# Everything that goes up must come down.

Is that correct? Prove it and get 50 points.

And please, this is a simple classical mechanics problem. So no quantum theory or general relativity or anything like that.
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Commented:
A real "proof" requires a descent (ascent?) into calculus, so here goes.

I'm sure we will all agree to define "up" as away from the center of the earth.

The force on the object "going up" is:

GMm/(d^2)  (Newton's law of universal gravitation)

where d is its distance fromn the earth's center.
M is the mass of the earth
m is the mass of the object
G is the universal gravitational constant

(I'm using the symbol ^ to indicate an exponent: ^2 is squared, ^3 is cubed, etc.)

We'll be using Newton's second law: F = ma (Force = mass * acceleration)

The calculus starts off with the definition of acceleration as the time derivative of velocity:

a = dv/dt

and continues with:

a = dv/dt = (dv/dx) (dx/dt)

and since dx/dt is the definition of velocity:

a = v(dv/dx)

a dx = v dv

Here, we'll substitute for a, using the Law of gravitation and remember that the acceleration is in a direction opposite to that of increasing distance from the earth:

-GM(1/x) dx = v dv

Now we'll integrate both sides using the following limits:

R is the initial distance from the center of the earth
X is the current distance from the center ov the earth
V is the initial upward velocity
v is the current velocity

The result of the integration is (sorry but I don't know how to represent integral signs on my keyboard):

-GM(1/r - 1/R) = (v^2 - V^2)/2

Being very careful to deal with the signs, we can rewrite:

v^2 = V^2 + 2GM(R/r - 1)/R

In the second term, r starts out at R so the term equals zero and the velocity is the initial velocity, as required.

As the value of r increases, the term contributes a negative value to the velocity.
If that negative value ever causes the velocity to become zero, the object stops rising and begins falling, increasing its velocity as it heads downward.
(Notice that when it reaches the starting point its velocity is the same as when it started, as it should be.

But, if r is allowed to get really, really big, the velocity approaches:

V^2 - 2GM/R

So if V starts out larger than the square root of 2GM/r, then the velocity will never drop to zero and the object will continue rising and

IT WON'T COME DOWN!
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Commented:
No. Of coursce the probability is very high, because there wasn't any counterevidence till now, but there is still a chance.

You only must have the time to wait for the chance. ;-)
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Commented:
I think it's an axiom and hence unprovable.  For every action (going up) there is an equal and opposite reaction (going down).
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Commented:
Couldn't you go up forever?? That would be cool..
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Commented:
physically, yes, the object comes down with the same amount of force and speed (9.8 m/s) as the object was thrown up.

but is it really correct? i can make a computer program that can make a circle float forever upward and never come down.
but even in space the object must come down.

maybe there is an exception for virtual worlds.
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Author Commented:
Some people say yes, some say no, nobody is sure...
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Commented:
Everybody is sure, whether (s)he says yes or no, but you are not sure, which one you should take...
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Author Commented:
Acrklor said: "but there is still a chance" - not sure

KenAdney said: "I think it's ..." - not sure

schachmann said: "Couldn't you go up forever??" - repeated the question

ice911 said: "but is it really correct?" - not sure

So, nobody is sure.

The answer is either yes or no. So, somebody already got it right. But I didn't take the answer because the explanation was not good. No real proof was given, based on the laws of physics. So people are just guessing.
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Commented:
What goes up faster than about 11km/sec (Earth escape velocity) may never come down.
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Author Commented:
As I said on the question, I want proof, not just a yes/no answer.
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Commented:
A real "proof" requires a descent (ascent?) into calculus, so here goes.

I'm sure we will all agree to define "up" as away from the center of the earth.

The force on the object "going up" is:

GMm/(d^2)  (Newton's law of universal gravitation)

where d is its distance fromn the earth's center.
M is the mass of the earth
m is the mass of the object
G is the universal gravitational constant

(I'm using the symbol ^ to indicate an exponent: ^2 is squared, ^3 is cubed, etc.)

