## 2 Movement Characteristics of Ore-Rock Block in the Ore Pass
### Movement Analysis of Ore Rock in the Ore Pass
#### 2.2.2 Ore-Rock Block Moving in the Ore Pass  
Due to the effect of the initial direction of the ore rock movement, the ore rock will collide with the ore pass wall [25]. When the velocity of the ore-rock block leaving the mine cart is ignored, the velocity of the ore-rock block entering the main ore pass, that is, the instantaneous velocity of the ore-rock block leaving point \(O\), can be calculated through Equation (1). \[v_{0}=\sqrt{2 gh\left(1-\mu\frac{1}{\tan\alpha}\right)} \tag{1}\]  
where \(\alpha\) is the inclination of the chute floor relative to the horizontal plane, g is the gravitational acceleration, 9.8 m/s\({}^{2}\), \(h\) is the height of the inclined chute, m, and \(\mu\) is the friction coefficient between the ore-rock block and the inclined chute floor. Based on the kinematic principle and the motion model given in Figure 2, assuming \(v_{0}\) is the instantaneous velocity of an ore-rock block entering the main ore pass from the unloading chute, \(\alpha\) and \(\beta\) are the angles of the displacement vector of the ore-rock block relative to the \(x\)-\(z\) and the \(x\)-\(y\) plane, respectively. Then, according to the kinematic principle, the velocity and the spatial position of the ore-rock block at time \(t\) before its initial collision with the ore pass wall can be obtained through Equations (2) and (3):  
\[\left\{ \begin{array}{l}v_{x}=v_{0}\ \cos\alpha\cdot\cos\beta\\ v_{y}=v_{0}\ \cos\alpha\cdot\sin\beta\\ v_{z}=v_{0}\ \sin\alpha+\mathrm{g}t\end{array} \right. \tag{2}\]  
\[\left\{ \begin{array}{l}x_{t}=v_{0}t\cdot\cos\alpha\cdot\cos\beta\\ y_{t}=v_{0}t\cdot\cos\alpha\cdot\sin\beta\\ z_{t}=v_{0}t\cdot\sin\alpha+\frac{1}{2}\mathrm{g}t^{2}\end{array} \right. \tag{3}\]  
where \(\alpha\) is the depression angle with a maximum value range of (\(0^{\circ}\), \(90^{\circ}\)), \(\beta\) is the initial direction angle ranging from \(-90^{\circ}\) to \(90^{\circ}\), \(x_{t}\), \(y_{t}\), and \(z_{t}\) denote the three-dimensional position coordinates of the block before the collision, and \(t\) is the duration of the movement preceding the impact. The remaining symbols are the same as above. When the ore-rock block moves through the ore pass, there is a relationship between the diameter \(D\) of the ore pass and the displacement \(x\) of the ore-rock block in the horizontal direction, i.e., \(D\geq x\). if \(x=D\), the moving ore-rock block will come into contact with the ore pass wall, and, if \(x>D\) (it is impossible to have \(x>D\) due to the constraints imposed by  
Figure 2: Spatial motion model of the ore-rock block entering the ore pass: (**a**) longitudinal projection along the length of the unloading chute; (**b**) a top view of figure (**a**). the ore pass wall), the ore-rock block will collide with the ore pass wall. Therefore, \(x\geq D\) constitutes the basic condition for the ore-rock block impact with the ore pass wall. When both \(\alpha\) and \(\beta\) are known values at initiation, \(D\) is the diameter of the main ore pass and the time \(t\) is:  
\[t=\frac{D\cos\beta}{v_{0}\cos\alpha} \tag{4}\]  
According to the above, we can obtain the coordinates of the collision point between the ore-rock block and the ore pass wall, as shown in Equation (5):  
\[\left\{ \begin{array}{l}x=D\cos^{2}\beta\\ y=\frac{1}{2}D\sin 2\beta\\ z=D\cos\beta\bigg{\{}\tan\alpha+\bigg{[}\frac{D\cos\beta}{4\text{h}\,\cos^{2} \alpha\big{(}1-\frac{D}{\text{H}\alpha\big{)}}}\bigg{]}\bigg{\}}\end{array} \right. \tag{5}\]  
Consequently, Equation (5) provides the precise coordinates necessary to locate where the collision occurs. At this particular time, when the impact takes place between the ore-rock block and the ore pass wall, the depression angle \(\alpha_{1}\) and the direction angle \(\beta_{1}\) of the ore-rock block coming into contact with the ore pass wall can be obtained as follows:  
\[\left\{