## 2 Analytical Methods  
### Classification of Methods  
The authors have proposed several analytical models over the years to describe the process of penetration mechanism of impacting objects on target media with suitability to material, impact velocity, and geometric properties of both the impacting objects and target media [27]. Analytical models collect insight into the phenomenon and penetration mechanics and provide tools for design studies, optimization, and risk analysis in the impact of object-target media interactions. The execution of analytical models is faster than other approaches, namely, numerical or experimental methods. [PERSON] et al. [12] presents the research summary of the analytical method applied to predict the response of granular media to projectile impact at the meso-scale and macro-scale. They outlined the research areas in the granular target media such as effect of saturation in soil media, role of particle crushing in impact energy dissipation, and rate of strain rate, that need further attention. The various proposed physics-based analytical approaches to analyze the response of targets to impacting objects are internal friction theory and cavity expansion theory (CET)-based models [28], model of orthogonal layers or discs [29], Poncelet Model [30], hydrodynamic and modified hydrodynamic-theory-based models [31], conservation of energy and momentum models [32; 33], Tate-Alekseevskii model [34; 35], dynamic plasticity and Ravid-Bodner models [36], and the Walker-Anderson model [37]. Many of the above-mentioned analytical models are applied to predict the impact penetration of ductile and homogenous targets with specific penetration problem and may be extended to study the projectile penetration process in soil media and non-homogenous media by developing them into a versatile model of projectile penetration of soil targets. The summary of underlying mathematical equations presenting the theoretical background of the analytical method is given in Appendix A. The analytical methods developed so far are basically based on \"[PERSON]'s second law of motion or conservation of momentum\" to study projectile-penetration and have validity over a wide range of projectile velocity. The analytical method developed on the law of conservation of energy has limited applicability because energy dissipation changes with impact velocity [38]. The cavity expansion theory-based formulation is used for studying low, and low to high transition velocity impact problems (3 km/s \(>\) v \(>\) 0.3 km/s) [Yu 2013], the Tate-Alekseevskii formulation is used for studying medium velocity impact problems (v\(\sim\)3 km/s) [35], and the fluid-dynamics formulation is used for studying high velocity impact problems (v \(>\) 3 km/s) [31]. The categorization of impact penetration processes based on the physical mechanisms that describe the nature of the penetration process are elastic, plastic, and thermal behaviors, as well as a reduced compressibility [14]. The resistive force or resistance function of a rigid projectile to penetration in a projectile-penetration process has three components, namely, dissipative, static, and dynamic components. The first component has a negligible effect on resistive force for granular media [32]. The static aspect of resistance force depends on penetration depth. The dynamic component depends on the projectile's velocity in the direction of penetration, cross-sectional area, and soil density. However, the first principle theory for explaining the role of granular resistance, i.e., predicting granular media's rheological response, is still a challenge, mainly on the edge of solid-liquid-like transition [39] at high-velocity impact. To calculate resistive force, stress at the impactor nose must be integrated over the applicable area incorporating yield criterion in terms of yield stress, projectile geometry, and the velocity in direction of impact. [PERSON] et al.