## 2 Mathematical Modelling of Ricochet
### Momentum Analysis of Ricochet Process  
The projectile and the target in the first impacting are analyzed as a whole, assuming that the projectile is rigid and without deformation while hitting the target. The axis of the projectile coincides with its initial flight velocity, and under the initial status and only the deflection force, the attacking angle of the projectile is zero. The coordinate system is set with the \(X\) axis parallel to \(MN\) and the \(Y\) axis perpendicular to \(MN\).  
On the basis of theoretical mechanics, the impacting analysis of the projectile is performed. Figure 2 displays the momentum in the process of first impacting, where the projectile with a mass of \(m\) and length of \(l\) obliquely penetrates the target with a velocity of \(v\) and incidence angle of \(\theta\). After the first impacting, the motion of the projectile is decomposed into a linear motion with a velocity consistent to that of its centre of mass and a rotation motion around its centre of mass. The MM equations of this system are expressed as:  
\[mv^{\prime}_{cx}-mv_{cx}=\sum I_{x} \tag{1}\]  
\[mv^{\prime}_{cy}-mv_{cy}=\sum I_{y} \tag{2}\]  
\[J_{C}\omega_{2}-J_{C}\omega_{1}=\sum M_{c}\left(I^{(c)}\right) \tag{3}\]  
where \(v_{cx}\) and \(v_{cy}\) are, respectively, the \(X\) and \(Y\) axial component of velocity of the centre of mass (VCM) of the projectile before the first impacting; \(v^{\prime}_{cx}\), \(v^{\prime}_{cy}\) are the \(X\) and \(Y\) axial components of the VCM of the projectile after the first impacting, respectively; \(I_{x}\) and \(I_{y}\) are impulses along the \(X\) and \(Y\) axis in the first impacting, respectively; \(I_{C}\) is the inertia moment of projectile about its centre of mass; \(\omega_{1}\) and \(\omega_{2}\) are, respectively, the angular velocity of the projectile before and after the first impacting; and \(\sum M_{c}\left(I^{(c)}\right)\) is the vector sum of the angular impulse about the centre of mass (CM) of the projectile. Compared to its impulse, the gravity and aerodynamic drag force (moment) of the projectile can be neglected. Likewise, the projectile's drag force (moment) resulted from axially rotating, and its effects on the attacking angle and ballistic trajectory are negligible, because the projectile axis, deflecting force and velocity are all in the incident plane.  
Figure 1: Geometric diagram of the projectile obliquely penetrating the target.  
Before the impacting, the angular velocity \(\omega_{1}\) of the projectile is zero, and because the contact surface is smooth, the projectile impulse \(I_{y}\) along the \(Y\) axis in the first impacting is zero as well. The \(X\) axial component of the VCM after the first impacting is equal to that before the first impacting, and its corresponding expression is:  
\[V^{\prime}_{cx}=V_{cx}=V_{c}\cos\theta \tag{4}\]  
When the material restitution coefficient \(e\) is introduced, the above formula can be written as:  
\[ev_{c}\sin\theta=v^{\prime}_{cy}+l^{\prime}\cos\theta\cdot\omega_{2} \tag{5}\]  
Consequently, the analytical expressions of \(\omega_{2}\) and \(\beta\) can be derived as:  
\[\omega_{2}=\begin{array}{c}\frac{ml^{\prime}(e+1)v_{c}\sin 2\theta}{2(J_{c}+ ml^{\prime 2}\cos^{2}\theta)}\end{array} \tag{6}\]  
\[\beta=\arctan\frac{v^{\prime}_{cy}}{v^{\prime}_{cx}} \tag{7}\]