---
title: "Cycloid Challenge: Fastest & Isochronous Interactive Lab | Sandbox Physics"
description: "Which curve lets a bead slide down fastest and also makes beads from any height arrive together? The cycloid does both."
canonical_url: "https://sandboxphysics.com/labs/cycloid-challenge/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/cycloid-challenge/"
last_modified: "2026-10-27"
---

M046 · Energy and geometry

# Cycloid Challenge: Fastest & Isochronous

**Which curve lets a bead slide down fastest and also makes beads from any height arrive together?** The cycloid does both. Between two fixed points, a bead slides down a cycloid faster than along any other curve. A cycloid is also isochronous: beads released at different heights on the same cycloid reach the bottom at the same moment.

Race particles along straight, circular, polynomial and cycloid wires between the same two endpoints. Then release particles from three heights on one cycloid, and compare their raw arrival times with a circular track that has the same curvature at the bottom.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The particles slide without friction on wires that hold them from both sides. The fastest-descent race and the equal-arrival test are separate experiments. Numerically integrated times and analytic reference values are kept separate.

Physics tutorial

## A clock hidden in a curve

**Background** The rail fixes the shape of the path, and gravity supplies the energy. How hard the track pushes and how long the trip takes depend on the whole path.

**Why it matters** The track you see, the motion and the recorded readings all come from the same model quantities, so you can compare them directly.

Start with the essentials

- **Focus question**: Which curve lets a bead slide down fastest and also makes beads from any height arrive together?
- **One-sentence intuition**: To compare speed, keep the endpoints fixed. To test isochrony, keep the bowl fixed and change only the release height.

### Core mathematical model

#### Cycloid through both endpoints

$$
\begin{aligned}x&=a(\theta-\sin\theta)\\y&=H-a(1-\cos\theta)\end{aligned}
$$

Solve the endpoint ratio for the final angle, then find the radius of the generating circle. For wide spans, the cycloid can dip below the finish.

#### Travel time along a curve

$$
T=\int\frac{\sqrt{dx^2+dy^2}}{\sqrt{2g(H-y)}}
$$

The shortest path does not have to minimize this integral. The integrand blows up at a release from rest; a squared integration coordinate removes that singularity.

#### Exact travel time on the cycloid

$$
T_{race}=\theta_f\sqrt{a/g}
$$

This analytic value is independent of the numerical integration and checks its travel time for a cycloid that starts at its cusp.

#### Motion measured along the arc

$$
\begin{aligned}s_b&=4a\cos(\theta/2)\\y&=\frac{s_b^2}{8a}\\\ddot s_b&=-\frac{g}{4a}s_b\end{aligned}
$$

Measured by distance along the curve from the bottom of the bowl, the potential of the ideal sliding particle is exactly quadratic.

#### Same arrival time from every height

$$
T_{first}=\pi\sqrt{a/g}
$$

Three releases from rest, at different nonzero heights, all take the same quarter period. The circular reference has the same curvature at the bottom, but its timing depends on the amplitude.

### Common difficulties

#### An idealized ball

**Typical misconception** The ball on screen rolls, has suspension and behaves like real hardware.

**Better mental model** The model is a point particle that slides. The thickness of the rail and the marker is only for drawing.

### Run the experiment

1. 01

   #### Race between the same endpoints

   Change the horizontal span, the vertical drop and the polynomial sag.

   **What to observe:** The cycloid races the other curves from the same release point to the same finish.

2. 02

   #### Switch to the equal-arrival test

   Select Three release heights and look at the shared bottom.

   **What to observe:** The three cycloids have the same shape and gravity, and every particle starts at rest. Only the release heights differ.

3. 03

   #### Test equal arrival

   Compare Near-bottom releases with the circular reference.

   **What to observe:** For small swings, the circular reference approaches the same time. From a finite height, it is slower.

4. 04

   #### Record and refine

   Acquire arrivals with zero noise, then change the noise or the resolution.

   **What to observe:** The raw timing is shown separately from the analytic reference. Changing the apparatus clears the old readings.

Physical conclusion

Which curve is fastest and whether different starts arrive together are two different questions about the same curve.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)