## 3 Fragment Penetration Calculation Based on Forest Environment
The differential equation of fragment motion is:  
Figure 5:0  
where, \(C_{D}\) is the aerodynamic resistance coefficient; \(\rho_{a}\) is the air density; \(S\) is the windward display area of fragments; \(m\)\(v\) and \(t\) are fragment mass, fragment velocity and time respectively. The above formula is transformed into the differential form of velocity \(v\) and distance \(x\):  
Figure 5:1  
To integrate the above formula, there are:  
Figure 5:2  
So we get:  
Figure 5:3  
It can be seen from the above formula that the relationship between \(v_{x}\) and \(v_{0}\) should be linear when the thickness \(x\) of the target plate and the mass \(m\) of the projectile are constant. However, it can be seen from the simulation results that the growth of \(v_{x}\) is nonlinear with the increase in \(v_{0}\), as shown in Figure 5:4 Therefore, the velocity attenuation formula in air cannot be applied to the case of penetrating wood. According to the hypothesis of [PERSON] [14], the equation of penetration depth of a stable moving projectile varying with velocity can be obtained by integration:  
Figure 5:5  
The result is:  
Figure 5:6  
where \(s\) represents the distance along the straight trajectory. The speed after passing through the obstacle with thickness \(h\) is:  
Figure 5:7  
Figure 5:8 Relationship curve between \(v_{0}\) and \(v_{x}\) obtained by fitting. (**a**) is 6 mm fragment; (**b**) is 11 mm fragment. where \(v\) is the residual velocity; \(v_{0}\) is the initial speed; \(c_{1}\) and \(c_{3}\) are constants; \(A\) is the cross-sectional area of the projectile; \(h\) is the thickness of target plate; and \(m\) is the fragment mass. Since the prefabricated fragments are spherical, the above formula can be simplified as:  
Figure 5:9  
\(r\) is the radius of the fragment. According to the Poncelet formula, the difference between the two parameters \(c_{1}\) and \(c_{3}\) obtained from the initial and residual velocity data of 11 mm and 6 mm projectiles is too large, and the difference between \(c_{1}\) and \(c_{3}\) is more obvious. This may be caused by experimental errors, and there is a problem of an insufficient number of samples. Therefore, the simulation values of the two fragments are fitted to obtain the following relationship between \(v^{2}\) and \({v_{0}}^{2}\), as shown in Figure 6.0  
From the results of linear fitting, it can be seen that there is an obvious linear relationship between the two, so the \(c_{3}\) value of the two fragments can be calculated. Through comparison, it is found that the difference ratio between the two is not large, so the \(c_{3}\) parameter value of the dried pine wood is 325; However, the \(c_{1}\) value obtained by the Poncelet formula is too different and cannot be regarded as consistent. This should be the result of fragment parameters. Therefore, the fragment radius \(r\) in the latter half of the formula is modified and \(r^{2.2}\) is added to the denominator term and substituted into the latter half of the formula. At this time, the \(c_{1}\) value obtained is \(-2.6\). Thus, a modified formula about the residual velocity of wood based on the Poncelet formula is obtained:  
Figure 6.1  
After the residual velocity calculation formula is obtained, the influence of forest on fragment velocity can be obtained according to the scattering radius, stand density, and tree diameter.