The computation of rigorous enclosures of the reachable sets of nonlinear control systems is considered, with a focus on applications for which speed is crucial. Low computational costs make interval methods based on differential inequalities an attractive option. Unfortunately, such methods are prone to large overestimation and often produce useless results in practice. From physical considerations, however, it is common that some crude set is known to contain the reachable set. We establish a general bounding result, based on differential inequalities, which enables the effective use of such sets during the bounding procedure. In the case where this set is a convex polyhedron, an efficient implementation using interval computations is developed. Using readily available physical information from practical examples, this method is shown to provide significant advantages over alternative methods in terms of both efficiency and accuracy.