Numerical Simulation of Projectile Motion Using Euler’s Method

Overview
This project explores the mathematical modeling and numerical simulation of projectile motion. Analytical formulas for time of flight, range, and maximum height are implemented alongside a numerical Euler method. The simulation updates position and velocity iteratively over discrete time steps and compares results with the analytical solution. Trajectory, vertical displacement, velocity, and error analysis are visualized to demonstrate the accuracy and limitations of the Euler method. The project highlights the application of numerical methods for physics and engineering education.

Suitable For

• BS Physics / Applied Physics – mechanics, kinematics, and computational modeling

• BS Mathematics – applications of calculus, differential equations, and numerical methods

• BS Computer Science / Data Science – numerical simulation, Python programming, and data visualization

• BS Engineering – modeling, analysis, and interpretation of motion data

Technologies Used

• Programming Language: Python

• Libraries & Tools: NumPy, Pandas, Matplotlib

• Numerical Method: Euler’s Method

• Analytical Validation: Kinematic equations for projectile motion

• Evaluation: Trajectory comparison, vertical displacement, velocity, and error plots

Techniques

• Analytical calculation of time of flight, range, and maximum height

• Numerical simulation using Euler integration

• Stepwise iterative updates of position and velocity

• Error analysis between numerical and analytical solutions

• Visualization of motion through plots and tables

Visualization

• Euler simulation tables for projectile motion

• Trajectory y(x) comparison (Euler vs. analytical)

• Vertical displacement y(t) and vertical velocity vy(t)

• Error plots highlighting deviations of Euler method

Features

• Mathematical formulation of projectile motion as a differential equation

• Step-by-step Euler method implementation in Python

• Comparison of numerical results with exact analytical solutions

• Visualization and interpretation of simulation accuracy

• Error quantification to highlight method limitations

Deliverables

• Python source code implementing Euler simulation

• LaTeX project documentation with theory, methodology, and results

• Tables and plots showing trajectory, vertical displacement, velocity, and error

• Visual comparison of numerical and analytical solutions

Support

• Guidance on setting initial velocity, launch angle, and time steps

• Help interpreting discrepancies between Euler and analytical results

• Assistance with visualization and tabulation of simulation data

Benefits for Students

• Practical experience applying numerical methods to classical physics problems

• Understanding the link between analytical solutions and iterative numerical simulation

• Skills in Python programming, data analysis, and visualization

• Insight into accuracy, limitations, and error propagation in numerical methods