Pion decay (tell me if I'm approaching it correctly)

[quasar_4]

Homework Statement

A neutral pion may decay into two photons. A particular pion is traveling along the x-axis when it decays into two photos, the first going directly along the +x axis, the second going directly back along the -x axis. The energy of the photons is measured and it is found that E1 is n times more energetic than E2. Find the velocity u of the pion.

Homework Equations

1) relativistic momentum = gamma *m*u where u is velocity, m is mass
2) momentum of a photon is h/wavelength = h*frequency/c (I'm denoting frequency by nu).

The Attempt at a Solution

I tried to do this using conservation of momentum. I'd like to know if my solution is correct.

Pi = Pf (via conservation of momentum for isolated system)

so

gamma*m*u = h*nu1/c - h*nu2/c (the minus sign accounts for the fact that the photons move in opposite directions along the x axis)

but as given above, E1 = n*E2, so

gamma*m*u = h*nu2 (n - 1)/c

Then I solved for u using algebra (I didn't forget about the u in gamma, I squared everything and solved for u afterwards).

Can that be right? Or am I thinking way too simplistically here?

 

[tiny-tim]

Hi quasar_4!

(have a nu: ν )

It's a bit difficult to tell without seeing your final calculations,

but it looks as if you haven't used a conservation of energy equation.

You need a conservation of momentum equation and a conservation of energy equation, just as in the Newtonian case (and ν needn't come into it).

 

[quasar_4]

hello, tiny-tim! :-D

I need both? Hmm, I thought that I could do it just with conservation of momentum OR conservation of energy. We only have one unknown, so two equations and one unknown! What am I missing here? Why do we need to use both the conserved quantities?

(It's been a long, long time since I took any sort of modern physics class... I feel like such a dork for not remembering this!)

 

[kuruman]

There are three equations and three unknowns. The equations are

Momentum Conservation
Energy Conservation
E₁=n E₂

The unknowns are
u, E₁ and E₂.

Your answer should come out in terms of n, c and the rest-mass of the pion.

 

[kuruman]

Actually, the rest-mass of the pion drops out.