In the present work we formulate the relation between ballistic penetration depth versus impact speeds of rigid projectiles, having various head-shapes. The targets are semi-infinite rigid-perfectly-plastic metals. Each penetrator head-shape that was simulated in this work has a different geometrical convexity (named as `Rankine shapes`). The plastic flow that ejected out from the penetrated target is modelled via the potential flow theory with various length-to-width ratios. The upper-bound approach is used here to interrelate analytically the main physical parameters of the penetration, like the interfacial projectile/target friction, target`s yield strength, target`s density, plastic zone extension around the penetrator, etc. In the first part we compared the head-shapes in terms of mean indentation pressure during the non-steady indentation (similar to the situation in Brinnel hardness test), and later we enlarged the solution to the fully dynamic penetration path until the projectile is arrested (starting from impact velocities of at most 1400 m/sec).The test case of a head-shape with equal length and base radius (namely, spherical head) is shown to be favourably compared to previous analytical and experimental works taken from the open literature. The major result is that we have found the direct influence of head-shapes of projectiles on their whole penetration-depth path. It allows tracking the expenditure of energy of each penetration process for each head-shape. As a practical consequence of our solution we can offer an optimal head-shape (length to width ratio) of a Rankine type penetrator for maximum penetration efficiency.
