[Yahya (OP)]

when I throw a stone upward exactly vertical ( 90 degrees angle )with a particular initial speed, what is the duration at which its velocity is exactly zero? suppose it stops at a particular height, how long it will take while it is at this height ?

[yor_on]

de4pends on gravity. atmosphere, mass, speed, etc. You want to specify it? Maybe the idea is one of there being a 'equivalence' adapting those factors?

[zx16]

when I throw a stone upward exactly vertical ( 90 degrees angle )with a particular initial speed, what is the duration at which its velocity is exactly zero? suppose it stops at a particular height, how long it will take while it is at this height ?

Your question is deep.  It depends on whether the flow of time is smooth and indivisible.  Or whether time is made of discrete particles such as "chronotrons"

If time is made of chronotrons,  then your up-thrown stone would stay still -  at zero-velocity -  for precisely 1 chronotron.  Then it would reverse direction, and fall back down again.

[chiralSPO]

Trying to answer this very precisely would require more precise definitions. For instance, all of the atoms within the stone are moving randomly at incredibly high frequencies, so are we talking about the center of gravity of the stone, or the average velocity of all of the constituents of the stone? Also, whose perspective are we using as the frame of reference for determining the velocity of the stone?


I think it would be very hard to specify an amount of time that the stone actually had zero velocity. We could calculate some incredibly short window of time for which the momentum of the stone would be within the Heisenberg Uncertainty limit of zero given what is known about the location of the stone....

[Yahya (OP)]

I think the answer is : it spend at that point zero time , it spends nothing , a moving particle spend nothing when it is at a particular point apart from its velocity  , when we want to measure the time elapsed for this point , we set the time to be zero at the start point, because in fact the time is zero.
 

[evan_au]

I think the answer is : it spend at that point zero time

This is the traditional answer that Newton would have given.
- The acceleration of the stone is (almost) constant.
- the velocity starts off negative (upwards) and increases in an (almost) straight line until it reaches the same value, but positive (downwards)
- the amount of time spent crossing the v=0 axis is 0 seconds
- this answer doesn't really change if you add in the inverse square law of gravity, or air resistance

[chris]

This is a really nice question that highlights a common misconception. Many people, especially younger learners, get the idea that a ball or stone launched skywards "pauses" for a while at the top of its travel before it falls again.

Actually that pause is only instantaneous. The object is constantly slowing down as gravity accelerates it downwards. At some point the velocity is transiently zero before it begins to accelerate groundwards again.

The reason the object appears to pause is that as it approaches standstill it is moving progressively more slowly. As it begins to accelerate down again it is also initially travelling only very slowly.

The net result is that it looks like it's stopped in mid air for a bit.

[Yahya (OP)]

"everything should be as simple as it can be, but not simpler" Albert Einstein

[zx16]

This is a really nice question that highlights a common misconception. Many people, especially younger learners, get the idea that a ball or stone launched skywards "pauses" for a while at the top of its travel before it falls again.

Actually that pause is only instantaneous. The object is constantly slowing down as gravity accelerates it downwards. At some point the velocity is transiently zero before it begins to accelerate groundwards again.

The reason the object appears to pause is that as it approaches standstill it is moving progressively more slowly. As it begins to accelerate down again it is also initially travelling only very slowly.

The net result is that it looks like it's stopped in mid air for a bit.

Surely the object doesn't just "look like it's stopped".  It must actually have stopped going upwards, before it can start going downwards.

I mean the object can't be moving in two directions - upwards and downwards - at the same time. Surely there must be a "instant" in time at which the upward movement stops?  Before the downward movement can start.

And mightn't this "instant" be defined, in the particle-based physics which is absolutely required at present -  as 1 chronotron. 
Couldn't the chronotron  explain Time, just like the Higg's Boson explains Mass. The Higg's Boson was (supposedly) detected  by the LHC.

Would it possible for us to build a"Chronotron" detector?

[chiralSPO]

Surely the object doesn't just "look like it's stopped".  It must actually have stopped going upwards, before it can start going downwards.

