STUDY OF THE TRAJECTORY OF THE PROJECTILE FIRED AT A CRITICAL ANGLE TO THE HORIZON - Zdeshchyts, Valerii; Zdeshchyts, Anastasiia; Slyusarenko, Mykola (Physical and Mathematical Education, 2025)
The work is devoted to studying the laws of motion of a body released with an initial velocity at an angle to the horizon. This class of tasks includes, for example, shooting a projectile from a cannon. Formulation of the problem. If the projectile is fired vertically, it first moves away and then approaches the place of the shot. If you release the projectile horizontally, it will constantly move away. Therefore, a critical angle must separate these two modes of projectile flight. Materials and methods. The goal: theoretical determination of the dependence of the radial distance of the projectile on time, x-coordinates, and experimental proof of the validity of the discovered laws - was solved using an installation in the form of a spring and magnetic cannon with a ball projectile. The angle of the shot was adjusted within 0-900 and registered with an inclinometer as part of the Phyphox mobile application with an accuracy of tenths of a degree. With the help of an inclinometer, a ruler was also displayed horizontally. One smartphone was used in slow-motion mode to record the ball’s trajectory, and the other was used as a millisecond timer. Results. The presence of three types of flight trajectories of a projectile fired at an angle α to the horizon is theoretically substantiated. The phenomenon of finding the projectile three times at different times at the same distance from the point of the shot was revealed. It is shown that for launch angles exceeding the theoretically and experimentally determined critical angle of 70.53°, the projectile first moves away from the origin then moves back to it, and then moves away again. Conclusion. An interesting phenomenon related to projectile motion is considered. It is shown that for launch angles more significant than the theoretically and experimentally determined critical angle of 70.53°, the projectile first moves away from the origin then moves back toward it, and then moves away again, in defiance of common sense, which tells us that a projectile should only move away from the point of the shot. The presence of three types of flight trajectories of a projectile fired at an angle α to the horizon and the discovery of the phenomenon of finding the projectile three times at different times at the same distance from the shot point is an unexpected result for students studying mechanics.