# Slowest velocity possible?

• curt291
In summary, an object can never stop moving, but between any two velocities, there are an infinite number of velocities in between.f

#### curt291

I was wonder what the slowest velocity possible for an object is. According to mathematics, between any two numbers, there are an infinite number of numbers in between, so between a velocity of 0 and any arbitrary velocity there must be an infinite number of velocities. But from common knowledge and experience, objects don't slow down forever. So I suppose my question is: what is the smallest velocity an object can have before it is defined as being at rest? And why couldn't we just take that velocity and keep dividing in in half for example? Anyways, just some late night pondering any explanations would be terribly appreciated!

Curtis

As far as I know space and time cannot be (meaningfully) divided infinitely, I believe this is related to the Greek paradox of the runner and the tortoise.

Could it be a Planck length per infinite amount of time?

Though that wouldn't make a lot of sense, because any length per infinite amount of time will be the same.

Zero.

Zero.

HAHA, you beat me to it! Yes, zero is the slowest possible. I think what the person wanted to ask is what curt291 wanted to ask is what the next slowest possible velocity. I would assume it would be Lplanck per sec. But we don't really know if space-time is quantized, and if it is, we could kiss Lorentz invariance goodbye, since a discrete space-time is not consistent with length contraction.

... zero is the slowest possible. ...it would be Lplanck per sec.
zero v is not/ no velocity, and what about L planck per 2 sec?

As far as I know space and time cannot be (meaningfully) divided infinitely, I believe this is related to the Greek paradox of the runner and the tortoise.
That was a puzzle (not a true paradox) 2500 years ago. Zeno didn't have the mathematical tools to resolve his paradoxes of motion. Nowaday saying that Zeno's paradoxes are anything but a simple lack of descriptive tools it is pretty much nonsense.

Could it be a Planck length per infinite amount of time?

Though that wouldn't make a lot of sense, because any length per infinite amount of time will be the same.
Yeah, it's called zero.

There's nothing special about a velocity of exactly zero (with respect to what?) compared to a velocity of exactly any other value. Claiming knowledge of an exact velocity (better: an exact momentum) says that you know exactly nothing about position.

. Nowaday saying that Zeno's paradoxes are anything but a simple lack of descriptive tools it is pretty much nonsense..
what is nowaday the meaning of Zeno's?

The simplest explanation of all was given by Diogenes, a contemporary of Zeno. He got up and walked out of the room. The paradoxes were just as much nonsense 2500 years ago as they are today.

A less snarky answer is that Zeno did not have the necessary mathematical tools to explain his puzzles. That Zeno could not explain how a faster runner could overtake a slower one was understandable 2500 years ago. The concept of an infinite series was a ways off into his future. Now we know that 1/2+1/4+1/8+1/16+... is a uniformly convergent series. The arrow paradox is fully explained by the concept of a derivative. One again, Zeno did not have those tools on hand. That would have to wait for Newton, Leibniz, Weierstrass, and others.

Well to clarify, when an object is slowing down, theoretically it must hit every velocity between its initial velocity and zero, so it could never stop, it would just slow down forever. There is an infinite number of velocities between an arbitrary velocity and zero. So what is the slowest velocity before it is defined as having zero velocity? Or could it be that it never has zero velocity but it is just moving so slow that we can't detect any movement?

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Well to clarify, when an object is slowing down, theoretically it must hit every velocity between its initial velocity and zero, so it could never stop, it would just slow down forever. There is an infinite number of velocities between an arbitrary velocity and zero. So what is the slowest velocity before it is defined as having zero velocity? Or could it be that it never has zero velocity but it is just moving so slow that we can't detect any movement?
That is Zeno's arrow paradox, and it is nonsense.

There's nothing special about zero here. Any two distinct speeds will do. Suppose you're moving along at 1 m/s and want to speed up to 2 m/s. But in between those two speeds are an infinite number of other speeds, so therefore it takes an infinite amount of time to accomplish this goal. Motion itself is impossible by the same twisted notion. There are an infinite number of points between any two points on a trajectory. Therefore it takes an infinite amount of time to go from point A to point B. Diogenes was right. His response of standing up and walking out showed that motion is possible.

The resolution is with calculus. As you split the distance (or velocity) into finer and finer increments, it takes less and less time to move from one point to another / accelerate from one velocity to another. In the limit of infinitesimally small increments, you have the Lebesgue integral.

If the deceleration is uniform it will reach 0 velocity in a finite amount of time. If the deceleration decays exponentially then it will take an infinite amount of time to reach 0 velocity.

I didn't mean to bring up Zeno's paradox as a way of affirming it, more as a comparison between that debunked argument and the OP's.

"nonsense" is so generic that carries very little sense, meaning itself; same as "humbug" [Scrooge says] or "fiddlesticks" [Diogenes says with body language].That is only emotional reaction of disbelief or disdain. You dismiss with one generic word what a most reputable logician and mathematician defined "immeasurably subtle and profound"

Could you rationally, analytically define your disbelief, or disdain? what is really a paradox? Zeno's "tortoise" hasn't changed its definition after 25 centuries. Zeno of 'Elea' had an axe to grind, wanted to prove the "static universe" theory of 'Eleatics', and, trying clumsily to do that, found the truth 24 centuries before Planck and QM. If you accept that can you dismiss Zeno?.

The first thing a junior learns, when he starts algebra, is to prove that 2=1. Is that a paradox, is it different from Zeno['s], is it generic "nonsense" too, [what do you call it], a schoolboy prank, a joke, or you envisage something else? is Banach-Tarsky a paradox, is it "nonsense", too?.

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Thanks everyone for helping to explain! I wasn't trying to argue anything, I was just trying to understand haha, thanks again though!

"nonsense" is so generic that carries very little sense, meaning itself; same as "humbug" [Scrooge says] or "fiddlesticks" [Diogenes says with body language].That is only emotional reaction of disbelief or disdain. You dismiss with one generic word what a most reputable logician and mathematician defined "immeasurably subtle and profound"
We've advanced enough in 2500 years to call what might have been (moderately?) puzzling at the time a silly misunderstanding of a simple mathematical principle today.
Could you rationally, analytically define your disbelief, or disdain? what is really a paradox?
As said, the "paradox" assumes constant steps of time. But each time you cut the distance in half, the time it takes to make the step gets cut in half as well. A simple examination of the equation for distance at constant speed will show this - I wouldn't even bother using calculus for it.

This is my favorite demonstration of it's wrongness, though:
DH said:
The simplest explanation of all was given by Diogenes, a contemporary of Zeno. He got up and walked out of the room.
Obviously, even when Zeno proposed the "paradox", he had to know it was false as arrows were already hitting trees and people were already arriving at destinations they were walking to.
logics said:
The first thing a junior learns, when he starts algebra, is to prove that 2=1. Is that a paradox, is it different from Zeno['s], is it generic "nonsense" too, [what do you call it], a schoolboy prank, a joke, or you envisage something else?
I'd call it a prank, yes - or today, perhaps "troll math". 2 does not equal 1. Any "proof" that it does must have a flaw.

How about 1 Planck length per 13.7 billion years?
Currently 3.7 10-53 m/s.

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