= Antiproton mass from momentum and velocity
{title2=$m^2c^4=p^2c^2(\beta^{-2}-1)$}
For a singly charged track perpendicular to a uniform <magnetic field>, its curvature gives <magnetic rigidity> $p/|q|=Br$, and the bend direction gives the sign of <electric charge>. The <mass shell> relation and <four-momentum> relation between <energy> and <velocity> give
$$
E^2=p^2c^2+m^2c^4,\qquad \beta=\frac vc=\frac{pc}{E},\qquad m^2c^4=p^2c^2\left(\beta^{-2}-1\right).
$$
For a known path length $L$ and constant <velocity>, <time of flight> $t=L/(\beta c)$ therefore gives
$$
m^2c^4=p^2c^2\left[\left(\frac{ct}{L}\right)^2-1\right].
$$
This measures <rest mass> under an assigned <electric charge> magnitude. Curvature alone measures $p/|q|$, so independent evidence for $|q|=e$ matters. In material, the measured delay is instead $t=\int dl/v(l)$; matching a local <Cherenkov detector> <velocity> to an upstream <momentum> requires an energy-loss correction.
As an original illustrative calculation, choose $p=1.20\,\mathrm{GeV}/c$ and $L=12.0\,\mathrm m$, with rounded input <rest energies> $m_pc^2=0.93827\,\mathrm{GeV}$ and $m_\pi c^2=0.13957\,\mathrm{GeV}$. Constant-<velocity> propagation gives
$$
\begin{aligned}
\beta_{\bar p}&\simeq0.7878,&t_{\bar p}&\simeq50.81\,\mathrm{ns},\\
\beta_{\pi^-}&\simeq0.9933,&t_{\pi^-}&\simeq40.30\,\mathrm{ns}.
\end{aligned}
$$
Both particles have the same selected <momentum> and negative <electric charge>, yet their <time of flight> separation is about $10.51\,\mathrm{ns}$. These are calculated values for a deliberately simplified example, not measured discovery events.
\Image[/antiproton-momentum-velocity-selection-simulation.png]
{title=Simulated momentum and velocity tests distinguish antiprotons from negative pions}
{description=Original simulation with chosen populations and resolutions, not historical event data or abundances. The left panel assumes constant <velocity> over $12\,\mathrm m$. The right panel applies a toy $40\,\mathrm{MeV}$ energy loss before a local <velocity> reading and overlays historical C2 acceptance and C1 guard boundaries.https://escholarship.org/content/qt46p0z8w7/qt46p0z8w7.pdf[UCRL-3172, p. 5]{ref}}
For example, the toy energy loss changes the central <antiproton> <velocity> from $0.7878c$ to $0.7745c$. Directly substituting the slower downstream value together with the upstream <momentum> would overestimate the <rest mass>. The two panels thus describe different stages of a deliberately specified model; their numerical consistency depends on propagating <energy> between stages. Likewise, the scattered-<pion> markers represent an erroneous C2 response, not actual slow <pions>; an upstream C1 signal still rejects them.
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