🚀 Numerical Simulations of Motion – Python Project

This project features multiple simulations that model physical systems such as circular motion, spirals, rockets, and harmonic oscillators. The simulations employ numerical integration techniques including Euler's method, Runge-Kutta methods, Leapfrog, and Adams-Bashforth methods. All results are saved to output.txt for further analysis or plotting.

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📁 Simulation Scripts

🔄 Circular and Spiral Motion

• Euler's Circle Simulation.py
  Simulates uniform circular motion using Euler's method.
  ➤ Velocity function: [-sin(t), cos(t)]

• Circular Motion Simulation.py
  Simulates the same circular motion but with the more accurate Runge-Kutta 4th order (RK4) method.

• Euler's Spiral Simulation.py
  Models spiral motion where acceleration depends on both position and velocity, using Euler's method.

• Runge-Kutta Spiral Motion Simulation.py
  More accurate spiral simulation using RK4, incorporating position-dependent forces and damping.

• adams_bashforth_spiral_simulation.py
  Spiral simulation using Adams-Bashforth multistep method (up to 3rd-order). Begins with an Euler step.

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📈 Other Dynamic Systems

• Harmonic Oscillator Leapfrog Simulation.py
  Leapfrog integration of a spring-mass system (F = -kx), simulating a harmonic oscillator over 100 seconds.

• Rocket Altitude Simulation.py
  Models rocket flight over 10 seconds where the engine toggles on/off every second. Tracks altitude over time.

• Numerical Solution of Differential Equation.py
  Solves the logistic growth equation P'(t) = P(t)(1 - P(t)) from t=0 to t=10.
  ➤ Uses Euler's method, saves results every second.

• Numerical Solution of Logistic Equation.py
  Functionally similar to the above, included for practice and accuracy testing.

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🧪 Output Format

All simulations write to a file named output.txt (overwritten per script). Typical columns include:

• t: Time in seconds
• x: X-position or value
• y: Y-position (where applicable)

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▶️ How to Run

1. Make sure Python 3 and necessary packages (numpy, pandas, matplotlib) are installed.
2. Run any simulation like this:

   python "Euler's Circle Simulation.py"

3. Use the optional plotting code (if enabled) to visualize results.

🧠 Learning Outcomes Practice implementing common ODE solvers: Euler, RK4, Leapfrog, Adams-Bashforth.

Simulate physical systems like circular or spiral motion, spring-mass dynamics, and rocket flight.

Analyze and validate numerical results using file output and assertions.