Acceleration, distance and speed

[Dream Relics]

Okay, so admittedly my background and knowledge base is pretty weak compared to most of you guys on here. I have been reading as much stuff stuff as I can find because I find physics and cosmology so very interesting. I just found today a physics tutorial series on youtube. The beginning lessons are about the equations for figuring acceleration, distance and time and that sort of thing, and a little bit so far on vector and scalar. But while watching it a hypothetical situation occurred to me. But it may well be that there are a lot of established limits on certain things that I am unaware of so forgive me if this is a silly question.

The question is this. If the rate of acceleration from gravity is a constant at 9.8 m/s/s? Then presumably, if someone dropped something from whatever the highest height is above that Earth from which something would fall without just floating away or going into orbit or whatever else might happen, then from that height there is a known maximum velocity that can be achieved by the falling object, of course forgetting about wind resistance and that sort of thing. So then I wondered, is the height of an influencing gravity field around a massive object always the same no matter how massive the object is, or if the object is more and more massive and the therefore the gravity more intense does the height away from the surface of that object of the gravity at the point where it would pull an object in increase? If it does increase, then is it possible to have a gravity field with enough height away from the surface that a falling object would then have enough time as it is falling to accelerate all the way up to the speed of light? it seems like if acceleration is always constant under gravity and there was sufficient distant for the object to cover before it made contact then that could happen. or if not, why not? I am really looking forward to having it explained to me why that won't work. So please tear me apart.

Though this one other thought occurs to me, if that can happen, then maybe somewhere beyond the observable universe there are super supermassive black holes that are doing just that to the distant galaxies, and that is what is causing the acceleration of the expansion all the galaxies out there.

 

[benny_91]

Whoa...Pretty good question. I don't think this can be answered using classical physics. You might need relativity. Or may be this could just be possible. Regarding your final paragraph I think all black holes make other heavenly bodies revolve around them just like the sun makes the Earth revolve around it using its gravitational force. The Earth does not fall onto the sun. Similarly our solar system and the other stellar systems of the Milky Way revolve around a massive black hole which is at the the center of our galaxy. So nothing goes and falls on the other.
Regarding the first question I too am eagerly waiting for a General Relativity scholar to answer it!

 

[TahirGorgen]

Okay, so admittedly my background and knowledge base is pretty weak compared to most of you guys on here. I have been reading as much stuff stuff as I can find because I find physics and cosmology so very interesting. I just found today a physics tutorial series on youtube. The beginning lessons are about the equations for figuring acceleration, distance and time and that sort of thing, and a little bit so far on vector and scalar. But while watching it a hypothetical situation occurred to me. But it may well be that there are a lot of established limits on certain things that I am unaware of so forgive me if this is a silly question.

The question is this. If the rate of acceleration from gravity is a constant at 9.8 m/s/s? Then presumably, if someone dropped something from whatever the highest height is above that Earth from which something would fall without just floating away or going into orbit or whatever else might happen, then from that height there is a known maximum velocity that can be achieved by the falling object, of course forgetting about wind resistance and that sort of thing. So then I wondered, is the height of an influencing gravity field around a massive object always the same no matter how massive the object is, or if the object is more and more massive and the therefore the gravity more intense does the height away from the surface of that object of the gravity at the point where it would pull an object in increase? If it does increase, then is it possible to have a gravity field with enough height away from the surface that a falling object would then have enough time as it is falling to accelerate all the way up to the speed of light? it seems like if acceleration is always constant under gravity and there was sufficient distant for the object to cover before it made contact then that could happen. or if not, why not? I am really looking forward to having it explained to me why that won't work. So please tear me apart.

Though this one other thought occurs to me, if that can happen, then maybe somewhere beyond the observable universe there are super supermassive black holes that are doing just that to the distant galaxies, and that is what is causing the acceleration of the expansion all the galaxies out there.[/QU

Okay, so admittedly my background and knowledge base is pretty weak compared to most of you guys on here. I have been reading as much stuff stuff as I can find because I find physics and cosmology so very interesting. I just found today a physics tutorial series on youtube. The beginning lessons are about the equations for figuring acceleration, distance and time and that sort of thing, and a little bit so far on vector and scalar. But while watching it a hypothetical situation occurred to me. But it may well be that there are a lot of established limits on certain things that I am unaware of so forgive me if this is a silly question.

