Sectional curvature is related to tidal accelerations for small objects of nonzero rest mass. Generically, the magnification of tidal accelerations due to high speed goes as the square of the magnification of energy. However, some space-times have directions with bounded increases in tidal accelerations for relativistic speeds. These investigations also yield a characterization of null directions that fail to satisfy the generic condition used in singularity theorems. For Ricci flat four-dimensional space-times, tidally nondestructive directions are characterized as repeated principal null directions.