Apparent paradox with instantaneous change with velocity

There's a problem going round in my mind that i'm not realy sure it's possible to put forward in a way that will make sense, possibly because i have trouble defining the real point myself, but here goes.
It's a very simple thing. Take any object travelling or at rest and change it's velocity, the method to do so isn't important. Fine as it goes, but the problem I have is defining how this actually happens, due to what seems to me to be a bit of a paradox. An object, any object, cannot simply go from velocity A to velocity B, to do so would require an instantaneous change in velocity which in turn would require an infinite force.So now matter how small you make the gap between "A" and "B", no object (with mass) can ever be said to change from velocity A to velocity B. So how does anything ever change velocity?!!.

I'm sure there's a very obvious explanation that i'm just missing but this has been going around in my mind for months without a solution!

you shoud look up zeno's paradox's

i have thought of this myself at one stage but i reasoned thus:

if your gonna measure its acceleration, you measure its velocity at A, then at B, and see how it has changed. (B-A)/T , you then say this is force/mass, F/M . Because F=M*A

and so you have (B-A)/T=F/M

now what your doing is making T very very small and reasoning that
F/M must now get very very big, because M isnt changing(much) .you then reason that
F must be very very big.

BUT, if your going to reason that T is infinately small (instantaneous), then you must also reason that (B-A) is simularly small and so diving by the tiny tiny T wont yeild zero becasue (B-A) is simulary teeny weeny!

thats my reasoning on it anyhoo!
try not to think of it in terms of the situation at A and B
try instead to think of it as a Continuous process, the veocity dosnt jump from one value to another, it transitions with infinate decimal places!!

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This is an interesting question to me, given the apparent 'quantum' nature of our universe in which units of force are restricted to an almost digital-like computation versus analog behavior. Are infinite decimal places allowed?

There's a problem going round in my mind that i'm not realy sure it's possible to put forward in a way that will make sense, possibly because i have trouble defining the real point myself, but here goes.
It's a very simple thing. Take any object travelling or at rest and change it's velocity, the method to do so isn't important. Fine as it goes, but the problem I have is defining how this actually happens, due to what seems to me to be a bit of a paradox. An object, any object, cannot simply go from velocity A to velocity B, to do so would require an instantaneous change in velocity which in turn would require an infinite force.So now matter how small you make the gap between "A" and "B", no object (with mass) can ever be said to change from velocity A to velocity B. So how does anything ever change velocity?!!.

I'm sure there's a very obvious explanation that i'm just missing but this has been going around in my mind for months without a solution!

SansHalo,

We have agreed that, in science, we will use inductive reasoning where we observe numerous examples and try to state a general principle from those data. In the everyday world, we see lots of things smoothly accelerating and, from that, come up with the statement that finite acceleration is possible. I know of no experiment that contradicts this.

Sometimes, we do mental "what-ifs" where we say things like "If cows were spherical, how much milk could they produce?" But, those "what-ifs" can never contradict experimental reality; they can only be used for thinking.

f95toli
Gold Member
As phlegmy has already pointed out this is just a version of Zeno's paradox.

There is a formal mathematical solution to this problem. If you write down the equations for Zeno's paradox (which in the orignal formulation used a competition between Achilles and the tortoise, the "paradox" being that Achilles could never catch up with the tortoise if the latter got a head start) you will find that you end up with infinite but converging series.
Hence, the "pardox" is solved once we realize that "infinities" sometimes add up to finite values.

Thanks for the replies, and the pointer to Zenos's paradox - an interesting read and clarifies the issue perfectly for me. However if i read correctly, the point is made that the various mathematical solutions while providing a (approximated? - calculus is not my strong suite) numericall solution do not in fact resolve the paradox if the assumption that time and space are infinitely dividable is correct. Therfore motion is either an illusion or time and space are not infinitely divisible i.e are quantised. It's facinating to me that such a posibility can be arrived at from such a tangental issue.

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f95toli
Gold Member
No, in the mathematical solution of Zeno's paradox it is assumed that the distance is "infinitely dividable", otherwise you would not be able to so sum an series with an infinte number of terms (because you would have finite number of "intervalls") and get a finite answer. Moreover, the solution is exact.

There is nothing strange about this. The fact that an "infinite number" of intervalls can sum to a finite number is really just another case of convering series.

A well known (and easy to remember) example of a converging series is the following expansion of e:

$e=1+1/1!+2^2/2!+3^3/3!+.....$ (inifinite number of terms)

Dale
Mentor
2021 Award
However if i read correctly, the point is made that the various mathematical solutions while providing a (approximated? - calculus is not my strong suite) numericall solution do not in fact resolve the paradox if the assumption that time and space are infinitely dividable is correct.
The people who claim that the math does not resolve the paradox do not understand the math. There is nothing approximate about calculus. The mathematics of calculus are exact and give exact results. Time and space could still be infinitely dividable and the results would hold just fine.

