At a glance
This sim releases identical beads from rest on frictionless tracks that join the same two posts, and times each one. Johann Bernoulli posed the question it asks, which track is fastest, as a challenge in 1696, and the answer helped start the calculus of variations.
What you'll learn
- How a bead's speed depends on how far it has dropped.
- How the shape of a track sets its travel time.
- What shape the fastest track has, and how the two posts pick it out.
- How the gravitational acceleration g enters the race.
What's on screen
- Straight line (amber): the chord between the posts, with its length marked.
- Circular arc (blue): leaves the release post vertically and arrives at the finish post.
- Your trial (violet): a smooth curve through three ◆ handles that you can drag.
- Cycloid (cyan): a fourth track for the same posts, shown once it is revealed.
- Bead labels: each bead shows its elapsed time and speed. At the finish a ✓ appears, and the first bead to arrive is tagged 1st.
- Speed–time graph (below the stage): every bead's speed against time, with the dashed arrival-speed line.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Horizontal separation Δx | How far the finish post is to the side of the release post. You can also drag the finish post. | 0.5–4 m opens at 1.5 m | |
| Vertical drop Δy | How far the finish post is below the release level. | 0.5–2 m opens at 1 m | |
| Gravitational acceleration g | The strength of gravity that the beads fall under. | 1.6–25 m/s² opens at 9.81 | |
| Comparison tracks | Shows or hides the straight line and the circular arc. | on / off | |
| Speed–time graph | Shows or hides the graph under the stage. | on / off | |
| Trial track | Drag the ◆ handles to reshape the violet curve. A curve that climbs above the release level stalls its bead. Reset trial shape restores the opening curve. | 3 handles | |
| Reveal cycloid | Adds the cyan cycloid to the race. | button | |
| ⏱ Release beads | Puts every bead back at the release post and starts the race. | button | |
| Show | Turns the travel timers and the speed labels on the beads on or off. | on / off | |
| Travel times | Each track's time, filled in as its bead arrives; your best trial so far; the arrival speed. | panel | |
| Cycloid geometry | The cycloid's rolling radius R and roll angle θF, its length, and the chord length. | panel | |
| ▶ Play / ⏸ Pause | Runs or freezes the race. | top bar | |
| Speed | Plays the animation faster or slower on screen. The timers do not change. | 0.25×–4× opens at 0.5× | |
| ↻ Reset | Puts Δx, Δy and g back to their opening values and restarts the race. | top bar |
Fixed: frictionless sliding, released from rest, uniform gravity. The race is animated in slow motion; the timers show true physical time.
Speed comes only from the drop
On a frictionless track, the track's push on the bead does no work, so only gravity changes the bead's energy. A bead released from rest that has dropped a distance y below the release level has ½mv² = mgy:
Speed depends on the drop and nothing else: not on the mass, and not on the shape of the track. At the opening posts that is √(2 × 9.81 × 1.00) = 4.43 m/s at the finish, for every bead.
A track that dips below the finish level, like the blue arc here (by 0.08 m), lets its bead go faster than that on the way, then slow down on the climb.
Travel time weighs length against speed
A small step ds of the track takes ds/v. Adding up the steps gives the travel time, where y(x) is the track's depth below the release level and y′ = dy/dx:
A longer track adds steps. A bead that is already fast makes each step shorter in time. A steep start raises v early, which helps every later step, but it also lengthens the path. So the time depends on the whole shape of the track, not on its length alone.
| Opening posts, g = 9.81 m/s² | length | time |
|---|---|---|
| Straight line | 1.803 m | 0.814 s |
| Circular arc | 2.129 m | 0.710 s |
| Cycloid | 1.930 m | 0.693 s |
The straight line is the shortest of these tracks and the slowest.
The fastest track is a cycloid
Between two fixed posts, for a frictionless bead released from rest in uniform gravity, no track is faster than a cycloid. It is the path of a point on the rim of a circle of radius R that rolls along under the release level:
- The cycloid leaves the release post vertically, so the bead gains speed early, then flattens so the extra path does not run away.
- Any trial that differs from it is slower. At the opening posts the arc comes within 3% of it, yet it still loses.
The two posts pick which cycloid
Every cycloid has the same shape; they differ in size R. At the finish post, x = Δx and y = Δy. Dividing the two equations of the cycloid cancels R and leaves the roll angle θF to be found from Δx/Δy; R then follows:
At the opening posts, Δx/Δy = 1.5 gives θF = 175.8° (3.069 rad) and R = 0.501 m. Once Δx/Δy exceeds π/2 ≈ 1.571, θF passes 180°: the cycloid runs below the finish level, then climbs back to the post.
At Δx = 3 m the lowest point is 2R = 1.24 m below the release level, 0.24 m below the finish.
Gravity rescales the times, not the shape
θF, R and L are set by Δx and Δy alone, so g appears only under the square root. Changing g leaves every track where it is and multiplies every time by the same factor, 1/√g. Quadrupling g halves every time, so the order of arrival cannot change.
These sliders move the sim too.
Where it matters
- The calculus of variations. Bernoulli posed the problem in 1696 as a public challenge. Newton, Leibniz, l'Hôpital and his brother Jakob all solved it. The method, finding the curve that makes an integral stationary, was developed by Euler and Lagrange and underlies Lagrangian mechanics.
- Light and least time. Bernoulli's own solution treated the bead as a light ray in a medium where light travels faster the deeper it goes, and applied Fermat's principle of least time.
- Huygens' pendulum clock. The same cycloid, as a bowl, is also a tautochrone: a bead released anywhere on it takes the same time to reach the bottom. Huygens used cycloidal guides to make a pendulum swing in equal times.
Go deeper: Bernoulli's light-ray argument
- Cut the track into thin layers. In each layer the speed is v = √(2gy), and Snell's law says sin α / v stays constant along a ray, where α is the angle between the track and the vertical.
- So the fastest track obeys sin α / √y = constant. On a cycloid, sin α = sin(θ/2) and √y = √(2R) sin(θ/2), so the ratio is 1/√(2R) at every point.
Summary
Key points
- Speed is √(2gy), so every bead arrives at √(2gΔy), but at different times.
- The fastest track is a cycloid. Δx : Δy sets its roll angle and radius, and past 180° it dips below the finish.
- g multiplies every time by 1/√g and changes no shape.
Common mix-ups
- "The shortest track is the fastest." The straight line is the shortest track here and the slowest.
- "The steeper the start, the faster the race." A steep start gains speed early but adds path. The cycloid starts vertical and balances the two; a trial that plunges further is slower.
- "Stronger gravity changes the best shape." The shape depends only on Δx : Δy. Gravity rescales the times.
Things to try on your own
- Reshape the violet trial again and again and watch best trial so far in the Travel times panel. Compare it with the cycloid's time.
- Race at Δx = 0.5 m, Δy = 2 m, then at Δx = 4 m, Δy = 0.5 m. Compare how far apart the three times are in each case.
- Set g = 1.6 m/s², then 25 m/s². Multiply each cycloid time by √g and compare the two products.