Brachistochrone Time Calculator

Inputs, outputs, and units

To use the calculator, supply three quantities:

• Horizontal distance x – the horizontal separation between the start and end points (for example, in metres).
• Vertical drop y – how far the end point lies below the start point. Use a positive value measured downward (same length units as x).
• Gravity g – the magnitude of gravitational acceleration (for example, 9.81 m/s² near Earth’s surface).

You can choose any consistent length and time units. If you enter x and y in metres and g in m/s², the computed brachistochrone travel time will be in seconds. If you use feet and ft/s², the time is still in seconds because the time unit is determined by the acceleration unit.

The brachistochrone curve and key formulas

The brachistochrone between two fixed points is a segment of a cycloid. A convenient parametrization for the cycloid is

x = a (\theta - \sin\theta),
y = a (1 - \cos\theta),

where:

• a is a positive scale parameter that sets the size of the cycloid,
• \theta is a parameter that runs from 0 at the start point to some final value \theta_f at the end point,
• y is measured downward from the starting point.

For a bead sliding without friction under constant gravity g, the total descent time along this cycloidal arc can be written in the compact form

In many physics texts this is written symbolically as T = \theta_f \sqrt{a / (2g)}. The important point for users is that once the calculator has determined the cycloid parameters a and \theta_f, it can immediately compute the minimal travel time.

To match your chosen endpoints, the parameters must also satisfy

x = a (\thetaf - \sin\thetaf)
y = a (1 - \cos\thetaf).

Eliminating a leads to a single equation for the unknown angle \theta_f:

\dfrac{y}{1 - \cos\thetaf} = \dfrac{x}{\thetaf - \sin\thetaf}.

This is a transcendental equation (it mixes trigonometric and algebraic terms) and does not have a simple closed-form solution, so we solve it numerically inside the calculator.

How the calculator finds the time

The computation proceeds in three main steps:

1. Solve for . The tool uses an iterative numerical method to find the value of \theta_f that makes the ratio y / (1 - \cos\thetaf) match x / (\thetaf - \sin\thetaf) to high accuracy.
2. Recover the scale parameter a. Once \theta_f is known, the scale a follows directly from the vertical relation a = y / (1 - \cos\thetaf).
3. Compute the descent time. With a, \theta_f, and your chosen gravity g, the calculator evaluates the time formula and returns the minimal travel time along the brachistochrone between the two points.

Behind the scenes, a robust numerical solver refines an initial guess for \theta_f by repeatedly improving the estimate until changes are smaller than a specified tolerance. For typical geometries (end point lower than the start, and with reasonable aspect ratios) this converges in only a few iterations.

Worked example

Suppose you have two points separated by a horizontal distance of 5 m and a vertical drop of 2 m, under standard Earth gravity.

• x = 5 m
• y = 2 m (end point is 2 m below the start)
• g = 9.81 m/s²

You would enter 5 for horizontal distance, 2 for vertical drop, and leave 9.81 for gravity. The calculator solves for the cycloid parameters and then reports the brachistochrone travel time. For comparison, it is typically a noticeable fraction faster than a straight-line ramp connecting the same two points, even though the path length is slightly longer.

You can experiment by changing the horizontal distance while keeping the same drop. As the horizontal separation increases, the optimal curve becomes longer and the travel time increases, but it still beats simpler paths with the same endpoints.

Comparison with simpler paths

To appreciate why the brachistochrone is special, it helps to contrast it with a straight ramp or a "drop then slide" path between the same two points. The table below summarizes the qualitative differences.

The brachistochrone wins by letting gravity do extra work early on. Once the bead has picked up significant speed, it can cover the remaining distance quickly even along a relatively gentle slope.

Assumptions, limitations, and valid inputs

This calculator is based on an idealized physics model. When interpreting results, keep the following assumptions in mind:

• Frictionless motion. The bead is assumed to slide without friction along the track.
• No air resistance. Drag forces from the surrounding fluid or air are neglected.
• Rigid track following a perfect cycloid. The track is assumed to match the mathematical brachistochrone exactly, without imperfections.
• Constant gravitational field. Gravity g is treated as uniform along the entire path.
• End point is lower than the start. The vertical drop y should be positive (measured downward). If the end point is higher than the start, a brachistochrone of this type does not apply.
• Moderate geometry. Extremely large ratios of horizontal distance to vertical drop can make the numerical problem ill-conditioned, so very extreme inputs may fail to converge or may not represent a practical track.

In real experiments or engineering applications, friction, finite object size, and construction constraints will increase the actual travel time above the ideal value predicted here. Nonetheless, the brachistochrone calculation provides a useful theoretical benchmark for how fast a purely gravity-driven system could operate under perfect conditions.

Using the results

The computed time tells you how quickly an ideal bead would traverse the brachistochrone track between your chosen points. You can use this to:

• Compare theoretical performance of different layouts in physics demonstrations.
• Explore how changing the horizontal separation or vertical drop influences travel time.
• Build intuition for variational problems, where the quantity being optimized is time rather than distance.

If you design actual tracks or simulations, treat the calculator’s output as a lower bound. Any additional effects like rolling resistance, wheel bearings, or non-uniform gravity will lengthen the true travel time relative to the brachistochrone ideal.