is this true?

[emo_boy]

check out the second question..... just wondering you opinions...

http://abstractinfluence.com/forums/viewtopic.php?t=133

or in theory you would never touch the wall the problem is that you can never accomplish this because you could not move at such small intervalls?

[muckyMuckMan]

It takes you infinte amount of 'steps'

If it's going to take you an infinite amount of steps then you won't hit the wall. I say this because taking an infinite amount of steps means your going to be taking steps forever.

Physically, I'm not even too sure you can take steps that are infinitesimal in length, so I'm still going to have to say you won't hit the wall.

The equation you showed is the convergence of a geometric series. It says that the sum will converge to a certain value when 0<|r|<1. Here, converge means that the sum will tend to add toward a single value as the number of terms increases without limit.

[ericlin]

Start from 0 toward 100, taking half the distance as the step width, then it takes forever to get the taget. But not "never". It will "never" get the target at 101, but takes "forever" to get the target at 100;

Yes, it will get the target.

What is "forever" ? How big is "forever" ?

a=0;
b=100;
n=0;
while(a<b){
n++;
a+=(b-a)/2;
}
trace(n);

The loop exits when n=25; If I write it in java, the same thing results.

To test by codes:

If the step width is >= than half the distance, the loop exits soon. Usually less than 25 loops; If the step width is < than half the distance, the loop really goes infinitely. For example: "a+=(b-a)/5" will make it goes infinitely; I think this is because computer "rounds" the step width. At some point, computer round 0.5 to 1 and this make it jump to equal to the target. while computer round whatever less than 0.5 to be 0, and the step width is 0, so it will never get to the target becdause it just stand still.

[emo_boy]

awesome, thanks, any more?

[muckyMuckMan]

Haha, well I tried ericlin's code out for the heck of it, and I got n=54.

Anyways, this is where the problem lies, using simple code will usually give you erroneous answers when doing these summations. You have to understand that Flash, Java, C++, etc. will only display a certain amount of digits before it starts rounding. Unless you know how to output high precision numbers to extremely large sig. figs., you can't base your conclusions on the outcome of your code due to rounding. In Flash, I think the maximum amount of numbers displayed is 16 digits. After that, Flash will start rounding.

[emo_boy]

ah, i see... So does any one else want to voice their opinion on the question???

[bit-101]

this is zeno's paradox.
another example of it is:

two runners are having a race. one is much much slower, so he gets a head start.
runner one starts, and a few seconds later runner two starts.
but no matter how slow 1 goes, and how fast 2 goes, 2 can never pass 1.
say 1 has run 100 feet when 2 starts.
1 runs to the 100 foot mark. but by then, 2 has run another 50 feet.
so 1 runs another 50 feet, but now 2 has run another 25.
this can continue infinitely, the gap always closing, but he can never pass the leader.

you can also use this to prove that motion itself is impossible.
in order to move anywhere, you first have to travel half that distance. but before you can travel half, you have to travel half of that. and so on. so you can never move at all, because the recursion is infinite.

of course these are paradoxes, because they seem logical,but don't reflect what we see in the real world.

in a computer, you can't model these situations perfectly, because eventually you hit the limits of "smallness". a variable can't hold an infinitesimal amount.

it's possible that the universe itself is built like that. that there is a finite limit to how small you can get. after that, you don't move through a space, you just jump across it - it would be a very very tiny distance, but it would be impossible to define "half" of that distance, since that is the minimum distance of the physical universe.

[emo_boy]

i see,thanks a lot!

[aa]

yes, i would like to chime in here.

you would hit the wall, no question about it. why? because you are halving the STEP distance, not the distance to the wall. the question states that one would take a step, and each subsequent step would be half of that previous STEP (not distance to wall). what if my first step was 3 feet? i'm a tall guy, that's not out of the question. then it would take me merely 3 steps to smack into the wall.

if on the other hand i was stepping half the distance to the wall each time, i would never contact it and the asymptotic nature of the problem would indeed come into play.

[Penleeki]

Hehe.. I'd like to post my view on this too.
In going from A to B, if you first have to travel half that distance, in the normal world this will take half the time, and every time you halve the distance, you halve the time it takes to get there. So you would be travelling each infinitely small step in an infinitely small amount of time.
So as long as it takes half as long for every halving of the distance, you can get to the wall... I think

[Cyanide300]

About his second questiong. The one about reaching the wall.

I disagree with him. You can not reach the wall because you can never reach infinity. Infinity has no end, therefore you can not reach it. Therefore, you can never reach the wall.


I got another question. If you were traveling foward at the speed of a bullet with a gun pointing behind you. When you fired the gun, would the bullet have any lateral mostion relative to the ground? As in, when you fired the bullet, would it not simply stay in one place?

[aa]

i'm quite sure the bullet would simply drop to the ground while you and your gun fled the shooting scene at the speed of, well, a bullet.

[ericlin]

Will he reach 99.99999 ? yes !
Will he reach 100.00001 ? no ?
Will he reach 100 ? infinity!

Just answer me, Yes or No !
Well, you push me too far.

Infinity exists not in the real world, but exists in the abstract world of mathmatician.

