Theoretical Analysis of Particle Trajectory in a Two-Dimensional Plane Under Periodic External Force

This paper mainly talks about how to find the trajectory equation of a particle in a two-dimensional plane under periodic external force and gravity, and how the force amplitude and frequency change the shape and range of the path. Using the knowledge that middle or high school students learn, we analyze this problem and look at the motion of the particle under gravity and a horizontal periodic force, with the assumption that there is no air resistance. Using Newton’s Second Law and basic kinematics, we decompose the initial velocity into horizontal and vertical components. The horizontal acceleration is expressed as: $a_x = \frac{F_0}{m} \sin(\omega t)$, where $F_0$ is the force amplitude, and $\omega$ is the angular frequency, while the vertical acceleration remains constant $a_y = -g$. The velocity functions are obtained as: $v_x = v_{0x} + \frac{F_0}{m \omega} (1 - \cos(\omega t))$, and $v_y = v_{0y} - g t$. From these equations, we can calculate both the velocity and the displacement, and when the time variable is removed, the trajectory equation can be found. The result shows that the path is not the normal parabola anymore, but becomes a curve that waves or oscillates, sometimes even looks like a snake shape, depending on the value of the force. For instance, when the force amplitude $F_0$ is small compared to gravity, movement trajectory is the same as parabola. When $F_0$ is increase or $\omega$ approaches the natural frequency, the movement path shows that obvious oscillation, extending the horizontal range by up to 20-50\% in simulated cases. The analysis is still under classical mechanics. It can be helpful in understanding prediction systems or engineering problems, such as earthquake vibration or oscillating structures. For students, it is also a way to explore physics in non-uniform fields. This research connects basic kinematics and dynamic forces, improves educational tools and preliminary engineering designs. But the model is not perfect, because it does not consider friction and uses simple gravity assumptions. A more realistic situation can be explored by adding damping or creating three-dimensional models, if there are any further studies.