Physics-Sim

This repository collects small, self‑contained physics and numerical simulation projects intended for learning, teaching, and quick experimentation. Each subfolder contains one or more Jupyter notebooks plus a focused README that explains the underlying mathematics and how to run the code.

Projects Overview

• mProject_0/ — Projectile motion

  • Constant‑gravity projectile motion in 2D without air resistance.
  • The theoretical trajectory of a projectile launched with initial speed v0 and angle θ (from horizontal) is:
    • x(t) = v0 cos(θ) · t
    • y(t) = y0 + v0 sin(θ) · t − 0.5 g t²
  • The notebook compares numerical trajectories with these analytic expressions and explores how the range
    • R = (v0² · sin(2θ)) / g depends on launch angle.

• mProject_1/ — Random walk

  • 2D random walk (drunkard’s walk) with independent steps in x and y.
  • Demonstrates that the typical displacement after N steps scales like:
    • <r²>^(1/2) ∝ √N which is a fundamental result in diffusion and stochastic processes.

• Numerical Analysis/ — Numerical methods and dynamical systems

  • Numerical_Integration.ipynb: Approximates integrals of the form
    • I = ∫_a^b f(x) dx using the trapezoidal and Simpson rules, and compares numerical errors as the step size h is refined.
  • Root_Finding.ipynb: Solves nonlinear equations f(x) = 0 using bisection and Newton–Raphson iterations, for example
    • x_(k+1) = x_k − f(x_k) / f'(x_k).
  • Logistic Map.ipynb: Iterates the logistic map
    • x_(n+1) = r x_n (1 − x_n) and builds a bifurcation diagram showing period‑doubling and chaos as the control parameter r varies.

See the README in each subfolder for more detailed mathematical background and notebook‑specific instructions.

Requirements

• Python 3.8+
• numpy
• matplotlib
• A Jupyter environment (jupyterlab or notebook)

You can install dependencies either from requirements.txt (if present) or manually:

pip install numpy matplotlib jupyterlab

How to Run the Notebooks

1. Create and activate a virtual environment (recommended).
2. Install dependencies, e.g.

   pip install -r requirements.txt

   or

   pip install numpy matplotlib jupyterlab

3. Start Jupyter in the repository root:

   jupyter lab

   or

   jupyter notebook

4. Open the notebook of interest (e.g. mProject_0/Projectile.ipynb) and run the cells from top to bottom.

Each project README suggests experiments and parameter changes to help you explore the underlying physics or numerical method.