Question:Where will the man be "at last" ? We might answer: 100.

So, he "does" reach the point 100, right ?

[bit-101]

the bullet one is interesting. it demonstrates the general theory of relativity. from the shooter's viewpoint. the bullet would be travelling away from him, at the speed of a bullet. from a "stationary" observer's viewpoint, the shooter would be moving at bullet speed, and the bullet would simply drop to the ground. i quoted "stationary" because all motion is relative.

however, you get into special relativity with the speed of light. say a person was travelling at the speed of light, or close to it and shined a flashlight behind him. from his viewpoint, the light would be traveling away from him at the speed of light. but from a stationary observer's viewpoint, would the light simply leave the flashlight and stand still? no, it would also seem to be travelling at the speed of light.

if you can really visualize this, it becomes a complete paradox. since all motion is relative, the speed of the light should be relative. since speed is distance over time, and both observe the same speed over the same distance, the only factor that can change is the time, which is why time slows down as you approach the speed of light.

[Cyanide300]

The flashlight thing is interesting. However, there is one flaw in that comparison. The bullets speed is caused by a very rapid expansion of gasses caused when the gun powder in the cartrage is ignited. As with all explosions the gas will expand in the path of least resistance (this is how shaped charges work) which is in the direction that you are traveling. Since the force of the explosion is a constant it will apply a force to the bullet sufficient enough to cancel out the speed that you are traveling at. Thus making it appear to stay in one spot. However, photons travel at their speed under their own power, not because of an acting force. So, as soon as the photon is released from the heated element of the flashlights bulb it immediately accelerates to the speed of light.

BTW, there have been some studies that show photons do in fact have an incredibly small amount of mass. Which kinda screws with Einstiens theories. Which are only theories. Not laws.

[bit-101]

my point was that they ARE two different things, even though they appear the same at the outset.

[ArmedJimmy]

Back to question 2 you would eventually reach the wall, no doubt about it. THe question states that the steps half each time, so it you started 10feet away and your first step was 5 feet, you next would 2.5, the next 1.25, the next 0.625 the next 0.3125 0.15625 0.078125 0.0390625 0.01953125 0.00976 0.0048 0.002441 which if added up gives 10.00722975 which means I reached the wall.

I thought I agreed with whats been said about the bullet, but if you think about a car travelling at 70mph, and a man is flung out the back of the car, the observer doesn't see the man just drop down, he actually moves in the same direction as the car for a bit until the friction of the ground slows him down.

So I think the observer would see the bullet come out of the gun,then travel in the direction that you are travelling in. THe problem for you is if the force of the gun isn't enough to push the bullet fully out of the chamber of the gun.

[bit-101]

Originally posted by ArmedJimmy
Back to question 2 you would eventually reach the wall, no doubt about it. THe question states that the steps half each time, so it you started 10feet away and your first step was 5 feet, you next would 2.5, the next 1.25, the next 0.625 the next 0.3125 0.15625 0.078125 0.0390625 0.01953125 0.00976 0.0048 0.002441 which if added up gives 10.00722975 which means I reached the wall.

check your math...

9.99746975

[ArmedJimmy]

dammit! Its I err..have a dodgy calculator...honest!

So its impossible then. Thats my final answer!

[goodnightall]

Originally posted by bit-101
this is zeno's paradox.
another example of it is:

two runners are having a race. one is much much slower, so he gets a head start.
runner one starts, and a few seconds later runner two starts.
but no matter how slow 1 goes, and how fast 2 goes, 2 can never pass 1.
say 1 has run 100 feet when 2 starts.
1 runs to the 100 foot mark. but by then, 2 has run another 50 feet.
so 1 runs another 50 feet, but now 2 has run another 25.
this can continue infinitely, the gap always closing, but he can never pass the leader.

you can also use this to prove that motion itself is impossible.
in order to move anywhere, you first have to travel half that distance. but before you can travel half, you have to travel half of that. and so on. so you can never move at all, because the recursion is infinite.

of course these are paradoxes, because they seem logical,but don't reflect what we see in the real world.

in a computer, you can't model these situations perfectly, because eventually you hit the limits of "smallness". a variable can't hold an infinitesimal amount.

it's possible that the universe itself is built like that. that there is a finite limit to how small you can get. after that, you don't move through a space, you just jump across it - it would be a very very tiny distance, but it would be impossible to define "half" of that distance, since that is the minimum distance of the physical universe.

I have to say something on the first example.
Let say that 1 started when 2 reached 100m,when 1 reached 100m,as 2'speed is half of 1,he is now at 150......
Notice the time,if it took 5sec. to let 1 run to 100m,then it would take 2.5 sec to run to 150m,that means,the time this happening is getting smaller and smaller,may be 0.0000000001......
So,may I ask:if 1 took 5 sec. to run 100m,what will happen when 20sec is passed?
In fact,LET there is a point where 1 meets 2,and x meters is the distance 1 ran up till this point,since 2's speed is half of 1,he has ran x/2+100
x/2+100=x
2(x/2+100)=2x
x+200=2x
200=2x-x
x=200
So,in fact,in the above example(modifed by me),1 meets 2,at 200m after the start.