Ever diminishing distance

[Driller]

Another little thing like the speed of light thing that has been driving me mad:

Take 2 objects and bring them progressively closer together by halving the distance between them.

In my mind this can literally go on forever because however far away something is, if you approach at half this distance you can’t touch it.

My apologies if this now drives someone else bonkers but also for not understanding the pragmatic scientific explanations that will surely follow.

[MTech535]

They well get very close, but not touch.

How accurately can you measure?

[Cardinal Hips]

This was posed to me at school with a different method.

A frog has 1 metre of distance to cover, it's first jump is 50cm, the 2nd 25cm 3rd 12.5cm etc. If it carries on jumping and every jump is half the distance of the previous one will it ever reach the 1 metre mark?

The answer for me was a simple "meh, it's all bks".

(It will never reach 1 metre, apparently).

[99dndd]

When they get to the closest measurable distance they will be doing the slowest measurable speed.


I think.

[Dr Jekyll]

Cardinal Hips said:

This was posed to me at school with a different method.

A frog has 1 metre of distance to cover, it's first jump is 50cm, the 2nd 25cm 3rd 12.5cm etc. If it carries on jumping and every jump is half the distance of the previous one will it ever reach the 1 metre mark?

The answer for me was a simple "meh, it's all bks".

(It will never reach 1 metre, apparently).

An infinite series is not an infinite number.

We did it in feet and inches in my day, much easier.

The dog chases a tortoise, the dog runs 10 times as fast but the tortoise has a 3 foot head start. When the dog has run 3 feet the tortoise has gone 0.3 of a foot, the dog runs the extra 0.3 and the tortoise is 0.03 feet ahead. After the dog runs a total of 3.33 feet the tortoise goes another 0.003 feet. This carries on indefinitely so the dog has to run 3.333333333333333333333333333333333333333333 (recurring) feet to catch the tortoise. Which is exactly 3 feet and 4 inches.

[Moonhawk]

Driller said:

Another little thing like the speed of light thing that has been driving me mad:

Take 2 objects and bring them progressively closer together by halving the distance between them.

In my mind this can literally go on forever because however far away something is, if you approach at half this distance you can’t touch it.

This thought exercise assumes space-time isn't quantised

Also at some point - inter-molecular forces (and ultimately subatomic forces) would come into play.

Edited by Moonhawk on Wednesday 23 May 15:24

[Toltec]

Driller said:

Another little thing like the speed of light thing that has been driving me mad:

Take 2 objects and bring them progressively closer together by halving the distance between them.

In my mind this can literally go on forever because however far away something is, if you approach at half this distance you can’t touch it.

My apologies if this now drives someone else bonkers but also for not understanding the pragmatic scientific explanations that will surely follow.

Are you moving at a constant speed and noting each time the distance has halved? In which case the set of recorded times is infinite, but the objects simply touch in a time determined by the starting distance and velocity.

If you are halving the distance in a fixed time for each movement then mathematically they do touch, but the time taken is infinite.

The infinite series you are describing converges to a value of 1, therefore mathematically they do touch-

https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_...

[beerexpressman]

Sound like one of Zeno's Paradoxes...

https://en.wikipedia.org/wiki/Zeno's_paradoxes

[Driller]

I’m talking about just halving the distance ie moving in steps. So they never actually touch then?

I have a headache

[Toltec]

Driller said:

I’m talking about just halving the distance ie moving in steps. So they never actually touch then?

I have a headache

Eventually they will be less than a planck length apart, you'll have fun moving half of that. Well before then you will be at a scale where the surfaces cease to become easily definable in relation the the nominal distance apart. As a simple analogy how difficult would it be to tell if two peoples heads were touching if their hair was considered part of their heads. Add a static charge to one persons hair to make it even more interesting.

[DanL]

As I understand it, it’s basically the same as this in maths.

[Driller]

Toltec said:

Driller said:

I’m talking about just halving the distance ie moving in steps. So they never actually touch then?

I have a headache

Eventually they will be less than a planck length apart, you'll have fun moving half of that. Well before then you will be at a scale where the surfaces cease to become easily definable in relation the the nominal distance apart. As a simple analogy how difficult would it be to tell if two peoples heads were touching if their hair was considered part of their heads. Add a static charge to one persons hair to make it even more interesting.

Don't worry, we can still mould things to keep the headache going: when the objects get that close then you consider the 2 closest molecules, then atoms, down to the smallest existing particle

[Zad]

If you use the wrong abstract model, you'll get the wrong answer.

Any abstraction loses detail in order to model a real world system in a compact synthetic manner, but it also has a scope, beyond which the particular modelling methodology breaks down.

[menguin]

Either you want the mathematical answer or you want the real world answer. Mathematically, no, sure they won't touch. Practically a frog can't jump 0.0125cm so yes he will.

[Terminator X]

Cardinal Hips said:

This was posed to me at school with a different method.

A frog has 1 metre of distance to cover, it's first jump is 50cm, the 2nd 25cm 3rd 12.5cm etc. If it carries on jumping and every jump is half the distance of the previous one will it ever reach the 1 metre mark?

The answer for me was a simple "meh, it's all bks".

(It will never reach 1 metre, apparently).

Of course it won't as it is only closing half the distance every time.

TX.

[Atomic12C]

This vid is a good one for describing 'touching' (oo err missus)

https://www.youtube.com/watch?v=P0TNJrTlbBQ

[EddieSteadyGo]

Atomic12C said:

This vid is a good one for describing 'touching' (oo err missus)

https://www.youtube.com/watch?v=P0TNJrTlbBQ

That guy is a character.

Based on this information, the frog will eventually reach the other side, based on his definition of "contact".

[Atomic12C]

EddieSteadyGo said:

That guy is a character.

Based on this information, the frog will eventually reach the other side, based on his definition of "contact".

Not just his definition, this is a typical/common definition (if viewing things on the micro-level).

Main thing is that in order something can be described, a definition of some sort needs to be agreed. Otherwise there will be no common use of constraints/boundaries etc.

The OP's question is all about how you set up an explanation of a situation rather than how it occurs in 'reality'.
If you set up a 'problem' so that it can not have a zero point, ie. the graph plot tends to zero without getting there, then it becomes an impossible mission to explain how to actually reach zero.

Set it up as a typical speed, distance, time scenario and moving things will close a distance to zero (if allowed to) - and if looking at the micro-scale then a definition of zero distance could employ the one in the video.

[V8LM]

Driller said:

I’m talking about just halving the distance ie moving in steps. So they never actually touch then?

I have a headache

To say when two objects touch requires a definition of their surface and whether there is no part of either object in the gap between them. At the micropscopic and macroscopic scales they do not touch, but as they get ‘closer’ then you have to consider them at the nanoscopic and smaller, quantum level. As soon as you consider the atoms of the objects at the quantum level then you define touching as when their probability/wave functions overlap. These always overlap - you are touching the ‘surface’ of the furthest star in the Universe right now.

So, they are touching to start with, and can never not touch.