Brachistochrone Time Race

The Brachistochrone Time Race

The Brachistochrone curve problem asks for the shape of the wire along which a frictionless bead, starting from rest, slides from one point to another in the minimum possible time under gravity. Johann Bernoulli famously solved this in 1696, proving that the optimal path is an inverted cycloid rather than a straight line.

A straight line is the shortest distance between two points, but time is distance divided by speed. A particle on a cycloid drops steeply at the beginning. This singular initial vertical slope gives the particle a high initial acceleration, allowing it to build speed quickly. Even though the cycloid path is longer in distance than the straight line, the increased speed over the entire journey results in a shorter total travel time.

For a cycloid defined parametrically by:

1. x = a(θ - sin(θ))
2. y = a(1 - cos(θ))

The exact time to reach angle θ is:

For a straight line path to the same coordinates, the time is:

Experiment

Explore the Brachistochrone Time Race module to visualize this phenomenon:

1. Select a value for the "Cycloid scale" control to set the overall size.
2. Select a value for the "Angle over pi" control to determine the target endpoint on the curve.
3. Observe the calculated "Cycloid time" and "Straight time" to see the time difference.

Worked Example

Let the cycloid scale a = 1 and the angle parameter be θ = π (angle over pi = 1). The gravitational constant is g = 9.81 m/s².

First, calculate the coordinates of the endpoint:

1. x = 1 · (π - sin(π)) = π ≈ 3.14159265
2. y = 1 · (1 - cos(π)) = 1 · (1 - (-1)) = 2.0

Calculate the time taken along the cycloid path:

1. t_(cycloid) = √(1/9.81) · π ≈ 1.00303334 s

Calculate the time taken along the straight line path:

1. t_(straight) = √((2(π² + 2²))/(9.81 · 2)) ≈ 1.18904293 s

The time saved by taking the cycloid path is calculated, and the expected result is:

1. Δ t = 1.18904293 - 1.00303334 ≈ 0.18600959 s

The endpoint speed for both paths depends only on the change in vertical height y = 2.0:

1. v = √2gy = √(2 · 9.81 · 2) ≈ 6.26418391 m/s

Limitations

This simulation assumes a completely frictionless path and a point mass bead. In a physical setup, rolling friction, air resistance, and the rotational inertia of a rolling ball would affect the actual travel times. Additionally, this module focuses strictly on comparing the travel time between the cycloid and the straight line for fixed endpoints; it does not solve the variational calculus problem for arbitrary endpoints, nor does it simulate other curves.

References

The mathematical formulation and derivation of the Brachistochrone curve travel times can be found in the following course materials:

1. Calculus Full Book - MIT OpenCourseWare