The question is this. If the rate of acceleration from gravity is a constant at 9.8 m/s/s? Then presumably, if someone dropped something from whatever the highest height is above that Earth from which something would fall without just floating away or going into orbit or whatever else might happen, then from that height there is a known maximum velocity that can be achieved by the falling object, of course forgetting about wind resistance and that sort of thing. So then I wondered, is the height of an influencing gravity field around a massive object always the same no matter how massive the object is, or if the object is more and more massive and the therefore the gravity more intense does the height away from the surface of that object of the gravity at the point where it would pull an object in increase? If it does increase, then is it possible to have a gravity field with enough height away from the surface that a falling object would then have enough time as it is falling to accelerate all the way up to the speed of light? it seems like if acceleration is always constant under gravity and there was sufficient distant for the object to cover before it made contact then that could happen. or if not, why not? I am really looking forward to having it explained to me why that won't work. So please tear me apart.

Though this one other thought occurs to me, if that can happen, then maybe somewhere beyond the observable universe there are super supermassive black holes that are doing just that to the distant galaxies, and that is what is causing the acceleration of the expansion all the galaxies out there.

This is verry brainstorming stuf. Shall i explaine why the constance is 9.8?

Their particles in our atmophere with their own density. So particles with a lower density is above and particles with a higer density below the atmosphere. This does effect the G pull of matter. But due to Earth's rotation is the pull (as the sun. The G of the sun (i discovered) is also time in space=speed, it spinns at high. So near the sun all will accelerate faster.) it does not make a difference. Because the pull is near Earth at high.

You have to understand G first, brainstorm then.

 

[TahirGorgen]

Sorry i am new. For the replay with the hole quote.

Btw the constance is just near the surface of earth.

 

[TahirGorgen]

It's verry difficult to understand. As in coriolis you have an upside push and an towards pull. If there where no atmosphere, the density of other matter, te constance would not be set at 9.8 you would keep accerelating till the surface. Due to the G of Earth and the density of matter -without resistance-.

 

[TahirGorgen]

It's verry difficult to understand. As in coriolis you have an upside push and an towards pull. If there where no atmosphere, the density of other matter, te constance would not be set at 9.8 you would keep accerelating till the surface. Due to the G of Earth and the density of matter -without resistance-.

Higher knowledge: so the mass/density of an planet (without resistance) for it's orbitingspeed, does not matter, because kg/weight is just an illusion of G.

 

[Borg]

Then presumably, if someone dropped something from whatever the highest height is above that Earth from which something would fall without just floating away or going into orbit or whatever else might happen, then from that height there is a known maximum velocity that can be achieved by the falling object, of course forgetting about wind resistance and that sort of thing.

If you don't have any wind resistance, then the object continues to accelerate. There wouldn't be a maximum velocity because the object will continue to accelerate until it strikes the planet. In Earth's case, there is atmosphere so you cannot ignore wind resistance. Then there is a maximum velocity which is called the terminal velocity. That velocity is highly dependant on the shape of the object that is falling through the atmosphere.

So then I wondered, is the height of an influencing gravity field around a massive object always the same no matter how massive the object is, or if the object is more and more massive and the therefore the gravity more intense does the height away from the surface of that object of the gravity at the point where it would pull an object in increase?

There is no point where objects just "float away" as you've described. Gravity does not have a limit for it's ability to affect other objects - its effect just becomes weaker as you get farther away. If you had a universe with just the Earth and an object light years away, there would still be a very small effect on the object due to the Earth's gravity. The acceleration wouldn't be 9.8 m/s² but it wouldn't technically be zero either.

 

[jbriggs444]

If the rate of acceleration from gravity is a constant at 9.8 m/s/s?

