# Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden
## 2 Construction of Theoretical Calculation Model of Blast Throwing Energy
### Calculation Model of Spherical Cartridge Throwing Energy  
Provided that the explosive and rock properties remain unchanged and only the burden is considered in the blasting program, for spherical cartridge blasting, Equation (3) can be transformed into:  
\[v/\sqrt{\sigma_{c}/\rho_{r}}=f\Big{(}Q/W^{3}\Big{)} \tag{4}\]  
According to the theory related to [PERSON]'s bursting funnel, a power function is applied to the above equation, expressed as:  
\[v=k\Big{(}Q/W^{3}\Big{)}^{\alpha_{1}} \tag{5}\]  
where \(v\) is the maximum initial velocity of the spherical cartridge throw; \(k=\sqrt{\rho_{r}/\sigma_{c}}\), \(k\), \(\alpha_{1}\) are factors related to the rock and explosive properties, and \(k,\alpha_{1}\) can be obtained by a linear fit.  
Using the micrometric division method, taking the cone unit body in the rocks between the cartridge depth and the free surface, the angle of the radian angle is \(\alpha\), as shown in Figure 1. From [PERSON]'s theorem of motion, the instantaneous equation of motion of the unit during acceleration is:  
\[PS=ma \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Dimension** & \(\pi\)**-Term** & **After Sorting** \\ \hline Initial casting velocity \(v\) & \(LT^{-1}\) & \(\pi_{1}\) & \(v/\sqrt{\sigma_{c}/\rho_{r}}\) \\ Minimum burden \(W\) & \(L\) & \(\pi_{2}\) & \\ Explosive density \(\rho_{\rm b}\) & \(L^{-3}\,M\) & \(\pi_{3}\) & \(\rho_{\rm b}/\rho_{r}\) \\ Explosive quantity \(Q\) & \(M\) & \(\pi_{4}\) & \(Q/\rho_{r}W^{3}\) \\ Explosive burst velocity \(D\) & \(LT^{-1}\) & \(\pi_{5}\) & \(D/\sqrt{\sigma_{c}/\rho_{r}}\) \\ Tensile strength \(\sigma_{\rm t}\) & \(L^{-1}\,M\Gamma^{-2}\) & \(\pi_{6}\) & \(\sigma_{v}/\sigma_{c}\) \\ Compressive strength \(\sigma_{\rm c}\) & \(L^{-1}\,M\Gamma^{-2}\) & \(\pi_{7}\) & \\ Rock density \(\rho_{r}\) & \(L^{-3}\,M\) & \(\pi_{8}\) & \\ Modulus of elasticity \(E\) & \(L^{-1}\,M\Gamma^{-2}\) & \(\pi_{9}\) & \(E/\sigma_{\rm c}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters related to initial casting velocity.  
where \(P\) is the pressure to which the unit is subjected; throughout the explosion, the unit is subjected to pressure by the explosion shock wave, stress waves, and explosive gas composition. The pressure its changes with time and can be seen as a function of time, which is denoted by \(P(t)\); \(S\) is the area of the unit under pressure, \(S=\pi(r\alpha/2)^{2}\) (when \(\alpha<5^{\circ}\), \(\sin\alpha=\alpha\), and \(r\alpha=L\)). A function of time is denoted by \(r(t)\). \(m\) is the mass of the unit, \(m=\pi\rho_{r}W_{i}(\alpha W_{i}/2)^{2}/3\). \(a\) is the acceleration, which is denoted by \(dv/dt\). Equation (6) can be expressed as:  
\[\pi P(t)\big{[}\frac{\alpha r(t)}{2}\big{]}^{2}=\frac{1}{3}\pi\rho W_{i}(\frac {\alpha W_{i}}{2})^{2}\frac{dv}{dt} \tag{7}\]  
Simplification yields.  
\[dv=\frac{3P(t)r^{2}(t)}{\rho W_{i}^{3}}dt \tag{8}\]  
Suppose a particular unitary \(i\) starts accelerating at time point \(t_{0}\); at time point t, the initial casting velocity is reached. The initial casting velocity equation of the unitary body is obtained after the integral variation of Equation (8):  
\[v_{i}=\frac{1}{\rho W_{i}^{3}}\int_{t_{0}}^{t}3P(t)r^{2}(t)dt \tag{9}\]  
When the unit is in the direction of the minimum burden, \(Wi=W\) and the calculation equation is:  
\[v=\frac{1}{\rho W^{3}}\int_{t_{0}}^{t}3P(t)r^{2}(t)dt \tag{10}\]  
Combining Equations (8)-(10) yields the initial cast velocity equation for any unitary rock within the blast funnel range.  
\[v_{i}=\frac{W^{3}}{W_{i}^{3}}v=k\frac{W^{3}}{W_{i}^{3}}(\frac{Q}{W^{3}})^{ \alpha_{1}} \tag{11}\]  
Equation (11) is the model for calculating the initial velocity of a spherical cartridge blast funnel throw when only considering the burden. The model uses the prerequisites of assuming a continuous rock medium and unchanged explosive properties.