PhysicsLibrary CV17 computational resource

Brachistochrone Cycloid and Travel Time

A bead released from rest follows a cycloid under gravity. Compare its descent with a straight path to the same endpoint.

Reference endpoint

All reference cases use a vertical drop of Y = 1 m.

1

Cycloid and straight-line comparison

Depth increases downward.

time-s
x-m
depth y-m
speed-m/s

2

Travel-time comparison

cycloid-s
straight-s
saving-%

3

Numerical travel-time convergence

segments-
estimate-s
relative error-%

4

Endpoint-root solve

iteration-
midpoint theta-rad
bracket width-rad
|residual|-

Reference cases

Three reference brachistochrone paths: endpoints before, at, and after the bottom of the cycloidal arch.

The after-bottom case is the CV17 numerical example, X = 4 m and Y = 1 m.

Saved data and source

Download Julia projectProvenanceLicenseEndpoint-ratio sweep CSVTravel-time convergence CSVBisection history CSV

Generated with Julia 1.10.12.

Before bottom

Cycloid ending before the bottom
Download CSV

At bottom

Cycloid ending at the bottom
Download CSV

After bottom

Cycloid dipping below the endpoint before rising
Download CSV