The acceleration from gravity is only a approximately constant when your distance from the gravitating object is approximately constant. For everyday purposes, sitting where we do on the surface of the Earth, our distance from the center of the Earth is approximately constant. So we see the acceleration from gravity as approximately constant -- 9.8 m/s²

Gravity follows an inverse-square law. The farther away you get, the weaker the influence. Double the distance and you divide the force by four. Multply the distance by ten and you divide the force by one hundred. That law applies easily for point-like objects. For big objects, one can treat them as a collection of small pieces. But there is an important result -- Newton's spherical shell theorem. It turns out that for a spherically symmetric object like the earth, one can pretend that all the mass is concentrated in the center and the inverse square law will continue to work just fine (for objects that are not underground).

In mathematical terms, this becomes Newton's universal law of gravitation:

$F=G\frac{mM}{r^2}$

where F is the attractive force of gravity between you (mass m), the planet (mass M}, r is your distance from the center of the planet and G is Newton's universal gravitational constant. If you prefer, you can divide by your mass (m) to obtain:

$a = \frac{F}{m} = G\frac{M}{r^2}$

If you want to obtain the impact velocity when falling from a given height, that can be done. Start by taking the integral of gravitational force as r varies from your starting position (measured as a distance from the center of the planet) to its ending position at the surface of the planet. This is an integral of force times incremental distance. Force times distance is work -- a change in energy. This integral gives you the kinetic energy gained as the object falls. Evaluating the integral yields:

Impact Energy $= \frac{1}{2}mv^2 = GMm(\frac{1}{r} - \frac{1}{h})$

Impact velocity $= \sqrt{2GM(\frac{1}{r} - \frac{1}{h})}$

It turns out that the impact velocity falling from a point infinitely high up is still a finite value. Reversing the trajectory and playing it backwards, this impact velocity is the initial speed that you would need to escape to infinity. This is more commonly known as "escape velocity".

So then I wondered, is the height of an influencing gravity field around a massive object always the same no matter how massive the object is, or if the object is more and more massive and the therefore the gravity more intense does the height away from the surface of that object of the gravity at the point where it would pull an object in increase? If it does increase, then is it possible to have a gravity field with enough height away from the surface that a falling object would then have enough time as it is falling to accelerate all the way up to the speed of light?

Yes, one can have objects massive enough and compact enough in radius so that their escape velocity from the surface exceeds the speed of light. Yes, this means that no trajectory originating at the surface can escape to infinity. However, the above equations are not accurate in this regime. One needs to adopt a different model of gravitation where it is not a Newtonian force and is instead a feature of curved space-time in order to obtain accurate predictions. You would be talking about a black hole.

 

[PeroK]

A simple answer:

In classical physics, the gravity of a large object could accelerate an object to beyond the speed of light.

But, if the mass of an object is great enough and as the velocity of an object approaches the speed of light then classical physics is no longer accurate. A new model of gravity and motion (General Relativity) is needed.

In this case, the speed of a falling object would never exceed the speed of light.

 

[Khashishi]

Then presumably, if someone dropped something from whatever the highest height

There's no highest height. There's gravity up into outer space and beyond, though it does get weaker with distance.

...if the object is more and more massive and the therefore the gravity more intense

No, the gravitational acceleration is the same for a feather or for a lead ball. The mass of the object doesn't matter, assuming it is negligible compared to the mass of the Earth.

...would then have enough time as it is falling to accelerate all the way up to the speed of light?

No. A falling object won't accelerate above the escape velocity (if it was dropped). Google that if you are unsure. Unless you are falling into a black hole, escape velocity is less than c. (The case of falling into a black hole is more complicated, and would require us to digress about coordinates.)

Though this one other thought occurs to me, if that can happen, then maybe somewhere beyond the observable universe there are super supermassive black holes that are doing just that to the distant galaxies, and that is what is causing the acceleration of the expansion all the galaxies out there.

No. Black holes and mass in general should be causing a contraction of the universe, not an expansion. Something else is going on. (We call it dark energy, but it isn't really understood.)

 

[benny_91]

Someone mentioned a formula for impact velocity which is same as escape velocity. So can't we adjust the values of the terms in the formula (such as mass of the pulling body and its radius) to get the impact velocity equal to the speed of light? I have heard that mass increases with speed but even then that would not change its acceleration which means the nature of motion will be same even at such high velocities. I think to apply Newtonian mechanics in this case is inappropriate. If we try to answer this it should be using Relativity.

