Notes. m is the mass of the projectile, z is the depth of the projectile, g is gravity, and v_c is the impact velocity. f(z), b(z) and h(z) are coefficients related to the quasi-static friction, viscous, and inertial drag forces, respectively. f_o, b, h, κ, α, and η are constants. D is the diameter of the projectile, A is the cross-sectional area of the projectile, ρ_p is the density of the projectile, ρ_g is the density of the surface material, and λ is the characteristic length of the container. z_o is the penetration depth of the projectile when v_c = 0 and z₁ is the depth at which f(z) becomes constant. ⁽¹⁾A tested collision velocity range of 0 indicates that the authors placed the projectile directly on or in the material in order to isolate and study the quasi-static friction force. ⁽²⁾Reference provides an expression for penetration depth.