  At fast timescales, the self-similarity of random Brownian motion is expected
to break down and be replaced by ballistic motion. So far, an experimental
verification of this prediction has been out of reach due to a lack of
instrumentation fast and precise enough to capture this motion. With a newly
developed detector, we have been able to observe the Brownian motion of a
single particle in an optical trap with 75 MHz bandwidth and sub-{AA}ngstrom
spatial precision. We report the first measurements of ballistic Brownian
motion as well as the first determination of the velocity autocorrelation
function of a Brownian particle. The data are in excellent agreement with
theoretical predictions taking into account the inertia of the particle and the
surrounding fluid as well as hydrodynamic memory effects.
