Interactive Physics Problem Sets & Computational Labs

Lab 01 • mechanics

Projectile Motion with Quadratic Air Drag

Model aerodynamic deceleration using coupled ODEs

Model the trajectory of a spherical projectile under the influence of gravity and non-linear aerodynamic air resistance.

Governing Equations & Formulation

\frac{dx}{dt} = v_x, \quad \frac{dv_x}{dt} = -\frac{c}{m} v_x \sqrt{v_x^2 + v_y^2}$$\n$$\frac{dy}{dt} = v_y, \quad \frac{dv_y}{dt} = -g -\frac{c}{m} v_y \sqrt{v_x^2 + v_y^2}

Parameters: Mass $m = 0.145\text{ kg}$, initial speed $v_0 = 45\text{ m/s}$, launch angle $\theta = 45^\circ$, drag constant $c = 0.0015\text{ kg/m}$, $g = 9.81\text{ m/s}^2$.

Starter Solution Ready

import numpy as np
import matplotlib.pyplot as plt

m = 0.145
g = 9.81
c = 0.0015
v0 = 45.0
theta = np.radians(45)

dt = 0.001
x, y = [0.0], [0.0]
vx, vy = [v0 * np.cos(theta)], [v0 * np.sin(theta)]

while y[-1] >= 0:
    v = np.sqrt(vx[-1]**2 + vy[-1]**2)
    ax = -(c/m) * v * vx[-1]
    ay = -g - (c/m) * v * vy[-1]
    vx.append(vx[-1] + ax * dt)
    vy.append(vy[-1] + ay * dt)
    x.append(x[-1] + vx[-1] * dt)
    y.append(y[-1] + vy[-1] * dt)

R_vac = (v0**2 * np.sin(2*theta)) / g
print(f"Vacuum Range: {R_vac:.2f} m")
print(f"Realistic Range (Drag): {x[-1]:.2f} m")

plt.figure(figsize=(7, 3.8))
plt.plot(x, y, label='With Quadratic Drag', color='#06b6d4', lw=2)
plt.title('Projectile Motion with Air Drag')
plt.xlabel('Distance (m)')
plt.ylabel('Height (m)')
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()

Lab 02 • electrodynamics

Electric Field & Equipotential Contours of an Electric Quadrupole

2D electrostatic field line integration and potential mesh computation

Calculate and visualize the 2D electrostatic potential field $\Phi(x, y)$ and field vector gradient $\vec{E} = -\nabla \Phi$ for an arrangement of four point charges in a planar quadrupole configuration.

Governing Equations & Formulation

\Phi(\vec{r}) = \frac{1}{4\pi \varepsilon_0} \sum_{i=1}^{N} \frac{q_i}{|\vec{r} - \vec{r}_i|}, \quad \vec{E} = -\nabla \Phi = -\left( \frac{\partial \Phi}{\partial x} \hat{i} + \frac{\partial \Phi}{\partial y} \hat{j} \right)

Parameters: Charges $q = \pm 1\text{ nC}$ placed at $(1, 1), (-1, 1), (-1, -1), (1, -1)$. Mesh resolution $200 \times 200$.

Starter Solution Ready

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-3, 3, 200)
y = np.linspace(-3, 3, 200)
X, Y = np.meshgrid(x, y)

charges = [
    {'q':  1.0, 'pos': ( 1.0,  1.0)},
    {'q': -1.0, 'pos': (-1.0,  1.0)},
    {'q':  1.0, 'pos': (-1.0, -1.0)},
    {'q': -1.0, 'pos': ( 1.0, -1.0)},
]

V = np.zeros_like(X)
for c in charges:
    r = np.sqrt((X - c['pos'][0])**2 + (Y - c['pos'][1])**2)
    r = np.maximum(r, 0.15)
    V += c['q'] / r

Ey, Ex = np.gradient(-V, y, x)

plt.figure(figsize=(7, 4.5))
plt.contourf(X, Y, V, levels=30, cmap='RdBu_r', alpha=0.7)
plt.colorbar(label='Potential V (arb. units)')
plt.streamplot(X, Y, Ex, Ey, color='black', density=0.9, linewidth=0.7)
plt.title('Quadrupole Potential & Electric Field Lines')
plt.xlabel('x')
plt.ylabel('y')
plt.show()

Lab 03 • quantum

Finite Square Well: Bound State Energies via Transcendental Bisection

Numerical root finding for symmetric and antisymmetric quantum wavefunctions

Determine the bound-state energy eigenvalues for a particle of mass $m$ trapped inside a finite 1D square potential well of width $2a$ and depth $V_0$.

