Collagen is a key structural protein in the human body, which undergoes mineralization during the formation of hard tissues. Earlier studies have described the mechanical behavior of bone at different scales highlighting material features across hierarchical structures. Here we present a study that aims to understand the mechanical properties of mineralized collagen fibrils upon tensile/compressive transient loads, investigating how the kinetic energy propagates and it is dissipated at the molecular scale, thus filling a gap of knowledge in this area. These specific features are the mechanisms that Nature has developed to passively dissipate stress and prevent structural failures. In addition to the mechanical properties of the mineralized fibrils, we observe distinct nanomechanical behaviors for the two regions (i.e., overlap and gap) of the Dperiod to highlight the effect of the mineralization. We notice decreasing trends for both wave speeds and Young's moduli over input velocity with a marked strengthening effect in the gap region due to the accumulation of the hydroxyapatite. In contrast, the dissipative behavior is not affected by either loading conditions or the mineral percentage, showing a stronger dampening effect upon faster inputs compatible to the bone behavior at the macroscale. Our results improve the understanding of mineralized collagen composites unveiling the energy dissipative behavior of such materials. This impacts, besides the physiology, the design and characterization of new bioinspired composites for replacement devices (e.g., prostheses for sound transmission or conduction) and for optimized structures able to bear transient loads, e.g., impact, fatigue, in structural applications.