Brachistochrone, explained.
The brachistochrone is the curve of fastest descent under uniform gravity for an ideal particle starting from rest between specified endpoints.
Why it happens
For endpoints (0,0) and (πR,2R), the cycloid x=R(θ−sinθ), y=R(1−cosθ) has travel time π√(R/g) and length 4R. The straight ramp has time √(π²+4)√(R/g). Advance time manually to compare positions.
The brachistochrone is the curve of fastest descent under uniform gravity for an ideal particle starting from rest between specified endpoints.
Read the result
Compare arrival times and path lengths separately. Early steep descent buys speed that can outweigh the additional distance along the cycloid.
A worked example
A longer track wins the race
Use the two-metre drop and gravity 9.8. The cycloid length is four metres while the straight ramp is about 3.72 metres.
Advance in tenths of a second and compare particle positions, noting the cycloid's steep initial descent.
The cycloid arrives in about one second, before the straight ramp's roughly 1.19 seconds, under frictionless point-particle assumptions.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
For endpoints (0,0) and (πR,2R), the cycloid x=R(θ−sinθ), y=R(1−cosθ) has travel time π√(R/g) and length 4R. The straight ramp has time √(π²+4)√(R/g). Advance time manually to compare positions.
Where this idea is useful
A practical use
A route's length alone does not determine its duration when the path changes speed along the way.
A common misconception
“The shortest path must also be the fastest.”
Travel time depends on speed along the route. Here gravity changes speed differently on the two paths, so length alone is insufficient.
What this explanation leaves out
- Particles are frictionless points, not rolling balls; gravity is uniform and air resistance absent. The chosen endpoints match a half-cycloid. This does not solve arbitrary routing or transportation problems.
Would real rolling balls follow exactly these times?
No. Rolling adds rotational kinetic energy and practical tracks add friction and other effects. The activity models sliding point particles only.
When does a route change the speed available along it rather than only the distance travelled?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.