  We have measured absolute cross sections for ultrafast (fs) single-carbon
knockout from Polycyclic Aromatic Hydrocarbon (PAH) cations as functions of
He-PAH center-of-mass collision energy in the range 10-200 eV. Classical
Molecular Dynamics (MD) simulations cover this range and extend up to 10$^5$
eV. The shapes of the knockout cross sections are well described by a simple
analytical expression yielding experimental and MD threshold energies of
$E_{th}^{Exp}=32.5\pm 0.4$ eV and $E_{th}^{MD}=41.0\pm 0.3$ eV, respectively.
These are the first measurements of knockout threshold energies for molecules
isolated \emph{in vacuo}. We further deduce semi-empirical (SE) and MD
displacement energies --- \emph{i.e.} the energy transfers to the PAH molecules
at the threshold energies for knockout --- of $T_{disp}^{SE}=23.3\pm 0.3$ eV
and $T_{disp}^{MD}=27.0\pm 0.3$ eV. The semi-empirical results compare
favorably with measured displacement energies for graphene $T_{disp}=23.6$ eV
[Meyer \emph{et al.} Phys. Rev Lett. \textbf{108} 196102 (2012) and
\textbf{110} 239902 (2013)].
