Large-speed approximation for circular-trap kicks ($f_p\simeq f_T\Delta x/(2\pi R)$)

The exact separated-kick response is $f_p=f_T\Delta x/(C+\Delta x)$ for circumference $C=2\pi R$. Its relative correction to the independent-kick estimate is controlled by $\Delta x/C$. For a localized well with force-induced speed bounded by $ϵv_T$, $\Delta x/C\le (w/C){ϵ}^2/(1-ϵ)$. Thus high speed is sufficient when the support width is below one circumference. The local straight-force approximation also requires a narrow trap on the curved track.