For an object experiencing acceleration we write

$$\delta v = a\delta t$$ and then assume that

$$\frac{\delta v}{\delta t}$$ has a proper limit as $$\delta t \rightarrow 0$$.

ok, so the mathmatical solutions gives a precise, not approximate answer. However they do not define how it is possible to traverse an infinite number of steps(series) in order to get that answer. is it correct to say there can only be an infinite number of steps between any two points in space(or time?) as long as the distance of each step between points is zero

For any finite value of the (sub divided) distance between points there would obviously have to be a finite number of steps which obviously would take a finite amount of time. The only problem with this is that if the distance between sub divided points was in fact zero then the time to transverse any two points in space would be zero resulting in instantaneous travel - - the opposite of the zeno paradox.

SansHalo,

I'll throw in my two cents again. Keep in mind that I am a little outside the mainstream in that I strive for physical intuition above all else; thus I always try to reason my way to a solution before calculating it. That said, here goes.

Liebnitz (and, of course Newton or vice-versa) gave us the concept of CONTINUOUS change, rather than change by small increments. This applies to distance and time. Note that the universe might indeed be granular but not at the size and speed they were considering. If these quantities are smoothly continuous, then time steps are only a measurement convenience and not physically real. The driving concept is change rather than different places or times.

The test of this would be experimental. Does the arrow move? If it does, time and "exact space" do not come in small steps.

For the calculations, remember that the derivative goes hand in hand with the integral.

Dale
Mentor
2021 Award
ok, so the mathmatical solutions gives a precise, not approximate answer. However they do not define how it is possible to traverse an infinite number of steps(series) in order to get that answer.
Yes, they do. The whole point of the convergence of an infinite series is to show that you can add an infinite number of parts and still arrive at a finite whole. Once you have clear the fact that an infinite number of things can sum to a finite value then you realize that there is no paradox in movement or acceleration.

is it correct to say there can only be an infinite number of steps between any two points in space(or time?) as long as the distance of each step between points is zero
No, that is not correct. Consider the following situation: Points A and B are 1 m apart, so if we start from A and go halfway to B we have gone 1/2 m, if we go halfway to B we have gone another 1/4 m, if we go halfway to B we have gone another 1/8 m, ... Each of these steps is a non-zero distance, there is an infinite number of them, and they sum to a finite value of 1 m. This is why the proper term describing the distance is "infinitesimal" meaning infinitely small, rather than zero. There is no number between "infinitesimal" and zero, but "infinitesimal" is not the same as zero.

For any finite value of the (sub divided) distance between points there would obviously have to be a finite number of steps which obviously would take a finite amount of time. The only problem with this is that if the distance between sub divided points was in fact zero then the time to transverse any two points in space would be zero resulting in instantaneous travel - - the opposite of the zeno paradox.
If the distance between points A and B is 0 then A is B, they are not distinct. This is why the concept of infinitesimal is so useful. If A is not B then there is at least an infinitesimal distance between the two.

To sum up, the relevant points for me are:

1.As pointed out by TVP45 the obvious fact is that in the real world objects move and accelerate with no problems whatsoever, proving experimentaly the solution to the paradox.

2.It is possible to divide any given distance\time span into an infinite number of, non zero points so long as those points form part of a converging series.

3.Such an infinite sequence can be sumed to an exact, finite amount.

So, If space/time is infinitely dividable, there is no paradox as an infinite number of points of finite distance can be summed, i.e travelled, in a finite time. Likewise if space/time is not there is no paradox simply because there has to be a finite number of steps between any two points.

So problem solved, right?

Well, logically, experimentally, mathematically pretty much any way, there is no doubt. Things can and in fact do move on a regular basis.

So why the but? Well I guess it comes down to a problem at an intuitive level, something i cannot and probably will never get my head around, and it comes down to this.

You can SUM an infinite number of steps, but you could never walk them.

And and yet, we do, every day. 7 impossible things before breakfast?...easy street.

Dale
Mentor
2021 Award
You can SUM an infinite number of steps, but you could never walk them.
What do you mean by this? Every time you take a single step you walk an infinite number of points.

Perhaps you are thinking that each point takes a finite amount of time to traverse. It does not, it takes an infinitesimal amount of time to traverse each point.

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You can SUM an infinite number of steps, but you could never walk them.
And and yet, we do, every day. 7 impossible things before breakfast?...easy street.

As i said, we do everyday. The remark was a philsophicaly flippant one taken to mean a person could never walk an infinite number of actual STEPS , unless they're infinitesimal of course though in that case you'd never actually see them... 