Physics simulations
Four classic systems you can play with: change a slider and the motion, the graph and the numbers follow. Launch angle, pendulum length, air resistance, spring stiffness, damping, and gravity on four worlds. Press play, or drag the time slider frame by frame.
Without air resistance the path is a parabola and 45° gives the longest range. With resistance (drag ∝ v²) the range shrinks, the best angle drops below 45° and the ball falls more steeply than it rose.
T₀ = 2π√(L/g) holds only for small swings; the exact period grows with the amplitude (shown as a ratio). The simulation integrates θ″ = −(g/L) sin θ − b θ′, so large angles and damping are real, not approximated.
Without air every object falls alike: t = √(2h/g). With drag ∝ v² the speed approaches a terminal velocity √(g/k); the second ball shows the ideal fall for comparison.
ω₀ = √(k/m); the period does not depend on the amplitude. Damping ratio ζ = c / (2√(km)): below 1 the mass oscillates with a shrinking amplitude, at 1 it returns fastest without overshoot, above 1 it creeps back.
Ideal cases use the textbook formulas; air resistance, large-angle pendulums and damping are integrated numerically (Runge–Kutta 4). Everything is computed in your browser; nothing is sent or stored.
What the simulations cover
- Projectile motion with trajectory and velocity vector
- Air resistance and terminal velocity
- Pendulum with exact large-angle period
- Damping on every system
- Mass–spring oscillator and damping regimes
- Kinetic and potential energy bars
- Graphs of position and speed over time
- Play, pause, scrub and change speed
- Gravity on Earth, Moon, Mars, Jupiter
Projectile motion: range, flight time, maximum height, impact speed and angle, the full trajectory and the velocity vector at any moment; switch on air resistance to see the asymmetric path. Pendulum: the small-angle period, the exact period from the elliptic integral, a numerically integrated swing with damping, and kinetic / potential energy. Free fall: fall time and impact speed, drag and terminal velocity, ideal fall for comparison. Mass–spring: natural frequency and period, damping ratio and regime, position and velocity over time, energy. Gravity can be set to Earth, the Moon, Mars or Jupiter — or any value. More tools: Tools for students.