# Questions tagged [brachistochrone-problem]

the problem of finding the path between two points such that the transit time under specified conditions is minimized.

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### Brachistochrone Problem without Trigonometric Substitution

I'm trying to numerically reproduce the cycloid solution for the brachistochrone problem. In doing so, I eventually ended up with the following integral: $$x = \int{\sqrt{\frac{y}{2a-y}} dy}$$ ...
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### Can the brachistochrone curve be applied to a swimmer diving into water?

I want to determine a model for the optimal trajectory of the dive start in a swimming race. I would define the optimal trajectory as the one that maintains the most horizontal velocity. From my ...
158 views

### Brachistochrone to a vertical line [closed]

Just for fun, I am working through some problems in Mathematics of Classical and Quantum Physics by Byron and Fuller. Problem 2.13 reads: Prove that a particle moving under gravity in a plane from a ...
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### Can we solve the Brachistochrone problem using random walks and Markov chains?

I was reading a paper by Doyle and Snell titled Random Walk and Electric Networks. I saw that 'type' problems can be solved using electric circuits as analogs. I wonder if the Brachistochrone problem ...
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### Explanation for extra portion of Brachistochrone curve?

https://en.wikipedia.org/wiki/Brachistochrone_curve In this article, it points out that the curve taking the shortest time from point A to point B under constant acceleration has a different segment ...
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### Another Solution To Brachistochrone Problem

Recalling the statement of the problem : Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the ...
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### Descent on an inclined wavy frictionless track [closed]

The classical Brachistochrone was actually counterintuitive wherein the time of descent is lesser (the least) for the cycloid than that of the corresponding straight inclined track. Let an inclined ...
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### A subtlety in the Brachistochrone problem

The following is a specific instance of the brachistochrone problem, which I first encountered in grad school, and I have occasionally used as hw problem in teaching CM. A particle is started from ...
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### What is the direction of the velocity $v=\sqrt{2gy}$ vector? Is it tangential? If so, why?

I was trying to obtain the brachistochrone as a function of time, and I failed several times because I wrongly assumed that the $v=\sqrt{2gy}$ vector points downwards (vertical vector). However, in ...
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### What is the intuitive reason that the trajectory of a charged particle in a uniform crossed electro-magnetic field is a brachistochrone?

Cycloid is a type of trajectory which is traced by a point on the circumference of a planar circle rolling without slipping on a surface. It turns out that this is the solution to the Brachistochrone ...
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### Is a brachistochrone a straight line in curved space?

Please bear with me, and don't get upset if i have lack in knowledge about spacetime. Brachistochrone: Given two points A and B in a vertical plane, what is the curve traced out by a point acted ...
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### Brachistochrone Problem for Spacecraft

I understand how the Brachistochrone Problem works, and do understand how the friction is added to it, however I am unsure of how to use the Brachistochrone Problem for use in spacecraft. We know ...
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### Acceleration downhill, fastest trajectory for a ball

Given 3 ways of going downhill, like in this image: Would a ball behave like that in real life? Intuitively, it makes no sense. The shortest path here is not the fastest. Any hints to the math ...
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### Curve for fastest time down a ramp [duplicate]

I came across a physics experiment video showing three balls released from a point A, going down three different kinds of ramps leading to a point B (https://www.youtube.com/watch?v=61S0KW7e-rc) ...
127 views

### Maximising velocity at B when rolling down the curve between A and B

I would like to build a curve between two points A and B. A ball would roll down the curve in a gravitational uniform field (i.e., I'm actually going to build the thing here on Earth). My question ...
313 views

### Lagrangian Derivation of Brachistocrone? [closed]

I've been trying to derive the functional description of the brachistocrone. To do this, I used the Lagrangian to describe motion along a functional path. given a function $f(x)$ where $0<y<a$ ...
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### Brachistochrone problem with a limited slope angle and height

I'm trying to find the path of least time for a hollow sphere (a ping-pong ball) to roll from one point to another through an curve of interpolated points. The main problem here is that the ball isn't ...
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### Minimising the time of an object falling under gravity? [closed]

The first step of this task (the only step I am actually stuck with), is to prove that the time taken for an object to fall between two points, O and A, is given by: \Delta t~=~\frac{1}{\sqrt{2g}}\...
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### Fastest path for a gravity train [duplicate]

On wikipedia > gravity train, the time to go from any point A to any point B at the surface of the earth (assuming earth is a ...
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### Curved Slope faster than linear?

So I saw this gif the other day, and was wondering, is this real or fake? And supposing there is no energy dissipated by the friction, why does such thing occur?
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### Is it possible to bend/refract light 180° using glass sheets of different refractive index?

If possible, How many combination between the numbers of the different layers and the thickness of each layer will give me 180° refraction.
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### Tautochrone curve and Theory of Relativity

I just came across the concept of a tautochrone curve (or isochrone curve). My question stems from the intuition that a body starting from a higher point would need to be traveling at a greater ...
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### Ball on a slope with hollow

The experiment shown in the image suggests that ball B will reach the goal faster than ball A although the balls have identical properties and they start from the same height. The authors even ...
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### What is the position as a function of time for a mass falling down a cycloid curve?

In the brachistochrone problem and in the tautochrone problem it is easy to see that a cycloid is the curve that satisfies both problems. If we consider $x$ the horizontal axis and $y$ the vertical ...
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### Comparing Brachistochrone curve with a Hypocycloid curve

I want to compare the time that it takes to slide a particle in a frictionless hypocycloid curve, so time would be given by the arclength divided by the velocity So I need first compute the arclength ...
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### Optimal tunnel shape for travelling inside the earth [duplicate]

Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other (...
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### Why is a cycloid path the fastest way to roll a ball downward? [duplicate]

Possible Duplicate: Path to obtain the shortest traveling time I've been told that if one would want to make a ramp to get a ball from point A to a lower point B (at a certain horizontal distance ...
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### Other kind of Brachistochrone problem

We know the Brachistochrone problem that to find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the ...
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### Questions regarding solving the Brachistochrone problem using Lagrangian

brachistochrone problem: Suppose that there is a rollercoaster. There is point 1 ($0,0$) and point 2 ($x_2, y_2)$. Point 1 is at the higher place when compared to the point 2, so the rollercoaster ...
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### What shape of track minimizes the time a ball takes between start and stop points of equal height?

I was at my son's high school "open house" and the physics teacher did a demo with two curtain rail tracks and two ball bearings. One track was straight and on a slight slope. The beginning and end ...
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### Brachistochrone problem for 3 points

I wonder how I can solve the Brachistochrone problem for 3 points? The matter starts from point A that is the highest point and it must pass from B and must finish with point C. (No any friction in ...
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### Path to obtain the shortest traveling time

Asume we have a particle sitting at the point A(0,0) in a gravitational field. (g=9.81) It is going to move along some path to the point B(a,b) Where a>0 and b<0. What is the curve the particle ...
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### Is there an intuitive reason the brachistochrone and the tautochrone are the same curve?

The brachistochrone problem asks what shape a hill should be so a ball slides down in the least time. The tautochrone problem asks what shape yields an oscillation frequency that is independent of ...