Experimental model for the description of the behaviour of a 9-mm projectile at a target

Analysis of crime scenes involving single-fire-gun projectiles requires the determination of the direction of arrival of a projectile at the target and other factors to reconstruct events. The movement of a projectile can be analyzed by applying Euler's equations to a solid symmetrical rigid body. The present work starts from a Newtonian reformulation of these equations to show that, in the presence of a gravitational field, the system can be expressed with a complex variable nonlinear equation, where the inclusion of small nutation variables allows us to find possible solutions. As a particular case, we analyzed the movement of a 9-mm projectile fired from distances greater than 1 m to demonstrate that the direction of arrival of the projectile at the target cannot be traced by a stick placed in the target hole, as is usually performed in crime investigations. A series of shots were fired from distances varying between 1 m and 7 m. Impact data were recorded on Riemann planes of projection for the description of nutation and precession motions, allowing the observation of the motion dynamics of the projectile. We show that the direction of arrival at the target can be determined approximately from the analysis of the nutation and precession curves through Riemann planes of projection. The results presented in this work will allow more accurate judgements to be made in judicial investigations.