Trajectory simulations by the numerical solution of the point-mass equations of motion for 7.62 mm/.308” rifle bullets
Authors: Gangopadhyay Soham , Rohatgi Richa 
Published: August 2022

BACKGROUND: The understanding of the dynamics of the trajectory is important in ballistics to estimate the values of various flight variables accurately. The paper deals with the study of the fundamental principles of external ballistics, which allows to delve into the trajectory characteristics of the free flight trajectory of seven. 308 caliber bullets by numerically solving the point-mass equations of motion. Numerical solutions were performed by writing scripts in the Python programming language and using the Matplotlib library to plot simulated trajectories.
AIM: the three aims of the study were to observe the variation of CD with Mach number (Ma) of flight and calculate an average CD for each bullet under consideration. Further, solving the 3-DoF (Degrees-of-Freedom) Point-Mass trajectory equations of motion for the given bullets (along side observing the effects of range winds on the trajectory behaviour as a variable). And finally, solving the flat-fire approximation with analysis of the effects of a crosswind.
MATERIALS AND METHODS: Simulations of free-flight trajectories of seven different 7.62 mm/.308 rifle bullets (designated B0B6) have been carried out by the numerical solution of the equations of motion. The average drag force coefficients (CD) for B0B6 have been calculated by scaling the variation of CD with the Mach number of flight with reference to the G7 standard projectile. The Point-Mass trajectory model and its Flat-Fire approximation have been studied with and without the effect of range winds. The solutions of the systems of equations have been carried out by writing scripts in the Python programming language.
RESULTS: It is observed that an increase in the bullet weight and consequently the sectional density lowers the CD. As expected, it is seen that the bullet with the highest drag (B0) has the shortest range and lowest apogee, while lower drag bullets fly further and higher. The crossover of trajectories is observed at ~30 angle of gun elevation, which implies that the maximum range is not achieved when fired at 45, as is the case with vacuum trajectories. Flat-fire approximation of the point-mass model was also solved to observe trajectories and crosswind deflections of the bullets when fired at 5 angles of elevation.
CONCLUSION: This project presents the numerical solution of equations of motion of the Point-Mass model for a bullet fired from a gun to computationally simulate its trajectory. A group of seven 7.62 mm/.308 rifle bullets were chosen as samples to simulate free-flight trajectories. The programming language Python is well-equipped to carry out numerical solutions of systems of differential equations owing to its library of in-built functions which assists in writing an efficient script and reduces computational load. This method of solution can be applied with suitable modifications in the field of forensic ballistics for the reconstruction of bullet trajectories and to form a conclusion based on the available evidence from a crime scene.