# A thought experiment

1. Aug 8, 2008

### xavier_r

Consider, a particle at position 0 at time = 0
Lets say it moves to 0.5 after 0.5 seconds
It moves to $$0.75$$ after $$0.75$$ seconds
and in general...
It moves to $$1 - 0.5^n$$ after $$1 - 0.5^n$$ seconds

So where will the particle be after 1 second?

EDIT:

n goes like 0,1,2,3...

Last edited: Aug 8, 2008
2. Aug 8, 2008

### nealh149

What is the variable n? This seems very very unphysical.

3. Aug 8, 2008

### Staff: Mentor

It looks to me like you defined the problem to say that distance increases linearly at a rate of 1 unit per second.

d=s*t
s=1
d=t

Or am I missing something? I don't see where a variable "n" would fit in.

4. Aug 8, 2008

### peter0302

No, he means n = t.

He's saying

x(t) = 1 - .5^t

so:
x(0) = 0
x(1) = .5
x(2) = .75
x(3) = .875
etc.

You get infinitely close to 1 but never reach it.

Last edited: Aug 8, 2008
5. Aug 8, 2008

### Crosson

You haven't defined where the particle will be at when time = 1, since $1 - 0.5^n$ is less than 1 for all positive integers n = 0,1,2,3,...

6. Aug 8, 2008

### Defennder

A better expression for distance moved each turn would be $$0.5\frac{1}{2}^{n-1}$$ which is easily recognised to be a geometric series which sums to 1.

7. Aug 9, 2008

### xavier_r

Well it seems u guys are pretty confused,
I'm sorry for that... I'll explain more clearly whats in my mind

See,
Here the function for time t(n) = $$1-0.5^n$$
And the function for distance is x(n) = $$1-0.5^n$$
n is nothing but a parameter...

So after t(n) seconds the particle is at position x(n)...
And as n approaches infinity the particle does approach one
And at n = infinity, the particle will be (or perhaps it won't) at position 1 after1 second...

Here we are not concerned with n itself... But rather how is it possible, that the particle performs infinite amount of tasks in a given finite time, ie., 1 second... ?

Last edited: Aug 9, 2008
8. Aug 9, 2008

### Defennder

If time and space were quantized in discrete units called Planck time and length, then no paradox would arise.

9. Aug 9, 2008

### xavier_r

Planck time and space... very fascinating concepts indeed... thanks for sharing that...
Then what is the fatest event in the universe, such that no other event could occur any faster...??? Is it a photon moving through a distance equal to the Planck length ???

EDIT: Maybe this universe is digital... ;)

10. Aug 9, 2008

### Defennder

11. Aug 9, 2008

### Staff: Mentor

He doesn't define n=t, in fact he defines a distance function without a t in it (in two separate posts). He says "it moves to 0.5 after 0.5 seconds It moves to .75 after .75 seconds"

That's
x(t)=t
x(.5)=.5
x(.75)=.75
That's two hyperbolic functions x(n) and t(n), but a function x(t) would again be linear:

for n=1, t(n)=.5, x(n)=.5
for n=2, t(n)=.75, x(n)=.75
etc.
What you are doing is just proving a simple principle of math: there are an infinite number of points between any two points. Mathematically, you can always make an interval smaller. You are examining your linear system in smaller and smaller intervals.

Last edited: Aug 9, 2008
12. Aug 9, 2008

### dy-e

hi

So, first let's say that our physical equivalent of the equations you've made is simple:
$$x(t) = t$$
To make it more adequate, let's denote that t varies from [0,1].
Variable n doesn't have any physical meaning - it's a parameter, as one said. So, as i read through the topic, it was just a tool to show that vast infinity huh? Maybe clever but it has similiar function to descriptions of Zeno paradox. As russ_waters said, in fundamental physics we belive that space and time isnt' quantified.

Where will be the particle after first second? We don't know, your eq. don't say that.

13. Aug 9, 2008

### xavier_r

Yea, I agree!!! Mathematically, it is very easily evident!!!
But in physics, how can infinite number of tasks be done in a finite amount of time?

14. Aug 9, 2008

### xavier_r

I think Defennder is right

"If time and space were quantized in discrete units called Planck time and length, then no paradox would arise."

15. Aug 9, 2008

### Staff: Mentor

By making sure that as the number of tasks becomes infinite, the time per task becomes infinitely small.

16. Aug 9, 2008

### Staff: Mentor

...which is what you did, xavier!

17. Aug 9, 2008

### Crosson

18. Aug 9, 2008

### xavier_r

Thanks crosson...!!!

19. Aug 9, 2008

### HallsofIvy

Staff Emeritus
Looks straightforward to me. You are saying that the particle has speed of 1 distance unit per second. After 1 second, it will be at 1. The "n" is a red herring.