= Solution
Interpret the motion as \b[uniform rectilinear motion]: the charge is stationary in an inertial <rest frame> and has no radiation field. Constant <speed> alone is insufficient; a curved trajectory has <acceleration> and can radiate even in vacuum.
In a transparent medium, a charge can move faster than the <light> <phase velocity> $c/n$ without exceeding $c$. The induced polarization disturbance then adds coherently on a Cherenkov cone, with $\cos\theta_C=1/(n\beta)$. Thus
$$
\beta_{\rm th}=\frac1n,\qquad
\boxed{K_{\rm th}=m_pc^2\left[\frac1{\sqrt{1-n^{-2}}}-1\right]
\simeq321\,\mathrm{MeV}\quad(n=1.5).}
$$
This is the kinematic threshold limit for the specified <refractive index>; appreciable emission requires exceeding it, and dispersion selects which <frequencies> satisfy the condition.
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