Proving Massless Particle's Speed: 0.5c

[im_slow]

Homework Statement

Massless Particles
A neutral pion traveling along the x-axis decays into two photons, one being ejected exactly forward and the other exactly backward. The first photon has three times the energy of the second. Prove that the original pion had speed 0.5c.

Homework Equations

for m=0, E=p*c
conservation of Energy E^2=(c*p)^2+(m*c^2)^2
gamma=1/sqrt(1-Beta^2)
Beta = v/c
p=gamma*m*v
E=gamma*m*c^2

The Attempt at a Solution

sum(momentum photons) = sum (momentum pion)

 

[kamerling]

The photon energy is easy to compute in a frame where the pion is at rest.
Use the formula for the relativistic doppler effect to find the frequency of both fotons in a frame that moves with a speed v. The frequency of the forward photon must be 3 times the
frequency of the backwards photon.

 

[kdv]

Homework Statement

Massless Particles
A neutral pion traveling along the x-axis decays into two photons, one being ejected exactly forward and the other exactly backward. The first photon has three times the energy of the second. Prove that the original pion had speed 0.5c.

Homework Equations

for m=0, E=p*c
conservation of Energy E^2=(c*p)^2+(m*c^2)^2
gamma=1/sqrt(1-Beta^2)
Beta = v/c
p=gamma*m*v
E=gamma*m*c^2

The Attempt at a Solution

sum(momentum photons) = sum (momentum pion)

Are you familiar with four-momentum conservation?


You may use P_\pi = P_1 + P_2 where 1 and 2 refer to the photons. After squaring and using P^2 = m^2 c^4 , so that P_1^2 = P_2^2 = 0, you get

m_\pi^2 c^4 = 2 E_1 E_2 - c^2 {\vec p}_1 \cdot \vec{p_2}

now, using the fact that the two photons move in oppposite directions, you find that 4 E_1 E_2 = m_\pi^2 c^4. Using the fact that one photon has three times the energy of the other one, you then have the energy of each photon in terms of the pion mass.

Then use E_1 + E_2 = \gamma m_\pi c^2 to find the speed.

 

[bjewels]

I don't understand this. can you try explaining it in more detail?