# 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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### Find curve minimizing energy loss due to friction [closed]

I am looking for an ansatz of the following problem: Given a mass $m$ moving in a constant gravitational field along curves $C$ connecting two fixed points I want to find the curve $C_0$ that ...
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### Relating Brachistochrone problem to Fermat's principle of least time [closed]

When I came across the Brachistochrone problem, my teacher said we could relate it to Fermat's principle of least time. So, we could make many glass slabs of high $\mathrm dx$, and every slab has a ...
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1 vote
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### How to prove that the Brachistochrone problem could be reduced to finding a curve on a plane?

Given two points in space, the 2D Brachistochrone problem could be solved to give solution of a cycloid. I am wondering how could one prove that in arbitrary dimensions ($d\geq 3$) with a 1D uniform ...
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### Which block reaches the floor first?

There are two blocks, each starting at the top of an incline. The particular inclines are depicted in the image below. The height through which the blocks fall is the same, the table lengths are the ...
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1 vote
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### Inconsistency in solving the Brachistochrone Problem. Did I make a mistake? [closed]

Background: Equation of Motion Okay. First I want to see if my "Newtonian Mechanics" lens of the problem is correct. Let the particle's path be given by $\vec{r}(t) = (x(t), y(t))$ and just ...
1 vote
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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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