  Many self-propelled objects are large enough to exhibit inertial effects but
still suffer from environmental fluctuations. The corresponding basic equations
of motion are governed by active Langevin dynamics which involve inertia,
friction and stochastic noise for both the translational and orientational
degrees of freedom coupled via the self-propulsion along the particle
orientation. In this paper, we generalize the active Langevin model to
time-dependent parameters and explicitly discuss the effect of time-dependent
inertia for achiral and chiral particles. Realizations of this situation are
manifold ranging from minirockets which are self-propelled by burning their own
mass, dust particles in plasma which lose mass by evaporating material to
walkers with expiring activity. Here we present analytical solutions for
several dynamical correlation functions such as the mean-square displacement
and the orientational and velocity autocorrelation functions. If the parameters
exhibit a slow power-law in time, we obtain anomalous superdiffusion with a
non-trivial dynamical exponent. Finally we constitute the "Langevin rocket"
model by including orientational fluctuations in the traditional Tsiolkovsky
rocket equation. We calculate the mean reach of the Langevin rocket and discuss
different mass ejection strategies to maximize it. Our results can be tested in
experiments on macroscopic robotic or living particles or in self-propelled
mesoscopic objects moving in media of low viscosity such as complex plasma.