We'll be using Newton's second law: F = ma (Force = mass * acceleration)

The calculus starts off with the definition of acceleration as the time derivative of velocity:

a = dv/dt

and continues with:

a = dv/dt = (dv/dx) (dx/dt)

and since dx/dt is the definition of velocity:

a = v(dv/dx)

a dx = v dv

Here, we'll substitute for a, using the Law of gravitation and remember that the acceleration is in a direction opposite to that of increasing distance from the earth:

-GM(1/x) dx = v dv

Now we'll integrate both sides using the following limits:

R is the initial distance from the center of the earth
X is the current distance from the center ov the earth
V is the initial upward velocity
v is the current velocity

The result of the integration is (sorry but I don't know how to represent integral signs on my keyboard):

-GM(1/r - 1/R) = (v^2 - V^2)/2

Being very careful to deal with the signs, we can rewrite:

v^2 = V^2 + 2GM(R/r - 1)/R

In the second term, r starts out at R so the term equals zero and the velocity is the initial velocity, as required.

As the value of r increases, the term contributes a negative value to the velocity.
If that negative value ever causes the velocity to become zero, the object stops rising and begins falling, increasing its velocity as it heads downward.
(Notice that when it reaches the starting point its velocity is the same as when it started, as it should be.

But, if r is allowed to get really, really big, the velocity approaches:

V^2 - 2GM/R

So if V starts out larger than the square root of 2GM/r, then the velocity will never drop to zero and the object will continue rising and

IT WON'T COME DOWN!
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Commented:
Sorry about the second posting; my browser hiccupped!
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Author Commented:
you made a lot of mistakes here...

1:
-GM(1/x) dx = v dv
should be
-GM(1/x^2) dx = v dv

2: the integral of 1/x^2 is -1/x and not 1/x, so
-GM(1/r - 1/R) = (v^2 - V^2)/2
should be
GM(1/r - 1/R) = (v^2 - V^2)/2

3: "Being very careful to deal with the signs, we can rewrite"
hehe, you weren't. and since you made another sign mistake, you reached the right expression:
v^2 = V^2 + 2GM(R/r - 1)/R
note that this expression is not equal to the one you wrote before:
-GM(1/r - 1/R) = (v^2 - V^2)/2
In this case, two wrongs made one right ;-)

4: "X is the current distance from the center ov the earth"
you mean r.

Isolating the starting velocity:

V^2 = v^2 - 2GM(1/r - 1/R)

We want to know how high can it get. We want to know the point where the particle stops and starts falling (v=0):

V^2 = - 2GM(1/r - 1/R)

This expression relates the starting velocity (V), the starting height (R) and the final height (r) the particle reaches before starting falling.

If we take the limit r->infinity, the particle will only come back when it "reaches infinity", i.e., it will never come back. So:

V^2 = -2GM(-1/R)

V = sqrt(2GM/R)

This is the velocity which will make the particle stop when it "reaches infinity". If V is greater than that, it would "reach infinity" with still some speed. So this is the minimum velocity you must apply to a particle so that it never comes back again.

G = 6.67e-11 m^3/(kg*s^2)
M = 5.98e24 kg
R = 6.37e6 m

V = 11190.74 m/s

So, if you throw a baseball upwards a little faster than 11.2 kilometers per second, you will never see it again ;-P

(air resistance was not considered in this problem)
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Commented:
Question - Does a "proof" make any sense in physics?

I would contend that if it a mathematical proof, you can't assume physical laws (like gravitation for example) as they are outside the domain.

If it is a philosophical proof, you have a whole bunch of assumptions that will need to be proved or agreed upon. Including all the good existensial stuff.

And you can't "prove" anything in physics. Any "law" is only the most recent currently un-falsified theory and therefore can't be assumed to be formally true. Science can't be proved by induction.

Regards
Gordon

PS My answer to the original post would be "Nobody can prove that it is true of false".
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Commented:
Doh.
It was explicity stated to use "Classical mechanics"
That gives us the asumptions to use.

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Author Commented:
"Any "law" is only the most recent currently un-falsified theory and therefore can't be assumed to be formally true"

The physical theories are never wrong. They simply don't apply to all fenomena. For example, relativity didn't prove that classical mechanica was wrong. It only proves that classical mechanics can't be applied to very high speeds and very high gravitational fields. Quantum machanica didn't prove classical mechanics wrong. It only shows that the microscopic world behaves differently.

All theories are correct, but they can't be applied to all situations. You must apply the correct theory for each problem. It is a great challege today to find the "theory of everything". But this theory even might not exist at all...
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Author Commented:
"Does a "proof" make any sense in physics?"
"And you can't "prove" anything in physics."

