The Next Generation Simulation (NGSIM) trajectory data sets provide longitudinal and lateral positional information for all vehicles in certain spatiotemporal regions. Velocity and acceleration information cannot be extracted directly because the noise in the NGSIM positional information is greatly increased by the necessary numerical differentiations. A smoothing algorithm is proposed for positions, velocities, and accelerations that can also be applied near the boundaries. The smoothing time interval is estimated on the basis of velocity time series and the variance of the processed acceleration time series. The velocity information obtained in this way is then applied to calculate the density function of the two-dimensional distribution of velocity and inverse distance and the density of the distribution corresponding to the "microscopic" fundamental diagram. It is also used to calculate the distributions of time gaps and times to collision, conditioned to several ranges of velocities and velocity differences. By simulating virtual stationary detectors, it is shown that the probability for critical values of the times to collision is greatly underestimated when estimated from single-vehicle data of stationary detectors. Finally, the lane-changing process is investigated, and a quantitative criterion is formulated for the duration of lane changes that is based on the trajectory density in normalized coordinates. There is a noisy but significant velocity advantage in favor of the targeted lane that decreases immediately before the change due to anticipatory accelerations.