I will address the problem of how a particle (impurity) injected with a finite momentum into a gas of quantum particles of other type evolves with time. I will show that contrary to conventional picture of relaxation dynamics, the particle never comes to a complete stop. Instead, for initially supersonic velocity long-lived oscillations develop and the velocity saturates to a value almost independent of initial conditions. The methods used, namely Bethe Ansatz and numerics, will be discussed in detail along with the state-of-the-art experiments.