## 2 Materials and Methods
### Image Analysis and Estimations  
A high-speed video system was used to capture the full movement trajectory of the falling rock during the experiments. The system employed a Photron FASTCAM UX50 high-speed camera (manufactured by Photron Inc., in Tokyo, Japan), with a maximum frame rate of 2000 frames per second and a resolution of 1280 \(\times\) 1024 pixels. The minimum time interval between recorded frames was \(5\times 10^{-4}\) s, and the maximum recording capacity of the device can reach up to 160,000 fps.  
The camera was connected to and controlled via Photron FASTCAM Viewer (PFV, version 3.6.9.1) software, which enabled real-time video capture and playback of the entire movement process, as shown in Figure 6a. Prior to formal testing, the camera was calibrated to ensure horizontal alignment with the slope model. A trial recording was performed to verify the camera parameters and confirm the image clarity. During experiments, PFV enabled live monitoring of the falling rock movement, with zoom-in and zoom-out tools enabling on-site inspection of image quality. To ensure uniform lighting and minimize movement blur, a flicker-free LED illumination system was installed over the falling rock movement area. The videos were stored and later analyzed using Photron FASTCAM Analysis (PFA, version 1.4.3.0) software, as shown in Figure 6b. The movement analysis process involved grid-based segmentation and spatial calibration using fixed reference markers along the slope. A static coordinate system was established, and tracking points were manually defined on the falling rock specimens. Both PFV and PFA are commercial software [39]. PFV is used to control the camera and view the captured footage, and PFA enables detailed movement analysis, including calculations of displacement, velocity, and acceleration.  
The centroid of each falling rock block was tracked semi-automatically using PFA software. From this, the displacement, velocity, and acceleration at each time step were calculated. Impact and rebound points were identified frame by frame based on the contact event and change in direction. All movement parameters were exported using the software's built-in data output functions for subsequent analysis.  
\(R_{\rm n}\) was calculated based on the ratio of the normal components of the rebound and impact velocities, derived from the tracked movement data. An uncertainty analysis was also conducted: based on repeated trials, the positional uncertainty was estimated at \(\pm\)2 pixels (approximately \(\pm\)0.5 cm), velocity uncertainty at \(\pm\)0.05 m/s, and the resulting \(R_{\rm n}\) uncertainty at \(\pm\)0.03. These uncertainties were considered in the data interpretation and comparative analysis. \(R_{\rm n}\) was estimated as follows:  
\[R_{\rm n}=\frac{v_{\rm r,n}}{v_{\rm i,n}}, \tag{5}\]  
where \(v_{\rm r,n}\) and \(v_{\rm i,n}\) are the incident and rebound velocities in normal movement before and after the collision, respectively.  
In addition to basic motion parameters, the image analysis was also used to calculate the impact angle and rebound angle of the falling rock. These angles were determined by analyzing the trajectory vectors immediately before and after the moment of collision, based on frame-by-frame displacement data. The angles, along with velocity components, were used to evaluate the energy dissipation characteristics of different slope materials.  
Combined with the velocity vectors, these parameters contributed to a detailed understanding of the rock-slope interaction dynamics. All extracted data, including trajectories, velocity vectors, impact/rebound angles, and contact points, were used as inputs for calculating \(R_{\rm n}\) and establishing the multi-factor interaction regression model.