Physically Impossible?

Question

Let's say a ball is suspended in the air by a rope with a surface beneath the ball.

I then allow the ball to fall so that it is now half-way to the surface, and I continue doing this so that every time I let it fall, it is half-way closer to the surface than it was before. Common sense -- even the laws of physics -- tell you that eventually, that ball will have to touch the surface.

This is where I reach my dilema. If you keep allowing the ball to fall half-way to the surface, then it should be PHYSICALLY IMPOSSIBLE for that ball to touch the surface.

or...

Is there a point where there is no such thing as half? (The ball is so close to the surface that to drop it another half-way would be impossible because it has to touch eventually.)

Please, please help me!! I asked my physics teacher and he told me that he doesn't know and I'm dying to know!

Let's also say that human error is not a factor and that all of the measurements are perfect.

The surface is stationary and the ball continues to move toward it, however slow it may eventually go.

Answer 1

Short answer: The ball will touch because space can only be divided into bits of a certain small size and no smaller.

More Detail:

"Is there a point where there is no such thing as half? (The ball is so close to the surface that to drop it another half-way would be impossible because it has to touch eventually.)"

You guessed right, space is not continuously divisible, you cannot keep going 1/2 way between one point and another point in space indefinitely. Space is "quantized", or, comes in little chunks which are only so small and never smaller. "Loop quantum gravity, string theory, and black hole thermodynamics all predict a quantized space-time with agreement on the order of magnitude."1. So, in our discretized(not infinitely divisible) space-time, you would run out of half-ways and the ball and surface would touch. A second reason, in reality, that the ball and surface would touch is that things do not need to infinitely close (have 0 distance between them) to be "touching" (I explain this a little later).

Alternatively, if you wish to consider this problem in a "general relativity" like universe, where space-time is a smooth and infinitely divisible continuum, then the answer to your question lies in considering what is happening when things "touch".

Everyday matter is made of atoms with a nucleus in the middle and electrons which zip around the nucleus, surrounding it. The outer physical boundary of something like a book or your hand or a ball is occupied mostly by electrons. Electrons repel each other because they are of the same electric charge. So when two objects come very close together, their outer boundaries will repel each other. When you put a ball on a table or grab a coffee cup you and it are not really quite touching in that your matter is in direct contact, you are very nearly touching, but not quite, because you are both sheilded from each other by your outer layer of electrons.

If space-time were an infinitely divisible continuum with no such "smallest chunk" of space then you would eventually get close enough that the ball and surface would be "touching" just as much as things ever do touch. The distance between things does not get to 0 when they touch, it's just very small, but not infinitely small.

If you wish to consider this problem in a further way, similar to the way you may have been considering it originally, then let us ignore our best physical evidence and the theories built from them and say that space-time is infinitely divisible; there is no such thing as a "smallest bit" of space. Let us also assume, contrary to reality, that physical objects have static (unchanging and exactly defined in space) outer boundaries, such as a mathematical geometric shape might.

If this is the case, then your question was answered by Xeno in "Xeno's Paradox".

your question was considered by Xeno in a famous problem known as Xeno's 'Achilles and the Tortoise' Paradox, which can be rephrased:

"Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room."2

So we have the idea that you can never get somewhere because no matter how close you get there is another bit of ground left to cover that is 1/2 as long as the one you covered since you last checked. Remember, it is important that we are working with the assumption that we have been able to take infinite little hops at the end of this room crossing.

Let's assume that we are correct in saying that taking an infinite number of hops, each 1/2 as long as the last, will never get us to the spot we wanted to get to. Then we should be able to find out where we would end up by adding all the lengths of those hops together like this:

? = 1/2 + 1/4 + 1/8 + 1/16th ... (the ... means and so on forever)

Math lets us answer questions like this, something like this is called an infinite series. Depending on what the parts you are adding up look like you may not even have an answer at all; the more parts you add, the bigger or smaller the answer gets. Some others don't keep getting smaller or bigger, but keep wandering around. Others, like ours, do come to a single answer, a single number, when you add up every little bit (even though there are infinitely many bits). These are called convergent infinite series.

So here it comes... what does our infinite series add up to?
The answer is 1.

1 = 1/2 + 1/4 + 1/8 + 1/16 ...

It's true, 1/2 + 1/4 + 1/8 + 1/16 will keep being 0.99999999999 with more and more zeros the further we go, but, 0.99999999999... = 1

You can make it easy, if you want.
What is 1/3 + 1/3 + 1/3?
Sure it is not one! because 1/3 = 0.333333...
If we add
0.333333...
0.333333...
0.333333...

we get 0.999999... right? exactly!
We also get 1 because 1/3+1/3+1/3 = 3/3 = 1

0.999999... = 1

They are "one and the same".

Answer 2

Well things that get pulled over the event horizon and into a black hole are said to reach singularity, where they have reached the ultimate density. The cannot get any more denser. Now this is not exactly pertaining to your question, but the principle does. Eventually the ball will touch the surface. The reason is that in the beginning half is easy to figure. There is coming a point where a half may be in millionths. It will only increase until the half way point is right next to the surface. Microscopically, the ball's surface is not perfect and part of the ball is touching the surface even before it theoretically touches the surface. We're talking really micro measurements to ascertain half each time. It becomes impractical. We live in a physical world, hence physics, and there are certain theoretical exercises that can't be duplicated in real time. You can divide the number 1 so many times until the anwer is .000000000000000000000000000000000000000000000000000000000000000000000000 to infinity, plus digits. For all practical purposes the answer is zero. For all practical purposes the ball touches the surface.