Governing Equations & Formulation

\xi \tan \xi = \sqrt{\xi_0^2 - \xi^2} \quad \text{(Even Parity)}, \quad -\xi \cot \xi = \sqrt{\xi_0^2 - \xi^2} \quad \text{(Odd Parity)}$$\n$$\text{where } \xi = \frac{a}{\hbar}\sqrt{2m(E + V_0)}, \quad \xi_0 = \frac{a}{\hbar}\sqrt{2m V_0}

Parameters: Well half-width $a = 1.0\text{ nm}$, Well depth $V_0 = 10.0\text{ eV}$, electron mass $m_e$.

Starter Solution Ready

import numpy as np
from scipy.optimize import brentq
import matplotlib.pyplot as plt

V0 = 10.0 # eV
a = 1.0e-9 # m
hbar = 1.0545718e-34
m = 9.10938356e-31
e = 1.60217663e-19

xi0 = (a / hbar) * np.sqrt(2 * m * V0 * e)
print(f"Dimensionless parameter xi_0 = {xi0:.3f}")

def even_eq(xi):
    return xi * np.tan(xi) - np.sqrt(xi0**2 - xi**2)

roots = []
xi_vals = np.linspace(0.01, min(xi0 - 0.01, 3*np.pi), 500)
for i in range(len(xi_vals)-1):
    x1, x2 = xi_vals[i], xi_vals[i+1]
    if np.cos(x1)*np.cos(x2) > 0 and even_eq(x1)*even_eq(x2) < 0:
        try:
            r = brentq(even_eq, x1, x2)
            roots.append(r)
        except:
            pass

print(f"Found {len(roots)} even bound states:")
for idx, r in enumerate(roots, 1):
    E_bound = (r / xi0)**2 * V0 - V0
    print(f"  E_{idx} = {E_bound:.3f} eV")

Lab 04 • thermo

2D Ising Model Monte Carlo via Metropolis Algorithm

Spontaneous symmetry breaking, magnetization, and phase transition simulation

Simulate the magnetic phase transition in a 2D ferromagnetic square lattice using the Metropolis Monte Carlo algorithm.

Governing Equations & Formulation

\mathcal{H} = -J \sum_{\langle i, j \rangle} s_i s_j, \quad \Delta E = 2 J s_i \sum_{\text{neighbors}} s_j$$\n$$P(\text{flip}) = \min\left(1, e^{-\Delta E / k_B T}\right)

Parameters: Lattice size $L = 32 \times 32$, Coupling $J = 1.0$, $T_c \approx 2.269$.

Starter Solution Ready

import numpy as np
import matplotlib.pyplot as plt

L = 32
N_steps = 15000
T = 2.0

spins = np.random.choice([-1, 1], size=(L, L))

def step(spins, T):
    for _ in range(L * L):
        i = np.random.randint(0, L)
        j = np.random.randint(0, L)
        s = spins[i, j]
        nb = spins[(i+1)%L, j] + spins[(i-1)%L, j] + spins[i, (j+1)%L] + spins[i, (j-1)%L]
        dE = 2 * s * nb
        if dE <= 0 or np.random.rand() < np.exp(-dE / T):
            spins[i, j] = -s
    return spins

for _ in range(50):
    spins = step(spins, T)

plt.figure(figsize=(5, 5))
plt.imshow(spins, cmap='coolwarm', interpolation='nearest')
plt.title(f'2D Ising Lattice at T = {T:.2f}')
plt.colorbar(ticks=[-1, 1], label='Spin Orientation')
plt.axis('off')
plt.show()

Interactive Physics Assignments & Labs

Projectile Motion with Quadratic Air Drag

Governing Equations & Formulation

Electric Field & Equipotential Contours of an Electric Quadrupole

Governing Equations & Formulation

Finite Square Well: Bound State Energies via Transcendental Bisection

Governing Equations & Formulation

2D Ising Model Monte Carlo via Metropolis Algorithm

Governing Equations & Formulation