## 6 Determining the Likely Range of Vertical Impact Angles
### PG-7 Trajectory Model  
The trajectory of the PG-7 projectile was modelled using a two-dimensional point mass model (PMM). Only the drag and the gravity were taken into account. As a consequence, it was assumed that at any moment, the velocity vector is aligned with the projectile axis and hence that the yaw angle is always equal to zero. The trajectory model was based on the properties of the RPG-7 weapon system as described in [4] and shown in Figure 25. The specific propulsion of the projectile required the trajectory to be modelled in three distinct phases:  
* Booster phase: the tail part of the projectile (with the fins still folded), containing a booster charge, accelerates the projectile out of the RPG-7 launcher, giving the projectile an initial velocity of \(\pm 117\) m/s at a distance of \(\pm 11\) m. This phase starts with the ignition of the booster charge due to the firing of the weapon and ends with the ignition of the rocket engine. A constant accelerating force for the booster was assumed for this phase of the trajectory model, next to the gravity and drag forces. * Sustained flight phase: the rocket engine is ignited and further accelerates the projectile from its initial velocity of \(\pm 117\) m/s up to its maximum velocity of \(\pm 300\) m/s. The rocket engine is burned out after a flight distance of \(\pm 100\) m (corresponding to the bending point in the curve in Figure 26). A constant accelerating force was again assumed for this phase of the trajectory model next to the gravity and drag forces. Figure 24: Areal densities for the different considered cage armour systems. * Ballistic phase: on this part of its trajectory, the PG-7 projectile is no longer propelled by the rocket engine but continues following a purely ballistic trajectory only subjected to the gravity and drag forces. Figure 25: Available open-source data regarding the trajectory and projectile flight characteristics of the PG-7 projectile reproduced from [4]. The instantaneous horizontal and vertical accelerations \(a_{p,x}\) en \(a_{p,y}\) of the PG-7 projectile were finally calculated using the following basic set of equations of movement:  
\[m_{p}\cdot a_{p,x}=\left(-\frac{1}{2}\cdot\rho_{air}\cdot\frac{\pi}{4}\cdot D_{p }\cdot C_{D0,p}\cdot v_{p}^{2}+F_{booster}+F_{rocket\ engine}\right)\cdot cos\delta \tag{2}\]  
\[m_{p}\cdot a_{p,y}=\left(-\frac{1}{2}\cdot\rho_{air}\cdot\frac{\pi}{4}\cdot D_{ p}\cdot C_{D0,p}\cdot v_{p}^{2}+F_{booster}+F_{rocket\ engine}\right)\cdot sin\delta-m_{p}\cdot g \tag{3}\]  
With:  
* \(a_{p}\) [m/s\({}^{2}\)] = acceleration of the projectile
* \(m_{p}\) [kg] = mass of the projectile
* \(D_{p}\) [m] = maximum diameter of the projectile
* \(v_{p}\) [m/s] = instantaneous velocity of the projectile
* \(C_{D0,p}\) [/] = drag coefficient of the projectile
* \(\delta\) [rad] = angle between the projectile axis/velocity vector and the horizontal plane
* \(F_{hooster}\) [N] = accelerating force acting on the projectile generated by the booster
* \(F_{rocket\ engine}\) [N] = accelerating force acting on the projectile generated by the rocket engine
* \(p_{air}\) [kg/m\({}^{3}\)] = specific density of the air
* \(g\) [m/s\({}^{2}\)] = gravitational constant  
By multiple integration, Equations (2) and (3) finally give the instantaneous velocity and position as a function of time. The values for \(m_{p}\), \(D_{p}\), \(\rho_{\rm air}\) and \(g\) were assumed to be constants. This is strictly speaking not entirely correct for \(m_{p}\), as the mass of the projectile will diminish while the booster and rocket engine are burning. However, the final mass of the projectile is less than 10% lower than the initial mass, which was the rationale for taking \(m_{p}\) constant. Although \(D_{p}\) is a purely geometrical parameter and hence is constant, only taking the maximum diameter of the projectile body into account to calculate the drag force ignores the effect of the fins folding out after the burning of the booster.