OBSERVATION CHAMBER / ACTIVE
Brachistochrone & Tautochrone Observatory
Race a straight line, parabola, and cycloid between the same endpoints. Compare arrival times and the cycloid’s equal-time descent.
In equal-arrival mode, click the left branch to move start 1. Use ← / → while this field has focus, or use the start-angle input. Starts 2 and 3 stay at 1.2 and 2.4 radians.
Equations and numerical checks
Downward y is positive. Race: x=a(θ−sin θ), y=a(1−cos θ), T=θf√(a/g). Straight: x=Du², y=Hu². Fixed parabola: x=Du⁴, y=Hu²; dt/du=√((16D²u⁴+4H²)/(2gH)). Equal arrival: cos(θ/2)=cos(θ₀/2)cos(√(g/(4a))t), T=π√(a/g).
Save this observation
Observation report
OBSERVATION GUIDE / LAB 97
Two different questions
First compare arrival times from the same start to the same destination. The cycloid is the fastest admissible track in this ideal model; the fixed parabola is only one comparison track. Then switch experiments: three point masses start at rest at different positions on a single valley-shaped cycloid and reach the bottom together.
- Set D and H, then play or step. Each step advances physical model time by 1/120 second. Playback advances the same model time; it is not a stopwatch measurement.
- When θf exceeds π, the fastest track descends below the destination before rising to it. This is allowed: there are no floor or obstacle constraints.
- Equal-arrival mode enables a and θ₀ and disables D and H. Reduced-motion mode disables playback; stepping and reports still work. Hidden tabs stop and never resume automatically.
Point masses slide on prescribed tracks under uniform gravity without friction or air resistance. They are not rolling balls. Lengths are in metres and time in seconds. This ideal model is not an engineering measurement, track design tool, or experimental proof. Calculation and export stay in this browser.
OBSERVATION POLICY
This frictionless point-mass model runs in your browser under uniform gravity. Inputs, calculations, reports, and images stay local. It does not measure a real track or rolling ball.
Route
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