## 1 Introduction
The tests replicated the free-fall movement and impact behavior, allowing for more accurate modeling of energy dissipation during collisions. [PERSON] et al. [21] studied the effects of sandy soil thickness, impact angle, falling rock mass, and falling height on \(R_{\rm n}\). Recently, [PERSON] et al. [22] conducted laboratory falling rock free-fall tests to investigate the effects of impact velocity, incident angle, and slope surface material on \(R_{\rm n}\). These studies have collectively advanced our understanding of how various factors influence \(R_{\rm n}\). In on-site tests, [PERSON] et al. [26] validated the numerical simulation method based on on-site rockfall collision tests. [PERSON] et al. [24] obtained an empirical formula for \(R_{\rm n}\) related to the impact angle by fitting the on-site polyhedral falling rock test data. [PERSON] [25] collected several on-site spherical rockfall collision tests to fit the empirical formula between impact angle and \(R_{\rm n}\). [PERSON] et al. [23] verified the accuracy of the numerical simulation method by using the results of the on-site rockfall collision tests. [PERSON] and [PERSON] [27] calibrated the theoretical model calculation results by taking the on-site rockfall collision tests. [PERSON] et al. [28] determined \(R_{\rm n}\) required in the numerical simulation of the falling rock disaster by using the results of the on-site rockfall collision tests. It is generally recognized that \(R_{\rm n}\) is closely related to falling rock and slope surface characteristics. However, the factors affecting \(R_{\rm n}\) vary significantly across previous research. Most existing models for estimating \(R_{\rm n}\) predominantly focus on individual factors, leaving the underlying mechanisms of the main controlling factors poorly understood. In this study, falling rock refers to the rock that is released from a height and collides with the slope surface. Its behavior, such as velocity and energy dissipation during collision, is influenced by several physical parameters, including density, shape, surface roughness, and material composition. Slope surface characteristics encompass factors such as slope angle, surface roughness, and irregularities in the surface. These characteristics influence how the rockfall interacts with the slope surface during collision, affecting \(R_{\rm n}\) by altering the energy loss during the collision. Most existing models for estimating \(R_{\rm n}\) focus on individual factors in isolation, neglecting the interactions between these factors, which leads to significant deviations and contradictions in the results. This study addresses these gaps by investigating the interactions among the main controlling factors and improving the accuracy of \(R_{\rm n}\) prediction. For example, [PERSON] and [PERSON] [18] employed three empirical formulas to calculate \(R_{\rm n}\) with errors of up to 65%, and [PERSON] et al. [22] estimated \(R_{\rm n}\) empirical formulas to derive error results of up to 80%. Therefore, it remains a critical challenge to develop a practical and accurate estimation method that accounts for multi-factor interactions and can be validated in both laboratory and field conditions. This study aims to address this gap. This research employs laboratory physical model tests to better quantify the effect of the falling rock, slope, and kinematics characteristics on \(R_{\rm n}\). Seven relevant factors are identified from the aforementioned research results. Initial falling rock conditions are generated using a self-designed release apparatus, while kinematic data during rockfall-slope collisions are captured and analyzed with a high-speed recording and analysis system. The main controlling factors were identified, and their interaction mechanisms were elucidated. A new estimation model for \(R_{\rm n}\), incorporating multi-factor interactions, is proposed and validated through field tests.