Name: The Brachistochrone
Uploaded: 2020-03-28T17:15:53.000Z
Duration: 25 min 57 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

The second conditional

If every single one of us held hands together in a chain of unity around Earth, would there be enough of us to go all the way around the planet?

But remember that that many human bodies thrown together into one big pile would barely fill the Grand Canyon.

The physical bulk of all living human flesh on earth today would only make a cone about 7000 people tall and 2000 across.

What if we made a one dimensional single file line of people, each with, say, one meter of room and we stretched that around the planet?

Well, we would make it all the way around and still have 99.5% of the human population left.

If we made a circle that included everyone, the ring of people would be more than 2.4 million kilometers in diameter, dwarfing the orbit of our own moon.

Now that's not just a circle that is a sir cool.

Let's talk about circles today specifically something that they do role.

But what is rolling well rolling occurs whenever something moves with respect to something else and is always in contact with that something else.

And the contact points have instantaneous velocities of zero that is, there's no sliding in mathematics.

The path traced by a point on a rolling object is called a roulette French for little wheel.

The center of a disc produces a roulette that's just a smooth straight line while rolling on one.

A square, on the other hand, would be bumpy, but square centers can make straight smooth.

This is the principle behind square wheels, which I recently had the pleasure of enjoying.

At the Museum of Mathematics in New York City, Stan Wagon wrote a fantastic and famous article about wheels, which I've linked down in the description below.

He also contributed a fantastic interactive tool to the Wolfram demonstration project that allows you to build a wheel and then find the corresponding road shape that allows it to roll smoothly.

The roulette, traced by a point on a disk as it rolls on a straight line, has a special name.

It's called a tro coin from the Greek word for real tro cose.

Now TRO coins could be curtained or pro late, depending on whether the tracing point is inside or outside the circumference.

If the point is on the circumference, the resulting curve is called a psych Lloyd Psych Lloyds are very special, and they are the star of this episode now.

I've been working with Adam Savage a lot lately as we gear up for brain candy live our 40 city tour that I hope to see you at recently, I asked Adam for some help with rule ETS Adam, you have a favorite polygon.

I just literally, actually, you can't like your Children.

I do have a favorite thing that you could do with.

If I take one of the Vertex is of this square and I start rolling this square and you follow where that Vertex Oh, shoot.

The actual path it describes occur described here.

And as you pick polygons with more and more sides, you get closer and closer to what I want from you today.

Now, I could have just done this in photo shop and done looking animation like I normally do.

But you have a shop we can actually build, and you could build things.

With only gravity to move you, what's the fastest way to roll or slide from a point to some other point below, but not directly below.

Well, that would certainly be the shortest path.

But when you fall, gravity accelerates you and falling vertically a lot right away would mean having a higher top speed during more of the journey.

And that can more than make up for the fact that a path like this is much longer than a straight line, but of these two considerations accelerate quickly and don't have too long of a path.

The path of least time is called the brick east a crone problem, and it's been around for a while.

Galileo thought the answer was a path that's just a piece of a circle.

And in 16 97 Johann Bernoulli came up with the answer, using a very clever approach to see how he solved it.

You are standing in some mud, and you want to run to a ball in the street as quickly as possible.

Now a straight line would be the path of shortest distance.

But if you can run on pavement much faster than you can run in mud, the path of shortest time would be one in which you spend less time moving slowly and mud and more time on the surface.

You're fast on the angle you should enter the pavement at depends on how fast you move on both surfaces.

As it turns out, when the ratio between the signs of these two angles equals the ratio between your speeds across both surfaces.

The resulting path will be the optimal quickest route.

Light always obeys Snell's law when it changes speeds like when it leaves a material in which it moves more slowly like water and enters one in which it moves more quickly like air, it always refract according to Snell's law.

In other words, Light always follows the route that is the fastest for it to take brutally used this fact to tackle the Burke East.

A crone problem like changing speed, is analogous to a falling object changing speed.

But of course, a falling object doesn't just speed up.

Once its speed is always increasing, it's accelerating to mimic this using light, which Bernoulli knew would always show him.

The fastest possible route I only had to do was ADM or and more thinner and thinner, laters, in which the speed of light was faster and faster and faster and well, what do you know?

There it is the brick east, a crone curved path of least time rolled down a track like this, and you will beat anything rolling down any other path every time Bernoulli was clever enough to realize that this curve can be described in another way as a roulette.

Specifically, he noticed that it was a psych Lloyd, the path traced by a point on a circle rolling along a line.

A cycling satisfies Snell's law everywhere.

To see why I highly recommend watching this video on the brick East.

This channel is fantastic, by the way, on the huge fan, the visuals and explanations are top notch.

Anyway, a psych Lloyd provides the perfect balance between keeping the journey distance short and picking up speed early.

Now, I told Adam all of this, and I told him that it would be really fun to have a cycle lead curve.

We could roll things down and he said, Clearly, we should start building.

And then we're going to use that circle to trace amount cycle occur.

So you're gonna look, if we're gonna do like, if you have a point A here and point B here and you say that there's some kind of curve That's better than a straight line in terms of something traveling between those two.

Then I want to also make a straight line from a to B.

Yeah, and maybe also a really extreme curved like that.

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The Brachistochrone

material

practice

matter

straight