# Use of Digital Technology in Integrated Mathematics Education
## 3 Methods
The calculation of the trajectory is made considering the corresponding variations of the incidence angle and the drift angle, obtained by the separate study of the longitudinal motion and of the lateral motion; most of the time, however, to determine the trajectory, the well-known hypothesis of ballistic motion is admitted, without incidence, thus disregarding the effect of lift, the variation of which is conditioned, through incidence, by the motion around the centre of gravity. To determine the trajectory, a solution is to use Matlab R2024a [30] for modelling the equations of the trajectory [31] (Figure 9). The digital solution consists of solving a system of 12 non-linear ordinary differential equations (ODE), Equations (28)-(39). The nonlinearity is given by the aerodynamic forces' Expressions (16)-(24). The solving process of each individual equation represents the determination of the projectile translations and rotations, meaning the behaviour of the aerospace vehicle on the trajectory. The interface developed in Matlab [32] offers the possibility of choosing different characteristics of the projectiles and the possibility of defining the parameters of the aircraft. These parameters can be defined by their physical characteristics, performance capabilities, and operating limitations (Figure 10). Establishing these parameters is important for ensuring safe and efficient operations, as well as for regulatory compliance and maintenance planning (Figure 11). One of the main challenges is the identification of the initial conditions. Usually, the initial conditions are obtained by the measurements of the ballistic parameters. These measurements are influenced by different disturbances such as environmental factors, accuracy of the measurement devices, and human factors. All those perturbations could be seen as random numbers governed by a normal distribution, in that way being considered within the modelling process. Figure 8:7 (**a**) Matlab computing software for determining the trajectory; (**b**) Matlab software menu for defining the parameters of the firing aircraft and the target on which the firing is performed. Figure 8:8 Matlab computing software for determining the trajectory. Figures 12-14 present the evolution of the main parameters that describe the motion during the trajectory. Figure 8:9 Matlab software menu for defining the parameters of the firing aircraft and the target on which the firing is performed. Figure 7:0 Longitudinal trajectory of an aerospace vehicle. Figure 7:1 The ammunition velocity during the trajectory. In 40 s, the aerospace vehicle travels a horizontal distance of about 9 km. During this period, the speed first increases (corresponding to the active period when the rocket engine is running). This period is followed by the active period, where, up to 25 s, the speed decreases due to the effect of the drag force, after which it shows a slight increase as an effect of the gravitational acceleration. The angle of incidence (as the difference between \(\varphi_{V0}\) and \(\theta_{V0}\)) has an oscillating damped evolution. The damping of the oscillation takes place in about 1 s (Figure 15). The pitch angular speed has an evolution similar to that of the angle of incidence. Scenario 2--Determine the velocity of a projectile using a ballistic pendulum  
A ballistic pendulum is a device used to measure the speed of a projectile by capturing it in a pendulum and measuring the height to which the pendulum swings. It is often used in physics labs to demonstrate the principles of conservation of momentum and energy. The ballistic pendulum works by allowing a projectile to strike and become embedded in a large block of wood or other material attached to a pendulum. The change in momentum of the projectile can then be used to calculate its initial velocity. Mathematical solution for Scenario 2  
The bullet is fired into the pendulum, which then moves upwards a certain distance and at an angle that will be measured. From the height reached by the pendulum, its potential energy can be calculated.