  A mobile impurity particle immersed in a quantum fluid forms a polaron - a
quasiparticle consisting of the impurity and a local disturbance of the fluid
around it. We ask what happens to a one-dimensional polaron after a kick, i.e.
an abrupt application of a force that instantly delivers a finite impulse to
the impurity. In the framework of an integrable model describing an impurity in
a one-dimensional gas of fermions or hard-core bosons, we calculate the
distribution of the polaron momentum established when the post-kick relaxation
is over. A remarkable feature of this distribution is a two-sided power-law
singularity. It emerges due to one of two processes. In the first process, the
whole impulse is transferred to the polaron, without creating phonon-like
excitations of the fluid. In the second process, the impulse is shared between
the polaron and the center-of-mass motion of the fluid, again without creating
any fluid excitations. The latter process is, in fact, a Bragg reflection at
the edge of the emergent Brillouin zone. We carefully analyze the conditions
for each of the two processes. The asymptotic form of the distribution in the
vicinity of the singularity is derived.
