Penetration of fast projectiles into resistant media: from macroscopic to subatomic projectiles

The penetration of a fast projectile into a resistant medium is a complex process that is suitable for simple modeling, in which basic physical principles can be profitably employed. This study connects two different domains: the fast motion of macroscopic bodies in resistant media and the interaction of charged subatomic particles with matter at high energies, which furnish the two limit cases of the problem of penetrating projectiles of different sizes. These limit cases actually have overlapping applications; for example, in space physics and technology. The intermediate or mesoscopic domain finds application in atom cluster implantation technology. Here it is shown that the penetration of fast nano-projectiles is ruled by a slightly modified Newton's inertial quadratic force, namely, $F \sim v^{2-\beta}$, where $\beta$ vanishes as the inverse of projectile diameter. Factors essential to penetration depth are ratio of projectile to medium density and projectile shape.