# Selected Problems in a Two-Vehicle Impact Collision Modeling
## 3 Identification of the Parameters Crucial for the Analytical Calculations
Taking into consideration the given angles a simple formula to determine the \(\alpha\) can be used:  
\[\alpha_{1}=(180^{\circ}-\gamma)+\beta \tag{1}\]  
Both \(\beta\) and \(\gamma\) are known as they are the angles of the initial positions of vehicles no. 1 and 2 determined by V-SIM. Therefore, for the discussed example, assuming that the positive angles of the vehicles are measured clockwise, they are:  
\[\beta=10.6^{\circ},\]  
\[\gamma=172.4^{\circ}.\]  
and the \(\alpha\) will then be:  
\[\alpha_{1}=(180^{\circ}-172.4^{\circ})+10.6^{\circ}=18.2^{\circ}.\]  
This angle will be very important in further considerations. Next, the necessary parameters have been attached to the initial locations of the vehicles. Taking into account the collision with the tangential impulses of the impact force in Figure 5, the physical model of the crash is presented for the case of the locked front wheels during the collision with the positive directions marked. The impulses of the collision forces for each vehicle (\(S_{n1}\), \(S_{t1}\), \(S_{n2}\), and \(S_{t2}\), respectively) are presented and are marked as scalars because they are the components of the resultant impulses acting on each vehicle. Figure 5: The mutual location of both vehicles at the beginning of the collision with the positive directions marked. Additionally, the necessary dimensions, the masses, and the moments of inertia of both vehicles are marked in Figure 5.  
In Figure 6, the angle \(\alpha_{1}\) used in the distribution of the speed into the tangential and the normal velocity is presented as well as the angular velocities, respectively, for vehicles no. 1 and 2. These components have been used to determine the kinematic state of both vehicles after the collision. If a forensic expert was to verify the simulation results, a general model of collision could be used to conduct some simple analytical calculations, with the use of the following formulas:  
* in the direction normal to the impact plane (along the \(O_{n}\) line): \[m_{1}\big{(}v^{\prime}_{1n}-v_{1n}\big{)}=-S_{n1},\ m_{2}\big{(}v_{2n}-v^{ \prime}_{2n}\big{)}=S_{n2}\] (2)
* in the direction tangential to the impact plane (along the \(O_{t}\) line): \[m_{1}\big{(}v^{\prime}_{1t}-v_{1t}\big{)}=-S_{t1},\ m_{2}\big{(}v_{2t}-v^{ \prime}_{2t}\big{)}=S_{t2}\] (3)
* in the direction of rotation on a plane of the road, in accordance with the selected angular velocities: Figure 4:0 (4) As a result, the kinematic state of motion at the end of the collision is determined: Figure 4:1 (5) Figure 4:2 (6) Figure 4:3 (7) where:
* \(S_{n1}\), \(S_{t1}\), \(S_{n2}\), and \(S_{t2}\)--components of the force impulses of vehicles no. 1 and 2, respectively. * \(v_{1n}\) and \(v_{2n}\)--the normal velocity of vehicles no. 1 and 2, respectively, before the collision. * \(v_{1t}\) and \(v_{2t}\)--the tangential velocity of vehicles no. 1 and 2, respectively, before the collision. Figure 4:4 The mutual location of both vehicles at the beginning of the collision with the angles and the components of the velocities marked. \(\omega_{1}\) and \(\omega_{2}\)--the angular velocity of vehicles no. 1 and 2, respectively, before the collision. * \(n_{1}\), \(t_{1}\), \(n_{2}\), and \(t_{2}\)--coordinates of the center of mass of each vehicle in relation to the geometric center of the collision (point O). \(\alpha_{1}\) is the angle between the speed of vehicle no. 1 and the \(O_{n}\) axis (Figure 6). The markings adopted in this chapter refer to the symbols used in the equations of the analyzed collision. The index '1' refers to the parameters of vehicle no. 1 (Figures 5 and 6) and '2' specifies the parameters of vehicle no. 2 (Figures 5 and 6 as well). The indexes '\(n\)' and '\(t\)' denote the normal and the tangential directions, respectively.