 

[Dream Relics]

There's no highest height. There's gravity up into outer space and beyond, though it does get weaker with distance.

Ah' so does that mean if you wear in space. say at the orbital distance of our moon above the Earth and you let go of something, it would fall to the earth? Not on the moon of course, then it fall to the moon. But just out that far, and on the opposite side of the Earth from where the moon is so that the moon's gravity isn't affecting in a big enough way. But, then if you were out that far and threw the ball, the angular momentum might balance with the gravitational pull, and the object would orbit instead of falling to the earth. Right. But if you were in a jet airplane, or even up there sitting next to Felix Baumgartner as he was about to parachute from space, throwing a ball at the correct angle wouldn't send it into orbit, it maintain that momentum, but also fall. So by highest height, I guess I mean the point at which an object falls to earth, unless that point can be infinitely far away from the earth. but I don't think that is true, because as you said, the gravity gets weaker, so what I mean is that point where the gravity is still strong enough to pull something down and not just exert an influence. Does that make sense.

No, the gravitational acceleration is the same for a feather or for a lead ball. The mass of the object doesn't matter, assuming it is negligible compared to the mass of the Earth.

I didn't mean the mass of the falling object, I know that bit, I mean the mass of the object (planet, black hole) that our object is falling toward.

No. A falling object won't accelerate above the escape velocity (if it was dropped). Google that if you are unsure. Unless you are falling into a black hole, escape velocity is less than c. (The case of falling into a black hole is more complicated, and would require us to digress about coordinates.)

Okay, so you're saying that maximum impact velocity can never exceed the escape velocity? I don't know what you mean about coordinates.

No. Black holes and mass in general should be causing a contraction of the universe, not an expansion. Something else is going on. (We call it dark energy, but it isn't really understood.)

 

[Dream Relics]

I have to figure out how to do that point by point excerpt quotations that I see other people doing in replies. How does that work?

 

[Dream Relics]

The acceleration from gravity is only a approximately constant when your distance from the gravitating object is approximately constant. For everyday purposes, sitting where we do on the surface of the Earth, our distance from the center of the Earth is approximately constant. So we see the acceleration from gravity as approximately constant -- 9.8 m/s²

Gravity follows an inverse-square law. The farther away you get, the weaker the influence. Double the distance and you divide the force by four. Multply the distance by ten and you divide the force by one hundred. That law applies easily for point-like objects. For big objects, one can treat them as a collection of small pieces. But there is an important result -- Newton's spherical shell theorem. It turns out that for a spherically symmetric object like the earth, one can pretend that all the mass is concentrated in the center and the inverse square law will continue to work just fine (for objects that are not underground).

In mathematical terms, this becomes Newton's universal law of gravitation:

$F=G\frac{mM}{r^2}$

where F is the attractive force of gravity between you (mass m), the planet (mass M}, r is your distance from the center of the planet and G is Newton's universal gravitational constant. If you prefer, you can divide by your mass (m) to obtain:

$a = \frac{F}{m} = G\frac{M}{r^2}$

If you want to obtain the impact velocity when falling from a given height, that can be done. Start by taking the integral of gravitational force as r varies from your starting position (measured as a distance from the center of the planet) to its ending position at the surface of the planet. This is an integral of force times incremental distance. Force times distance is work -- a change in energy. This integral gives you the kinetic energy gained as the object falls. Evaluating the integral yields:

Impact Energy $= \frac{1}{2}mv^2 = GMm(\frac{1}{r} - \frac{1}{h})$

Impact velocity $= \sqrt{2GM(\frac{1}{r} - \frac{1}{h})}$

It turns out that the impact velocity falling from a point infinitely high up is still a finite value. Reversing the trajectory and playing it backwards, this impact velocity is the initial speed that you would need to escape to infinity. This is more commonly known as "escape velocity".Yes, one can have objects massive enough and compact enough in radius so that their escape velocity from the surface exceeds the speed of light. Yes, this means that no trajectory originating at the surface can escape to infinity. However, the above equations are not accurate in this regime. One needs to adopt a different model of gravitation where it is not a Newtonian force and is instead a feature of curved space-time in order to obtain accurate predictions. You would be talking about a black hole.

