  We consider an active Brownian particle moving in a disordered
two-dimensional energy or motility landscape. The averaged
mean-square-displacement (MSD) of the particle is calculated analytically
within a systematic short-time expansion. As a result, for overdamped
particles, both an external random force field and disorder in the
self-propulsion speed induce ballistic behaviour adding to the ballistic regime
of an active particle with sharp self-propulsion speed. Spatial correlations in
the force and motility landscape contribute only to the cubic and higher order
powers in time for the MSD. Finally, for inertial particles two superballistic
regimes are found where the scaling exponent of the MSD with time is $\alpha=3$
and $\alpha=4$.
  We confirm our theoretical predictions by computer simulations. Moreover they
are verifiable in experiments on self-propelled colloids in random
environments.
