# Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden
## 4 Discuss Test Results
### Casting Velocity Analysis  
#### 4.2.1 Casting Velocity Distribution Mechanism  
We utilized the previously built calculation method to obtain the initial casting velocity and throw kinetic energy. Then, we divided the broken rock into 20 micro-elements along the axial direction of the blast hole. We took the midpoint of each micro-element as the characteristic point. Finally, a relative coordinate system was constructed with the background grid as the size reference, as shown in Figure 9. At the same time, we used the tracking function of hotspot/cc processing software to track and capture the characteristic points of different models. Finally, on the basis of conducting five burden model experiments, the position and shape of the bulge movement of the free-face rock for every millisecond in the 0 ms to 19 ms period were obtained, as shown in Figure 10.  
Figure 8: Space position of rock at different time. Figure 9: Relative coordinate system. Due to the differences in the initial bulging time in different burden situations, the first-millisecond morphology of Models 2 and 3 and the first and second-milliseconds morphology of Models 4 and 5 are not presented. From the high-speed photography process and the analysis results in Figures 8-10, it can be seen that the start of cracking and bulging movements in the free surface rock is about 2 to 3 ms after explosive detonation. With time, the displacement of the rock movement in the middle of the step is more extensive than in the upper and lower ends. The overall movement morphology is convex in the middle and low at both ends. This indicates that the rock in the middle part moves faster than at the two ends. To accurately obtain the rock casting velocity, the spatial position relationship of the different models in the model test at different time points thrown rocks is calculated using the following equation:  
\[v_{ji}=\sqrt{\left(x_{t}-x_{t-1}\right)^{2}+\left(y_{t}-y_{t-1}\right)^{2}}/t \tag{23}\]  
where \(v_{ji}\) is the velocity of the mass point \(j\) at the time point \(t\), m/s; (\(x_{t-1}y_{t-1}\)) and (\(x_{t}\),\(y_{t}\)) are the coordinate positions at time t \(-\) 1 and time t, respectively. Aiming at the fluctuation of broken rock in the process of throwing, the maximum velocity is chosen as the casting velocity in the calculation model. The specific calculation equation is:  
\[v_{j}=\text{max}\bigg{(}\sqrt{\left(x_{t}-x_{t-1}\right)^{2}+\left(y_{t}-y_{t- 1}\right)^{2}}/t\bigg{)} \tag{24}\]  
Figures 8 and 10 show the real spatial motion law of blasting broken rock obtained from the high-speed photography. According to Equation (23), the space-time relations of the five burden-free surface broken rock masses at different moments are calculated. When the broken rock on the free surface reaches the initial casting velocity, the distribution law of the casting velocity of broken rock masses at different step heights is thus obtained. The experimental values of the mass velocity distribution are shown in Figure 11.  
Where the maximum casting velocity occurs in five burdens, using the Equation (24), the relationship between the velocity of the particle and the time is obtained, as shown in Figure 12. Calculation results show the following. Figure 10: _Cont._  
Figure 10: Spatial distribution of particle positions. 1. In addition, a comparative analysis of the bulge morphology of each model reveals that the timing of the initial bulge movement varies among models; the bulging time of Model 1 is about 1 ms, while Models 2 and 3 are about 2 ms, and Models 4 and 5 are about 3 ms. This indicates that as the minimum burden increases, the penetration time of the blast crack will be delayed and the free surface bulge start time is lagging. During the same movement time, the displacement of the broken rock on the free surface decreases with the increase of the burden. This indicates that the casting velocity shows a decreasing trend with the increasing burden;
2.