Unified force law for granular impact cratering

Experiments on the low-speed impact of solid objects into granular media have been used both to mimic geophysical events(1-5) and to probe the unusual nature of the granular state of matter(6-10). Observations have been interpreted in terms of conflicting stopping forces: product of powers of projectile depth and speed(6); linear in speed(7); constant, proportional to the initial impact speed(8); and proportional to depth(9,10). This is reminiscent of high-speed ballistics impact in the nineteenth and twentieth centuries, when a plethora of empirical rules were proposed(11,12). To make progress, we developed a means to measure projectile dynamics with 100 nm and 20 mu s precision. For a 1-inch-diameter steel sphere dropped from a wide range of heights into non-cohesive glass beads, we reproduce previous observations(6-10) either as reasonable approximations or as limiting behaviours. Furthermore, we demonstrate that the interaction between the projectile and the medium can be decomposed into the sum of velocity-dependent inertial drag plus depth-dependent friction. Thus, we achieve a unified description of low-speed impact phenomena and show that the complex response of granular materials to impact, although fundamentally different from that of liquids and solids, can be simply understood.