# 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 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 ...
2answers
80 views

### 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 ...
2answers
267 views

### 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) ...
1answer
74 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 ...
2answers
94 views

### Lagrangian Derivation of Brachistocrone?

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 $0<y<a$ ...
1answer
170 views

### 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 ...
1answer
74 views

### 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}}\...
0answers
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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 ...
1answer
1k views

### 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?
1answer
249 views

### 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.
1answer
271 views

### 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 ...
3answers
1k views

### 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 ...
3answers
1k views

### 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 ...
1answer
134 views

### 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 ...
0answers
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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 (...
1answer
5k views

### 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 ...
1answer
478 views

### 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 ...
1answer
840 views

### 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 ...
1answer
531 views

### 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 ...
2answers
2k views

### 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 ...
2answers
6k views

### 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 ...
2answers
950 views

### 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'...
1answer
2k views

### Brachistochrone Problem for Inhomogeneous Potential

This recent question about holes dug through the Earth led me to wonder: if I wanted to dig out a tube from the north pole to the equator and build a water slide in it, which shape would be the ...