Pi meson decay (relativistic momentum)

[RyanP]

Homework Statement

A charged π meson (rest mass = 273m_e) decays into a neutrino (zero rest mass) and a μ meson (rest mass = 207m_e). Find the kinetic energies of the neutrino and the mu meson.

Homework Equations

E = m_oγc²
K = m_o(γ-1)c²
v = pc²/E
p = m_oγv

The Attempt at a Solution

In the rest frame of the pi meson (S frame),
E_i = 273m_ec²
E_f = 207m_eγc² + K where γ is the gamma factor of the mu meson and K is the energy (kinetic) of the neutrino.

Since the neutrino has no rest mass, it travels at c, so p_v = E_v/c = K/c.

I couldn't figure out how to carry on in this frame, so I switched to an S' frame where the mu meson is at rest.

In this frame,
p'_i = -273m_eγv where v is the speed of the pi meson initially (equal to the speed of the mu meson in the pi rest frame).
p'_f = -E'_nu/c since the neutrino is the only thing moving after the collision in this frame.

E'_i = 273m_eγc²
E'_f = 207m_e + E'_nu

How do I proceed from here?

 

[phyzguy]

Go back to the rest frame of the pion. Since the pion is initially at rest, the initial momentum in this frame is zero. Since momentum is conserved, the final momentum is zero also. So what do you know about the momenta of the neutrino and the muon?

 

[RyanP]

Go back to the rest frame of the pion. Since the pion is initially at rest, the initial momentum in this frame is zero. Since momentum is conserved, the final momentum is zero also. So what do you know about the momenta of the neutrino and the muon?

The momentum of the neutrino is equal and opposite to the momentum of the muon - should I solve for the speed of the muon?

 

[phyzguy]

You should be able to write four equations for the four unknowns, Enu, Emu, pnu, pmu.

 

[RyanP]

Ended up with a quadratic equation for v (speed of the muon), in terms of both masses and c. Brute-forced it with the quadratic formula and got the right answer.