Particle at 0 at Time 0: Where After 1 Sec?

[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...

 

[nealh149]

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

 

[russ_watters]

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.

 

[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.

But what's your question?

 

[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,...

 

[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.

 

[xavier_r]

Well it seems u guys are pretty confused,
I'm sorry for that... I'll explain more clearly what's 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... ?

 

[Defennder]

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

 

[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... ;)

 

[Defennder]

Anyway, there is a Wikipedia page on how Zeno's paradoxes may be resolved:
http://en.wikipedia.org/wiki/Zeno's_paradox_solutions

 

[russ_watters]

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.

But what's your question?

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

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

See,
Here the function for time t(n) = 1-.5^n
And the function for distance is x(n) = 1-.5^n
n is nothing but a parameter...

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.

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... ?

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.

 

[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 similar function to descriptions of Zeno paradox. As russ_waters said, in fundamental physics we believe that space and time isnt' quantified.

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

 

[xavier_r]

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.

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?

 

[xavier_r]

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

I think Defennder is right

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

 

[Doc Al]

But in physics, how can infinite number of tasks be done in a finite amount of time?

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

 

[russ_watters]

...which is what you did, xavier!

 

[Crosson]

But in physics, how can infinite number of tasks be done in a finite amount of time?

This is called a http://plato.stanford.edu/entries/spacetime-supertasks/" .

 

[xavier_r]

This is called a http://plato.stanford.edu/entries/spacetime-supertasks/" .

Thanks crosson...!

 

[HallsofIvy]

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...

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.