# 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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### Euler-Lagrange on Brachistochrone - clarification on a solution

I have essentially the same question asked by the person in Another Solution To Brachistochrone Problem - why do time-minimizing curves look like straight lines in $\theta - t$ plane? The solution ...
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### For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?

I am having trouble understanding why $$\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0.\tag{7.132}$$ you can see my explanation leading up to it below. I would greatly ...
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### Is this idea/concept very useful to alternatively tackle a brachistochrone problem?

As we know initially we would be given two points on a 2D surface (consider a plane perpendicular to ground ) we would like to run a small ball from higher level point to lower level point from a ...
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### Hypocyloid Integral in Polar Coordinates

I've been working on the classic problem of finding the path through which a body travels in least time between two points on the surface of the Earth, assuming that the body is allowed to fall ...
1 vote
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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 ...
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### 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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### 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 ...
1 vote
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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) ...
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### 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 ...
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### 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 ...
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### Brachistochrone problem in general relativity

This question Brachistochrone Problem for Inhomogeneous Potential has the obvious extension. Namely the same question, when gravity is treated according to general relativity. To make it specific let'...
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