The standard explanation is that it has stopped instantaneously before reversing directions. But I believe what chris was referring to is the illusion that an object hangs for a finite moment before reversing course. For instance, if one counts g as 9.8 m/s², an object would spend about 200 microseconds (a far cry from instantaneous) with a speed less than 1 mm/s (a far cry from 0, but maybe not distinguishable by a casual observer)

I mean the object can't be moving in two directions - upwards and downwards - at the same time. Surely there must be a "instant" in time at which the upward movement stops?  Before the downward movement can start.

This depends on your definitions of "moving," "object," and "same time." :-)

[zx16]

It's Zeno's arrow all over again - after 2,000 years!

[nilak]

The object will stay exactly as newtonian mechanics would predict- an infinitesimal amount of time.
In newtonian mechanics you can throw an object perfectly vertical and then at some point it stops and stays there for  an infinitesimal amount of time.
If I interpret this using my theory, there is no such a thing as an object at rest. This also violates time asymmetry. However you can theoretical assign a centre to a structure with a certain degree of precision (you can't have perfect precision not even theoretically, sounds a bit like the uncertainty principle but this is something else) and that center will at some point stop and stay for an instant like in newtonian mechanics.
The best way to answer  your question in the real context is by the motion or a beam of light in a gravitational field. If you imagine sending a beam of light at 1 degree instead of 90deg, it will reach a certain height then it will go down following a parabolic trajectory . You can see that it never actually stop,
but it will stay at the top for an instant of time.

[Yahya (OP)]

time being infinitely small violates velocity definition , if there is distance elapsed then there is time taken and vice versa, if there is not any distance elapsed( supposing this is the definition of this stone at zero velocity because if it take any tiny distance downward its velocity will start to increase ) then there is not any time taken, this for any object moving, if its distance is zero and it takes time then its instantaneous velocity is 0/t =0 , for a moving object its instantaneous velocity can not be zero.
we can apply this rule for this stone because when its velocity is zero , that mean it is in an exact point, think of a number line , to the right are positive numbers and to the left are negative numbers, where is the zero ? it is at the exact point between the two.
you can think of the stone as a particle moving through the number line from the right to the left.

[nilak]

v=dx/dt    But dx and dt are not 0.

[Yahya (OP)]

they are. 0/0 can equal any number according to the situation , according to the smallest number the denominator  and numerator were  before they became zero, if the ratio equals a specific number, and the nominator decrease and the denominator decreases as well but keeping the same ratio , then they will end up in 0/0 but still the ratio the same  in this case 0/0 equals the instantaneous velocity (instantaneous velocity is constant) , remember also that : 0/0=any number, because 0=0*anynumber. and in the real world they are.

[nilak]

You can never divide by pure zero.

[Yahya (OP)]

You can never divide by pure zero.

it's just my personal explanation.

[nilak]

You need to learn calculus to understand the mathematical concepts. However we don't know how infinity and infinitesimal quantities occur in reality.
An example of an infinitesimal number can be 1-0.(9)    0.(9)is 0.999... basically you add an infinite amount of 9 decimals.

For the speed of light c=3*10^8 m/s = x/t=dx/dt
x=3*10^8
t=1
dx=x/n=3*10^8/n
dt=t/n=1/n
where n is a very large number.
Even if n tends to infinity dx/dt remains the same 3*10^8m/s

If we take a number r0=1 we can't really say what is the next real number because there is no next number.
If n tends to infinity then the next number after r can be thought as r1=r0+1/n then r2=r0+1/n+1/n=r0+2/n
All of these are only mathematical solutions.

[evan_au]

v=dx/dt    But dx and dt are not 0.

In calculus, v is normally expressed as "the limit as dt approaches 0. "

For a well-behaved function (like an object lobbed in a gravitational field), v is well-behaved as dt approaches 0, and when dt = 0. See L'Hopital's rule.

[nilak]

v=dx/dt    But dx and dt are not 0.

... and when dt = 0. See L'Hopital's rule.

dt can never be pure zero. That is the purpose of "d"