part (a) I tried to solve for the velocity of the ∏° meson by using the 9:1 photon energy relationship using the doppler shift equation.

Doppler shift is a relation of the frequency of photons in different reference frames, but that is a complicated way to solve it (as both photons are doppler-shifted).

If you know 4-momenta: They are the easiest way to solve the problem.

 

[vela]

The Attempt at a Solution

Not sure if this is along the right tracks, but I did:
E'=9*E=E*sqrt(1+β/1-β)
β=0.976
v=0.976*c
For part b I am not even sure where to start...

This isn't correct. The Doppler shift formula relates the energy E of a photon in one frame with its energy E' in a frame moving with velocity β. You're trying to use it to compare the energies of two different photons.

You can use the formula to solve this problem, but you have to apply it correctly.