Well yes, I was talking about black hole, or at least I was talking about anything with enough mass, like a black hole, or something maybe with slightly less mass than a black hole but still a lot of mass. I guess the way I am visualizing this, which may be wrong, is a sort of analogy between the way the atmosphere of the Earth defines a line between the Earth and space. That in a similar fashion, we know that gravity decreases in strength as you move higher up away from the earth. That then there must be sort of an event horizon for gravity in the sense that the pull becomes weak enough at some distance to not pull something down the way it does for us here on earth. even the highest flying jets, and satellites can crash down to the earth, but where is the point where the gravity becomes weak enough that an object won't get pulled back to earth. And if there is such a point does the distance where that point occurs relative to the surface of the massive body creating the gravity increase the more massive the body?

 

[jtbell]

I have to figure out how to do that point by point excerpt quotations that I see other people doing in replies. How does that work?

Insert a [/quote] tag followed by a [quote] tag at the position where you want to put your comment, and put your comment in between them. I suggest you also hit return/enter after the new [/quote] and after your comment, to make it easier for you to see what you're doing. (The forum software will format it in its own way when you post it.)

I suggest you also delete the parts of the quoted material that you're not responding to directly. If people want to refresh their memory of the complete post, they can simply look at it up-thread.

 

[jbriggs444]

That then there must be sort of an event horizon for gravity in the sense that the pull becomes weak enough at some distance to not pull something down the way it does for us here on earth

There is no such boundary. Gravity becomes weaker with distance but never vanishes entirely.

 

[benny_91]

Well yes, I was talking about black hole, or at least I was talking about anything with enough mass, like a black hole, or something maybe with slightly less mass than a black hole but still a lot of mass. I guess the way I am visualizing this, which may be wrong, is a sort of analogy between the way the atmosphere of the Earth defines a line between the Earth and space. That in a similar fashion, we know that gravity decreases in strength as you move higher up away from the earth. That then there must be sort of an event horizon for gravity in the sense that the pull becomes weak enough at some distance to not pull something down the way it does for us here on earth. even the highest flying jets, and satellites can crash down to the earth, but where is the point where the gravity becomes weak enough that an object won't get pulled back to earth. And if there is such a point does the distance where that point occurs relative to the surface of the massive body creating the gravity increase the more massive the body?

There is no such point where gravitational field of a body becomes completely 0. We can only say that the distance of that point from the center of the body under consideration tends to infinity. However far we go from that point we will always experience a force that gets smaller and smaller in magnitude with distance but never becomes zero. This is similar to the notion that if we continuously divide a 1 meter line segment into unequal parts of finite length and then continue this same process with the smaller part we get after each division, we will never get to point where the length of the smaller part is 0. Always finite but tends to 0.

 

[Borg]

I have to figure out how to do that point by point excerpt quotations that I see other people doing in replies. How does that work?

There are two ways. Selecting the Reply link on the bottom right corner of a post will populate your post with a section like the one here.

If you left-click and select a section of a post, you will get an option like this:

Selecting that Reply option will quote just that section like this:

how to do that

 

[Dream Relics]

Insert a [/quote] tag followed by a [quote] tag at the position where you want to put your comment, and put your comment in between them. I suggest you also hit return/enter after the new

and after your comment, to make it easier for you to see what you're doing. (The forum software will format it in its own way when you post it.) [/quote] [/quote]

[/quote]his is a test. And thank you. Everyone is so nice on here and quick to respond. It is frustrating to be interested in physics as just a layman. There are so many moving parts. In fact I think they are all moving parts. You think you have a picture of something that is clear. And then you realize, nope. [/quote]

hhmm' when you say quote tag you don't mean """".

 

[Dream Relics]

Selecting that Reply option will quote just that section like this:

Ah, that method seems to work. Thanks.

 

[jtbell]

Yes, the select-and-popup menu is easier. If you want to quote more than one chunk, simply repeat, and they'll end up as separate quotes in the editing box. Then add/insert your text wherever appropriate.

That method became possible only after a forum software upgrade a couple of years ago, so I still tend to think of the "old fashioned" methods first. Especially late at night when my brain is half dead.