Answer 3

This is Zeno's paradox, first formulated in ancient Greece. The answer is that the ball does reach the surface in a finite time because an infinite series of terms can add up to a finite sum.Draw a line 1 metre long on a sheet of paper. Mark the halfway point 50 cm. long, then 3/4 of the way 75 cm. from the left, then 7/8 of the way and so on. This is a practical demonstration that the infinitely long series 1/2 + 1/4 + 1/8 + 1/16... adds up to 1.

Answer 4

well, actually, there IS a very clear answer to your question. In fact, your physics instructor should have known this easily.

There is no such force as "touch." what we think of as a push or touch is actually the action of the electromagnetic force caused by the electrons in the atoms which make up objects. As the electrons are brought closer and closer together, the electromagnetic repulsive force gets stronger and stronger. Eventually, this force becomes so strong that we cannot push the atoms of the object closer together. This is what we title "touching," though in truth no matter is actually in contact with any other.

The same thing goes with your question. As the ball gets closer and closer, the electromagnetic interaction increases (every time you halve the distance, i believe its four times the repulsive force). Eventually you get to a point where the electromagnetic force is so strong it causes your ball to stop (even though the ball never REALLY contacted the surface). so you can't continually get half the distance because at a certain point you can't overcome the force needed to get another half amount closer. the objects are now considered touching.

Answer 5

It is in fact impossible to ever get to the surface by ONLY MOVING HALF THE DISTANCE. It is jus that every time you go halfway you are cutting down how far the next move will be. As you keep going half, the distance you will be moving next continues to decrease until you get to a level so small it would almost as if the ball didn't move at all. This could, theoretically, go on forever. However the reason you yourself can not accomplish this is because there will always be inaccuracies in your measurements due to human error.

It is never impossible to move half the distance it is just that there is a point where half the distance is so small that you can not see it, magnify it by 100x or 1000x or eveb 10,000x and that same are that seems impossible becomes HUGE! try using MS paint to draw a line and start to cut it in half like you talk about, when the line gets so small that you can not cut it anymore go to "View," select zoom and then select "Large Size." see how big that same area looks now? this could go on forever.

Hope this helped out a little. Happy Thanksgiving!

Answer 6

If you could remove the space between the ball and the flat surface. they still would never touch. Nothing can measure the space of zero. So that tells us nothing ever touches anything else. the only way it could touch is to go through the surface , Like a ball going through a class window. So then even if the ball bounces off the surface it never made contact.

Answer 7

In theory, it would never touch. In fact, not in theory at all. In reality. Your question is self-explanatory. You are actually SAYING that the ball will never touch the surface because you are always moving it half the remaining distance. There is always such a thing as halfway. You can't envision it as a human, because you cannot see or measure it. But if you were the size of an atom, say, for example, an oxygen atom, you would view the distance much differently than you do as a human. Consider this real-world example: Engine oil is designed to provide a protective film that is perhaps only one molecule thick between moving parts of an engine, for example, crankshaft bearings and the crankshaft. Even though it looks and feels like the parts are actually touching, there is at least one molecule of space between them as a result of that molecular film of oil. If this molecule-size space was not there, the parts would touch, and the resulting friction would destroy them. If it were possible to decrease the size of this oil molecule to half its size, the parts would be half as close to touching, but still not touching. Engineers and manufacturers use elaborate means to design and build machinery and lubricants with absolute minimum clearance between moving parts in order to improve efficiency and reliability. They do it by asking the exact same question you ask here. "Can we move this part closer to that part and still not let them touch?" And just when they think they have reached the limit of how close two parts can be, along comes some new technology that allows them to make the parts closer. I hope this helps you.

Answer 8

The fallacy is introduced when you ignore the time taken for each step also goes to 0. Let's say each step takes one second: in the first second you drop it 1/2 way, in the next second you drop it another 1/4, in the next second 1/8, and so on. In this case, in the ideal situation you postulate, it will never reach, you know after n seconds it will be 2^-n units above the surface.

If, however, the first step takes 1 second, the second 1/2, the third 1/4, and so on, the time series also sums to a limit, so after 2 seconds the ball will have reached its destination of the surface.

Your physics teacher should know this!

Answer 9

Calculus might not answer this question. Calculus assumes that you can divide any length into an infinite amount of smaller lengths and when you add them all up you get the original length.

String theory tells us that there IS a minimum length, it is on the order of the Planck length if I recall correctly. So, you can not keep going down half the distance. Eventually you will get down to moving a distance of the Planck length and you can not divide this in half and lower the ball by this length. Of course, this argument only works if you believe that this assumption made by string theory is correct.

Answer 10

If you can drop the ball in infintely tiny increments you can continue to drop the ball by half for eternity and it will not actually touch the surface. You would also have to be able to see on the sub atomic level to see that the ball is not touching the surface.