Where can you prove stuff then? Religion? Philosophy?

"Science can't be proved by induction."

What? So where should we use induction? Religion? Philosofphy?

Physical laws are not assumed. They are proved! You go to the lab and test them. Men reached the moon by using classical mechanics. How can it be wrong???
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Commented:
Well, actually, anyone who's had formal schooling
in mathematics and physics should know that most
physical laws are NOT proved. Tests cannot prove
laws, they can only show that a particular instance
is one way or another. "Proof" is a specific term that
means you show that it MUST be true. Not "every time we measure it" or "how can it be any other way".
Sometimes, it's easier to prove that all other possibilities are false, and this this one is true in
an instance of it.

This is why we have a "Theory of Gravity" which encompasses the formula  F=GMm/rr (I think i forget something in it ?) It is only a Law if you accept the
Theory.

Likewise the 3 Laws of Thermoblodynamics
are laws IF you accept the theory.

It is accepted that virtually all physical equations are
laws only if you accept the underlying theories.

(Such as "Using classical mechannics" - that says we'll accept the theories generally comprising and accepted as part of classical mechanics to be true for the meaning of the subsequent discussion)

As far as reaching the moon with classical mechanics - that doesn't matter - it's just an instance...

We can use something without it being true all the time!
Or even with no clue (how long was it before we knew how tunnel diodes worked???)

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Commented:
Well, actually, anyone who's had formal schooling
in mathematics and physics should know that most
physical laws are NOT proved. Tests cannot prove
laws, they can only show that a particular instance
is one way or another. "Proof" is a specific term that
means you show that it MUST be true. Not "every time we measure it" or "how can it be any other way".
Sometimes, it's easier to prove that all other possibilities are false, and this this one is true in
an instance of it.

This is why we have a "Theory of Gravity" which encompasses the formula  F=GMm/rr (I think i forget something in it ?) It is only a Law if you accept the
Theory.

Likewise the 3 Laws of Thermoblodynamics
are laws IF you accept the theory.

It is accepted that virtually all physical equations are
laws only if you accept the underlying theories.

(Such as "Using classical mechannics" - that says we'll accept the theories generally comprising and accepted as part of classical mechanics to be true for the meaning of the subsequent discussion)

As far as reaching the moon with classical mechanics - that doesn't matter - it's just an instance...

We can use something without it being true all the time!
Or even with no clue (how long was it before we knew how tunnel diodes worked???)

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Author Commented:
If you propose a law saying that every object regardless of the mass falls with the same speed, and then you prove it experimentally, then it is true. If eventually you find a situation where it doesn't apply, for example free fall in the presence of the atmosphere, the law is still true, but you now know the scope of the law.

As I said before, no physical law applies to all fenomena. A physical law is considered to be true if it can be proved experimentally to at least one situation. We then extrapolate and interpolate the physical law until we find a situation where it is not experimentally valid.

Unlike religion or philosophy you dont ACCEPT a theory. You TEST it. You can TEST gravitation.

In science you are not supposed to let you personal opinion or point of view to interfere in the scientific process. You are not supposed to ACCEPT or DENY a theory. You are supposed to TEST it.
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Author Commented:
"Not "every time we measure it""

Do you know the scientific method? Do you know its steps?

"it's easier to prove that all other possibilities are false"

No, it is not! There are infinite other possibilities. Prove to me that a number divided by itself (except the zero) equals one? Are you going to test all possibilities?
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Author Commented:
"most physical laws are NOT proved"

So, give me an example of a physical law that is proved and one that is not.
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Author Commented:
"As far as reaching the moon with classical mechanics - that doesn't matter"

That is not an argument. If it is, then everything you said doesn't matter.

" - it's just an instance..."

What you mean?

"We can use something without it being true all the time"

That is what I said! Physical laws are not true all the time.

"Or even with no clue (how long was it before we knew how tunnel diodes worked???)"

Just because we don't know everything it doesn't mean that what we know is not true.
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Commented:
acerola,

Thanks for pointing out my errors.  I will admit that I was not very careful in trying to transfer my scribbled calculations to the keyboard.

But please, people.  This is a "science and math" section.  Does every discussion have to degenerate (and I use the word with full impatience and bitterness) into a discussion of philosophy or metaphysics?  Can't you find a more appropriate group for that kind of debate?

