Something amazing to tell you concerning physics and motion

[Jenson71]

This blows my mind.

Okay, say you have two ball bearings. You hold one in your hand and drop it. The other you put in a gun, point it straight forward, as level as the one in your hand, and pull the trigger, sending it hundreds of yards.

Which one lands first?

The one dropped in your hand right? RIGHT!?!?

No, actually they both hit the ground at the same time.

This is because the vertical component of motion is the same for both objects. They both fall vertically the same distance at about 10 meters per second squared (gravity, actually more like 9.8).

But Jenson, Jenson! What about the horizontal component of the gun-fired ball bearing? Actually, when air resistance is small enough to ignore as it is in this case (being that the bearings are exactly alike), the horizontal and vertical components of a projectile's velocity are completely independent of one another.

And, in this experiment, the gun-fired bearing is constantly falling from the moment it is fired. Yes, it looks like it is holding up in a straight line, but, it's actually falling and it's just hard to see with our eyes. And again, it is falling at exactly the same speed as the ball bearing you released in your hand -- the speed of gravity.

Post more cool science things in this thread.

I think I will later post Aristotle vs. Galileo and gravity, for a little history spin on science.

[Buck]

Originally Posted by Buehler445

Why the **** did you even look this up?

I've known about it for years, but I've never understood it. I thought somebody here might have some insight.

[Jenson71]

Originally Posted by BuckinKaeding

If anyone can explain this to me in plain english, I'll rep the shit out of you.

http://en.wikipedia.org/wiki/Banach%...Tarski_paradox

That's just depressing, really.

[Zebedee DuBois]

I just want to say:
F=MA
We use that every day.

[MagicHef]

Originally Posted by BuckinKaeding

If anyone can explain this to me in plain english, I'll rep the shit out of you.

http://en.wikipedia.org/wiki/Banach%...Tarski_paradox

Mathematics: Not necessarily applicable to the real world.

[kcxiv]

Here is what i can tell you about it. EA and their football games have no clue how to implement them properly in their football games. lol

[orange]

Originally Posted by BuckinKaeding

If anyone can explain this to me in plain english, I'll rep the shit out of you.

http://en.wikipedia.org/wiki/Banach%...Tarski_paradox

I found the answer:

[Buck]

I was just researching the Taos Hum.

http://en.wikipedia.org/wiki/The_Hum

Pretty crazy.

[Buck]

Bump

[orange]

In the 1920’s two mathematicians proved a very interesting but at the same time very puzzling theorem, known as the Banach-Tarski Paradox. In plain English, the theorem states that it is possible to divide a solid ball into a few pieces and reassemble those pieces together to make two balls, each of which has the same size as the original ball that was divided. A more striking consequence of their theorem is (you may want to sit down before you read this) a solid ball the size of a small pea can be cut into a number of pieces and reassembled into a new ball the size of the sun! Strange as it may sound, it is a valid mathematical argument, not a myth. (We have to note that it is not something one can do at home using a knife and a cutting board, because some of the pieces have no volume!) The analogy that we would like to establish under SMT is to map that solid ball of the theorem to a small amount of water which could quench the thirst of an army of about 30,000 in Tabuk in year 631. This was a miracle given to Prophet Muhammad, peace be upon him. A similar miracle of Prophet Jesus, peace be upon him, is described in the Bible, Matthew 14:21 (Volker Runde, in the Sky 2 (2000), 13–15).

I think the zero-volume pieces are the key.

[Buck]

Originally Posted by orange

In the 1920’s two mathematicians proved a very interesting but at the same time very puzzling theorem, known as the Banach-Tarski Paradox. In plain English, the theorem states that it is possible to divide a solid ball into a few pieces and reassemble those pieces together to make two balls, each of which has the same size as the original ball that was divided. A more striking consequence of their theorem is (you may want to sit down before you read this) a solid ball the size of a small pea can be cut into a number of pieces and reassembled into a new ball the size of the sun! Strange as it may sound, it is a valid mathematical argument, not a myth. (We have to note that it is not something one can do at home using a knife and a cutting board, because some of the pieces have no volume!) The analogy that we would like to establish under SMT is to map that solid ball of the theorem to a small amount of water which could quench the thirst of an army of about 30,000 in Tabuk in year 631. This was a miracle given to Prophet Muhammad, peace be upon him. A similar miracle of Prophet Jesus, peace be upon him, is described in the Bible, Matthew 14:21 (Volker Runde, in the Sky 2 (2000), 13–15).

I think the zero-volume pieces are the key.

HOW IS THIS EVEN POSSIBLE

[orange]

A Layman’s Explanation of the Banach-Tarski Paradox
Posted on December 8, 2010 by Sean Li

The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.

—Wikipedia

Actually, regarding math topics, wiki often makes you more confused than you already were. But this one is not bad. (Not *too* bad.)

The Banach-Tarski Paradox as a topic was chosen by Patrick K, who attends SMU. The idea of the paradox is simply that you can double the volume of a 3-dimensional set of points without adding any new points. Why is it a paradox? Well, it defies intuition because in our everyday lives we normally never see one object magically turning into two.

It’s because it’s not possible in our physical world. The mathematical version of the paradox uses the concept of an immeasurable set. Every object in real life is measurable, because it is the set of a finite number of atoms taking up a finite amount of space. Mathematically, even when finite becomes infinite, you still usually have measurable sets. You really have to try very hard in order to create an immeasurable set.

The Banach-Tarski paradox splits the sphere into a finite number of immeasurable sets of points. The key word is finite. In fact, it can be shown that it can be split into just FIVE pieces, one of them being the point at the center. So with the other four pieces, we can separate them into two groups of two, and create an entire sphere out of each group, each the same size as the original sphere.

Though this is impossible to do in real life (because we are bounded by atoms), it is possible to make a real life analogy. This analogy will require basic knowledge of the gas laws, namely, that pressure and volume are inversely related. Here we go.

Consider an easily stretchable balloon with some volume of gas inside it. Now release the gas into some container and divide the gas in container to fill two balloons. Each new balloon will have one-half the volume of the original. But we’re going to introduce a trick. We’ll reduce the pressure of the room by half. This causes the balloons to each expand to double its size, so that each is as big as the original. We have reconstructed the paradox!

But wait, you say! Even though each new balloon has the same volume as the original, it has only one-half the density. So they’re not the same balloon as the original.

That objection is correct for the physical world. But in mathematics, we CAN get two identical spheres out of one. Here’s the catch. The mathematical sphere has infinite density. When you cut an infinite density in half, the new density is still… infinity. This explains the paradox.

Also, to have this paradox, you need this thing called the Axiom of Choice. You can check out the wiki article on it if you want; however, prepare to encounter some real math.

Sorry for the lateness of this post; I’ve had a busy day including a lot of writing. I got somewhat burnt out on writing by 5 pm, and by the time I started this post, I really did not feel like writing. To figure out some things about Banach-Tarski I did some actual research, both online and in my math textbook, which had a blurb about Lebesgue measure and a sentence about the Banach-Tarski paradox. Also, again there is NO hidden meaning or content in this post.

http://nargaque.wordpress.com/2010/1...arski-paradox/

[orange]

I'll take that rep, please. Mr. Li explains it very well.