The motion of an impurity atom in a degenerate Tonks-Girardeau gas is investigated in the regime of weak impurity-host coupling. It is shown that given some initial momentum $p_0$ an impurity relaxes to a steady state with a non-vanishing average momentum $p_\infty.$ The function $p_\infty(p_0)$ is calculated explicitly in the limit of vanishing coupling constant. Further to this, an impurity driven by a weak constant force is shown to exhibit pronounced momentum oscillations in agreement with earlier theoretical predictions.