 

[jbriggs444]

and after your comment, to make it easier for you to see what you're doing. (The forum software will format it in its own way when you post it.) [/quote] [/quote]

[/quote]his is a test. And thank you. Everyone is so nice on here and quick to respond. It is frustrating to be interested in physics as just a layman. There are so many moving parts. In fact I think they are all moving parts. You think you have a picture of something that is clear. And then you realize, nope. [/quote]

The [quote] and [/quote] tags are used in pairs. One prior to the quoted text and one after. If you have extra [/quote] tags that do not match up, they are shown verbatim.

It is difficult to discuss tags such as [quote] and [/quote] because they are interpreted and not displayed. A way around this is to use the [plain] tag. However, I have not figured out a good way to render a [/plain].

Edit: *doh*. It's obvious. If you want to render a [/plain], do not prefix it with a [plain].

hhmm' when you say quote tag you don't mean """".

Yes, the word "tag" is used this way when talking about various markup languages like HTML.

 

[russ_watters]

Someone mentioned a formula for impact velocity which is same as escape velocity. So can't we adjust the values of the terms in the formula (such as mass of the pulling body and its radius) to get the impact velocity equal to the speed of light?

That's basically the definition of a black hole (event horizon). It's where the escape velocity equals the speed of light.

 

[Dream Relics]

There is no such point where gravitational field of a body becomes completely 0. We can only say that the distance of that point from the center of the body under consideration tends to infinity. However far we go from that point we will always experience a force that gets smaller and smaller in magnitude with distance but never becomes zero. This is similar to the notion that if we continuously divide a 1 meter line segment into unequal parts of finite length and then continue this same process with the smaller part we get after each division, we will never get to point where the length of the smaller part is 0. Always finite but tends to 0.

Sure I get that. Every body influencing every other body. So not zero. But there must be a point at which the effect becomes negligible. For instance. If you were on the international space station. and you wanted to fire a rocket from there toward the sun. Does that rock still need the same escape velocity that you would need leaving from the Earth's surface, or since you are already considerably removed from the center of Earth's gravity. Does that required velocity go down?

 

[jbriggs444]

Sure I get that. Every body influencing every other body. So not zero. But there must be a point at which the effect becomes negligible.

For any non-zero value you pick for "negligible", there is a point where the effect gets less than that and stays less than that. Yes.

But there is no agreed upon threshold for "negligible". No specific "horizon" beyond which gravity cannot penetrate.

For instance. If you were on the international space station. and you wanted to fire a rocket from there toward the sun. Does that rock still need the same escape velocity that you would need leaving from the Earth's surface, or since you are already considerably removed from the center of Earth's gravity. Does that required velocity go down?

The escape velocity from an object depends on your starting distance from the object, yes.

 

[nasu]

"Negligible" is a relative term. It may be negligible for some purpose and non-negligible for others.

The space station is about 400 km above surface. The radius of the Earth about 6400 km.
So it is not so far from the center, compared with the surface. The reduction in escape velocity is "negligible". But it is not zero, for sure.
You can see this easily if you use the formula.

 

[Dream Relics]

You can see this easily if you use the formula.

Oh yes, if I could use the formula. But currently, when I see the equations it is all meaningless gibberish. I don't know what the letters stand for at all.
Though I am getting there, maybe, very slowly from this nifty physics tutorial series I found on youtube. But I am slow at math, even simple multiplication requires me to stop and scratch my head and struggle to remember things like 6 x 8, and use my fingers. I am a stupid dummy head. But I want to understand the equations. Things like string theory, where they are saying the math points to more dimensions and stuff like that. It is all well and good to talk about, but it seems to me that talking about what math is telling you without actually speaking and knowing the language of math is like looking at sheet music but never hearing the symphony.

So I have just now learned about terminal velocity because of wind resistance. and apparently even in a vacuum there is a sort of terminal velocity because of the limits of the energy input and the way that acceleration causes an increase in mass. Which does pretty much answer my question about whether or not an object of mass could accelerate to the speed of light. Although maybe the jury is still out on that when referring to an object falling toward a black hole.

