Issue calculating the velocity of pions from K-meson decay

[mountevans]

Homework Statement

We were supplied with several images of hydrogen bubble chamber tracks, from which we measured the length of the (invisible) K⁰ track, angles of departure from the K⁰ meson's track and curvature. As a way to limit the amount of data we're talking about, I'll only discuss the first particle. The length of the K⁰'s track was 0.016 m, the radius of curvature of the π⁺ was 0.65 m, the π^- 0.50 m, the angles of departure (π⁺,π^-) were 44 and 24 degrees respectively, and the magnetic field was 1.4 Teslas.

The momentum of the positive k meson tracks is given as 535 MeV/c. Rest mass of positive and negative pions is 139.57 MeV/c². c = 3*10⁸ m/s, of course.

Homework Equations

p = q*r*B
T = 1/2 mv²
γ = 1/√(1-v²/c²)
p [in MeV/c] = 3 ∗ 10² ∗ r [in m] ∗ B [in T]
F = mv²/r

The Attempt at a Solution

Using the formula for p above, 3 ∗ 10² * (0.65 m) * (1.4 T), I get 382.6 MeV/c for the π⁺ and 294.3 MeV/c for the π^- total momenta.

However, I'm now at the step where I think I need to determine the total energy of the pions so I can apply a gamma correction to their speed. When I take the above equations,

F = mv²/r and p = mv = q*r*B

I find that v must equal q*r*B/m, where m is the mass(-energy?) of the pions. So, here appears to lie my mistake: I divided 382.6 MeV/c by 139.57 MeV/c², and got 2.74, which I would assume is in units of c (which can't be right for v, since nothing exceeds v least of all a pion). So it seems like I'm missing a factor here - should I have converted the momenta in MeV/c to eV/c?

For the later components of this homework, we simply use 2D kinematics to find the momentum vectors of the pions - the sum of the components in-axis with the path of the kaon should equal the momentum of the neutral kaon, and the total momentum in the normal axis should sum to zero.

 

[vela]

F = mv²/r and p = mv = q*r*B

Those equations are non-relativistic. You can use the relationship $E^2 = (mc^2)^2 + (pc)^2$ where $E$ is the total energy of the particle and $p$ is the relativistic momentum. Also, your expression for the kinetic energy is should be $T=(\gamma-1)mc^2 = E - mc^2$.

 

[mountevans]

Those equations are non-relativistic. You can use the relationship $E^2 = (mc^2)^2 + (pc)^2$ where $E$ is the total energy of the particle and $p$ is the relativistic momentum. Also, your expression for the kinetic energy is should be $T=(\gamma-1)mc^2 = E - mc^2$.

Is $p=γmv$ correct in this instance? (where m = rest mass). Would the rest energy of the pion be 139.57 * (3*10⁸)² MeV, rather than the mass (in MeV/c²)? When going from mass to energy, if I'm not using c=1, I need to actually multiply by c², right?

 

[mountevans]

It was simply a problem with converting between MeV and J, if you can believe it. This page was very helpful.