  We consider the long-time behavior of a fast, charged particle interacting
with an initially spatially homogeneous background plasma. The background is
modeled by the screened Vlasov-Poisson equations, whereas the interaction
potential of the point charge is assumed to be smooth. We rigorously prove the
validity of the \emph{stopping power theory} in physics, which predicts a
decrease of the velocity $V(t)$ of the point charge given by $\dot{V} \sim
-|V|^{-3} V$, a formula that goes back to Bohr (1915). Our result holds for all
initial velocities larger than a threshold value that is larger than the
velocity of all background particles and remains valid until (i) the particle
slows down to the threshold velocity, or (ii) the time is exponentially long
compared to the velocity of the point charge. The long-time behavior of this
coupled system is related to the question of Landau damping which has remained
open in this setting so far. Contrary to other results in nonlinear Landau
damping, the long-time behavior of the system is driven by the non-trivial
electric field of the plasma, and the damping only occurs in regions that the
point charge has already passed.
