Physics Sim: Projectile, Pendulum, Spring, Orbit & E×B

The Physics Sim runs six classical initial-value problems in the browser: projectile motion with optional linear or quadratic drag, a large-angle pendulum, a double pendulum, a spring–mass–damper, a Kepler two-body orbit and a charged particle in crossed E and B fields. The tabs share one engine, one integrator selector and one conservation-drift plot; each tab has its own address (/science/physics/kepler) and preset chips such as Drag-free 45° or Eccentric e=0.9. Units are SI, drag is off by default, and g = 9.80665 m/s² in the three gravity systems.

Updated 2026-10-01Reviewed by CalcX Editorial Team6 sources5 min read

How to work a run

Pick a systemfrom the tab bar or open its URL; a preset chip fills a known case, and the equation card states the model and its conserved quantity.
Set the parameterswith sliders or typed fields: speed, angle in degrees, mass and g; pendulum length and rate; damping and stiffness; eccentricity, periapsis and μ; charge, fields and initial velocity.
Choose the integrator and resolution.Explicit Euler, Classic RK4, Adaptive RK45 (Dormand–Prince), and Velocity Verlet (symplectic) on the separable systems; Compare integrators runs them all. Except for the projectile, the run lasts a number of characteristic periods (gyroperiods for E×B) with steps per period setting the step.
Play, scrub, read.Play at 1× to 16×, drag the time bar, toggle the trail; result cards carry the headline numbers and the confidence chip names the method, its residual and any amber warning.
Reuse the result.Publish a card, pin it, add it to a report, or send the trajectory as a table and the main series as a vector; open with these inputs restores the snapshot. An expression sent here becomes a custom spring forcing F(t).

Worked examples

Every readout comes from these panels under the default integrator (RK4, 400 steps per characteristic period).

Panel · input	Readout
Projectile · Drag-free 45°, v₀ 30 m/s	range 91.7745 m, max height 22.9436 m, flight 4.32629 s; deviation from v₀²sin2θ/g 2.63e-15
Projectile · Quadratic drag, 60 m/s at 40°, c 0.012	range 104.524 m, impact 22.694 m/s at -64.71°, terminal speed 28.587 m/s; drag-free range 361.52 m (−71.1%)
Pendulum · 60°, L 1 m	exact period 2.153242 s, small-angle 2.006409 s (-6.819%)
Spring · m 1 kg, c 0.4 N·s/m, k 16 N/m	ω_n 4 rad/s, ζ 0.05, ω_d 3.995 rad/s, c_c 8 N·s/m, Q 10
Kepler · ellipse e 0.6, r_p 1, μ 1	period 24.83647 s, measured 24.83645 s, a 2.5, r_p/r_a 1 / 4, Kepler-III deviation 7.43e-7
Double pendulum · Strong chaos 180°/90°	normal modes 2.6215 / 1.0859 s, λ 1.255 /s (R²=0.945), 1/λ 0.7966 s
E×B drift · q 1 C, m 1 kg, Eₓ 1 V/m, B_z 2 T	v_d (0, -0.5) m/s, measured (-5.0401e-10, -0.5) m/s, ω_c 2 rad/s, gyroperiod 3.14159 s, gyroradius 0.559017 m

On the shared values — R = v₀²sin2θ/g, T = 2π√(a³/μ), v_d = E×B/B² — theory and simulation agree to 1e-7 or better (2.63e-15 on the drag-free range), and the panel prints the residual.

Assumptions and limits

One model per tab.: The projectile starts at height h₀, drag off by default (-bv linear or -c|v|v quadratic when on), and the closed-form range assumes a return to launch height. The pendulum keeps sin θ — small-angle T₀ is only a comparison row. The spring is a linear damper driven by F₀cos ωt or a typed F(t); Kepler is an ideal planar two-body problem; E×B uses uniform static fields with B along z.
Steps set the precision.: Fixed-step Euler and RK4 under-resolve fast passages; the drift plot and the amber warning quantify it and name the fix — adaptive RK45 or more steps per period. Readouts round on display (mostly six significant digits), and the forced spring's numeric steady amplitude is only meaningful once the transient has decayed.
What it doesn't do.: No rigid bodies, collisions, fluids or N-body, no unit conversion, no uncertainty budget: it integrates point masses in SI. Error bars belong to Uncertainty, signals and controllers to the Engineering Lab, and typed expressions or symbolic work to the scientific calculator and Math Tools.

The integrator trap: an orbit that closes early

Load Eccentric e=0.9 in the Kepler tab (r_p 1, μ 1) and leave RK4 at 400 steps per period: the measured period is 190.7812 s against 2π√(a³/μ) = 198.6918 s — a Kepler-III deviation of 3.98e-2 — with a conic-fit residual of 1.36e-1 and the amber under-resolved-periapsis warning. The step cannot follow the fast periapsis passage, so the error appears as spurious precession and the ellipse closes about 4% early. Switch to Adaptive RK45 and it reads 198.6918 s, 1.85e-7, 2.53e-7; even the default e = 0.6 ellipse trips the warning under RK4 (residual 3.55e-6), which RK45 clears (1.98e-7).

Where the lab is used

Teaching and coursework

The pendulum makes the small-angle error visible: at 5° T₀ is only -0.048% off the exact 2.007365 s, at 60° it is -6.819% short of 2.153242 s, and at 179° the period is 7.82713 s, T/T₀ = 3.90107, error -74.366%. The drift plot teaches the integrators: over four periods Euler's energy drift reaches 4.43e-1, RK4 holds 5.13e-10, and Velocity Verlet oscillates within 6.51e-5 without climbing.

Ballistics and sport

Drag rewrites the flight: a 60 m/s, 40° projectile with c = 0.012 lands at 104.524 m instead of 361.52 m (−71.1%), arrives at -64.71°, loses 85.69% of its mechanical energy, and shows a terminal speed of 28.587 m/s.

Orbits and plasma

The e = 0.6 orbit reads a = 2.5, r_p/r_a = 1 / 4, ε = -0.2, h = 1.26491, and the symplectic run holds angular momentum to 4.39e-15. In the E×B tab the negative-charge preset keeps v_d = (0, -0.5) m/s — the drift does not care about the sign of the charge or the mass.

References

BIPM, The International System of Units (SI Brochure), 9th edition — the SI units used throughout the lab. bipm.org （访问日期：2026-10-01）
NIST Special Publication 811, Appendix B.8 — standard gravity 9.806 65 m/s², the default g. nist.gov （访问日期：2026-10-01）
Wikipedia, Pendulum (mechanics) — the exact period 4√(L/g)·K(sin θ₀/2) and the small-angle expansion. en.wikipedia.org （访问日期：2026-10-01）
Wikipedia, Kepler's laws of planetary motion — orbit classification by energy and the period T = 2π√(a³/μ). en.wikipedia.org （访问日期：2026-10-01）
Wikipedia, Guiding center — the E×B drift v_d = E×B/B², independent of charge sign and mass. en.wikipedia.org （访问日期：2026-10-01）
Wikipedia, Lyapunov time — exponential separation of nearby trajectories and the chaos horizon 1/λ. en.wikipedia.org （访问日期：2026-10-01）