Velocity profiles from the SPH simulation (points) and 1D semi-analytical wind solution (lines) in a single-star configuration for the low (Ṁ_AGB = 10⁻⁸ M_⊙ yr⁻¹, upper panel) and the high (Ṁ_AGB = 3 × 10⁻⁶ M_⊙ yr⁻¹, bottom panel) mass-loss rate models. The free-wind approximation (red SPH points and 1D solid line profile), the geometrical approximation (light-green and dashed line), the Lucy approximation (orange and dash-dotted line), and the attenuation approximation (blue and dashed-double dotted line) are shown. The geometrical, Lucy, and attenuation approximation overlap in the upper panel. The vertical lines in the lower panel indicate the corresponding dust condensation radius for the geometrical, Lucy, and attenuation approximation. The dotted line shows the escape velocity.