  We describe a new technique to estimate the mean square velocity of a
Brownian particle from time series of the position of the particle sampled at
frequencies several orders of magnitude smaller than the momentum relaxation
frequency. We apply our technique to determine the mean square velocity of
single optically trapped polystyrene microspheres immersed in water. The
velocity is increased applying a noisy electric field that mimics a higher
kinetic temperature. Therefore, we are able to measure the average kinetic
energy change in isothermal and non-isothermal quasistatic processes. Moreover,
we show that the dependence of the mean square time-averaged velocity on the
sampling frequency can be used to quantify properties of the electrophoretic
mobility of a charged colloid. Our method could be applied to detect
temperature gradients in inhomogeneous media and to characterize the complete
thermodynamics of microscopic heat engines.