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Commented:
I'm just going to lurk and watch the hot air over here, lol.
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Commented:
Ya know...
if you quote peoples entire sentences
(taking the word "Sometimes," out totally changes the meaning) people will think you less doltish.

'Nuf said.
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Author Commented:
"Sometimes" was present in only one of the sentences.

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Commented:
Gee, they bite well here don't they!
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Commented:
> Everything that goes up must come down

False, unless you ASSUME something that is NOT in the statement.  You must ASSUME
that the FORCE which started the action DOES NOT continue
to act indefinitely!
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Commented:
I have to disagree with pleasenospam.

As long as there is an "up", there must be a force which determines the difference between down (the direction in which the force acts) and up.  If the object stops going up, either it is coming down, or there is no longer an "up" because the forece has stopped acting on it.
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Commented:
Read the original question again.  There is nothing in it saying that the force STOPS.
Therefore, we can just as readily assume that the force CONTINUES and the object will
continue to go UP.
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Commented:
My remark was not in response to the original question, but to your point about the assumption embedded in the question.

My statement began with the conditional "if the object stops" not "if the force stops."

If the force does not continue, it may have stopped after the object had begun its return journey, in which case the object will return, but there is no longer a direction that can be defined as "up."  In fact the point of origin may have moved and the returning body may not return to its exact starting point even though its new direction is opposite to its original direction.  If the force continues indefinitely, the object may never return because it was launched with escape velocity.  In either case, it can be argued that the object does not come "down."
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Commented:
that statement does say "evetything"
that includes items such as:
light
moods
items placed on a shelf
ether-fueld spacecraft

Guess its false.
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Author Commented:
Guys, read the original question. The question is:

"Everything that goes up must come down. Is that correct?"

fontmaster proved that it is not correct. He proved that something that goes up may never come down again. So what is your point? It is false! And fontmaster proved it!
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Commented:
The major constituents of air - oxygen, nitrogen, carbon dioxide etc do not come down except to the extent they are captured by life forms or water bodies.  Certainly all those other gases that are lighter than air do not come down.  If they did there would not have been a problem with chlorofluorocarbon causing a hole in the ozone layer.
CFCs are eventually broken down and so do not come down as CFCs.

Heavier items such as dust particles created by volcanoes can take years to dissipate and while I cannot produce the proof I am willing to bet that a meteorologist could show that some small percentage of dust particles will stay up 'forever' where 'forever' is the effective life of the earth.
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Author Commented:

"> Everything that goes up must come down

False, unless you ASSUME something that is NOT in the statement.  You must ASSUME
that the FORCE which started the action DOES NOT continue
to act indefinitely! "

It is false either if you assume or not that the force which made it go up continues to act. It is false anyway and fontmaster proved it. What is your point???
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Author Commented:
We can find lots of examples of things that go up and never come down. Sattelites, spacecrafts, space probes are some examples.
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Commented:
Why are you so obnoxious to people, acerola ?
If it as already proven  by someone, then what is YOU point in responding to additional comments?
And what is YOUR point in bothering to mention things
like "We can find lots of examples......"
It seems that if YOU have objections of people re-stating

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Author Commented:

Mine is freedom of speech. And what is yours?

"YOU have objections of people re-stating things"

I don't. But I'm tired of people discussing ambiguity in the questions I post. The questions I post are well known to everybody, and their correct interpretations are well known as well.

If you post something here which I disagree, I will say so. That is the point of discussing. If that is beeing obnoxious, than that is how I am.

If you say something and everybody just accept quietly, there is no discussion.
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Author Commented:
"Well, actually, anyone who's had formal schooling in mathematics and physics..."

Isn't that statement obnoxious?
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Commented:
>What goes up faster than about 11km/sec (Earth escape velocity) may never come down.

Yup, that's about the size of it in the short term unless you consider it to have come down if it lands on the moon or similar.

Surely the answer in the long term depends on whether the universe will expand forever in which case what goes up doesn't necessarily come down or whether it ends in a big crunch in which case everything comes down. That's just as valid in classic newtonian mathematics as it is in relativity & quantum physics.
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Author Commented:
Everyone seems to agree with the answer, so here are the points. Congratulations fontmaster.
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