  Motion of active particles, such as catalytic micro- and nano-motors, is
usually characterized via either dynamic light scattering or optical
microscopy. In both cases, speed of particles is obtained from the calculus of
the mean square displacement (MSD) and typically, the theoretical formula of
the MSD is derived from the motion equations of an active Brownian particle.
One of the most commonly reported parameters is the speed of the particle,
usually attributed to its propulsion, and is widely used to compare the motion
efficiency of catalytic motors. However, it is common to find different methods
to compute this parameter, which are not equivalent approximations and do not
possess the same physical meaning. Here, we review the standard methods of
speed analysis and focus on the errors that arise when analyzing the MSD of
self-propelled particles. We analyze the errors from the computation of the
instantaneous speed, as well as the propulsive speed and diffusion coefficient
through fittings to parabolic equations, and we propose a revised formula for
the motion analysis of catalytic particles moving with constant speed that can
improve the accuracy and the amount of information obtained from the MSD.
Moreover, we emphasize the importance of spotting the presence of different
motion dynamics, such as particles with active angular speed or that move under
the presence of drift, and how the breaking of ergodicity can completely change
the analysis by considering particles with an exponentially decaying speed. In
all cases, real data from enzymatically propelled micro-motors and simulations
are used to back up the theories. Finally, we propose several analytical
approaches and analyze limiting cases that will help to deal with these
scenarios while still obtaining accurate results.
