Integrated computational problem solving

This lesson combines algorithms, data structures, numerical methods, and visualisation into complete scientific workflows.

Typical computational physics pipeline

Define problem → Model equations → Implement code
    → Validate → Run simulation → Analyse & visualise

Example: projectile with drag

import numpy as np
import matplotlib.pyplot as plt

g, b = 9.8, 0.1          # gravity, drag coefficient
dt, t_max = 0.01, 10.0
N = int(t_max / dt)

x = np.zeros(N); y = np.zeros(N)
vx = np.full(N, 20.0); vy = np.full(N, 30.0)

for i in range(1, N):
    speed = np.sqrt(vx[i-1]**2 + vy[i-1]**2)
    vx[i] = vx[i-1] - b * speed * vx[i-1] * dt
    vy[i] = vy[i-1] - (g + b * speed * vy[i-1]) * dt
    x[i]  = x[i-1] + vx[i] * dt
    y[i]  = y[i-1] + vy[i] * dt
    if y[i] < 0:
        break

plt.plot(x[:i], y[:i])
plt.xlabel("x (m)"); plt.ylabel("y (m)")
plt.title("Projectile with Air Resistance")
plt.grid(True); plt.show()

Workflow checklist

1. Model — Write the governing equations (here: Newton's second law with drag).
2. Discretise — Choose a numerical method and time step.
3. Code — Implement with clear variable names and modular functions.
4. Validate — Check limiting cases (e.g., $b=0$ should give parabolic trajectory).
5. Visualise — Plot results and compare with expectations.

Strategy: Start simple (no drag, no friction) and add complexity incrementally. Each addition should be validated before proceeding to the next.

Common pitfall: Premature optimization wastes more effort than slow code ever did: first make it correct and clear, then measure, and only then speed up the part the measurement actually indicts.