Positron momentum from π+ → e+ + νe decay at rest

[lee_sarah76]

Homework Statement

We are the given the rare decay:

π⁺ → e⁺ + v_e

Also, we are given m_π = 139.6 MeV/c², m_v ≈ 0, m_e+ = 0.511 MeV/c²

And we are given that the decay starts with the pion at rest, so I took that to mean that p_π = 0

Solve for the momentum of the positron.

Homework Equations

Conservation of Momentum
Conservation of Energy
E² = (pc)² + (mc)²
E = mc²

The Attempt at a Solution

I started with the fact that since the pion was at rest before decay, p_π = 0 so p_e + p_v = 0.

Then, also using the fact that the pion was at rest before decay I solved for its energy using the formula E = mc², substituting m_π = 139.6 MeV/c² getting that E_before = 139.6 MeV.

E_before = E_after = E_e + E_v

(E_e)² = (p_ec)² + (m_ec²)²

And because the neutrino is massless:
E_v = p_v*c

From before, since E_e + E_v = 139.6 MeV, I solved for E_e to get E_e =139.6 MeV - E_v.

Then I substituted p_vc for E_v to get the equation:

(139.6 MeV - p_vc)² = (p_ec)² + (m_ec²)²

The (pc)² would cancel each other out, leaving:

-279.2p_vc = -19487.8 MeV

So, p_v = 69.799 MeV/c leaving p_e = -69.799 MeV/c

My question is, is this right, or have I made some glaring mistakes in my method?

 

[TSny]

Looks good to me.