Though on being a dummy head. I got to say. I know this guy who is a math genius. He can do calculations in his head super fast for all kinds of things, and when he hears random numbers he can tell you stuff instantly about its factors or whether it's prime and all sorts of stuff like that. I was just talking to him this morning, and I was in fact suggesting that he check out this forum. Well during that conversation I learned that he has arrived, through his own calculations and deductions at some conclusions that sound very crack pot. Namely that the Earth is flat, the solar system is not heliocentric. That there is fact no south pole and huge government conspiracies suppress that knowledge. (that is one I never heard before) etc. I was very surprised by this. and of course my knowledge base is way too feeble to argue the points with him. Though that is the problem with transmitted knowledge and information. Having never directly proved most things personally I can neither confirm or deny most things except to trust in the reliability of the collective mass of people who do and have tested and verified. But I guess that gets into an issue of epistemology.

I am very intrigued right now also by something I just learned about the difference between pure mathematics and formulas that are used to describe and calculate physical things. Math can be used to describe reality. But apparently there is a lot of math going on that may just be math and has no connection to the physical world. Sort of like the same way words can tell stories that are not true, like fantasy novels. I find this notion unsettling, and I wonder if perhaps the case is actually more like one where. math being the language of reality, all math does describe something in nature, we just don't alway know what yet.

 

[benny_91]

Oh yes, if I could use the formula. But currently, when I see the equations it is all meaningless gibberish. I don't know what the letters stand for at all.
Though I am getting there, maybe, very slowly from this nifty physics tutorial series I found on youtube. But I am slow at math, even simple multiplication requires me to stop and scratch my head and struggle to remember things like 6 x 8, and use my fingers. I am a stupid dummy head. But I want to understand the equations. Things like string theory, where they are saying the math points to more dimensions and stuff like that. It is all well and good to talk about, but it seems to me that talking about what math is telling you without actually speaking and knowing the language of math is like looking at sheet music but never hearing the symphony.

So I have just now learned about terminal velocity because of wind resistance. and apparently even in a vacuum there is a sort of terminal velocity because of the limits of the energy input and the way that acceleration causes an increase in mass. Which does pretty much answer my question about whether or not an object of mass could accelerate to the speed of light. Although maybe the jury is still out on that when referring to an object falling toward a black hole.

Though on being a dummy head. I got to say. I know this guy who is a math genius. He can do calculations in his head super fast for all kinds of things, and when he hears random numbers he can tell you stuff instantly about its factors or whether it's prime and all sorts of stuff like that. I was just talking to him this morning, and I was in fact suggesting that he check out this forum. Well during that conversation I learned that he has arrived, through his own calculations and deductions at some conclusions that sound very crack pot. Namely that the Earth is flat, the solar system is not heliocentric. That there is fact no south pole and huge government conspiracies suppress that knowledge. (that is one I never heard before) etc. I was very surprised by this. and of course my knowledge base is way too feeble to argue the points with him. Though that is the problem with transmitted knowledge and information. Having never directly proved most things personally I can neither confirm or deny most things except to trust in the reliability of the collective mass of people who do and have tested and verified. But I guess that gets into an issue of epistemology.

I am very intrigued right now also by something I just learned about the difference between pure mathematics and formulas that are used to describe and calculate physical things. Math can be used to describe reality. But apparently there is a lot of math going on that may just be math and has no connection to the physical world. Sort of like the same way words can tell stories that are not true, like fantasy novels. I find this notion unsettling, and I wonder if perhaps the case is actually more like one where. math being the language of reality, all math does describe something in nature, we just don't alway know what yet.

Yes I agree. For instance you have an entire theory on 4 dimensional geometry (I found it once in wikipedia) but such a geometry is difficult to even imagine. May be it shows something in the physical world which we are still unfamiliar with. But I would love to meet that guy who can mathematically prove that the Earth is flat!

 

[jbriggs444]

Oh yes, if I could use the formula. But currently, when I see the equations it is all meaningless gibberish. I don't know what the letters stand for at all.

That's why I put the meaning of all the letters in the post where I gave the formula. If you still do not know what they mean then you should ask.

 

[Jenab2]

Jbriggs444 identified the Vis Viva equation as the key to understanding the classical dynamics of freefall for initially separated point masses. I'll go a little further.

Two point masses, M and m, are initially at rest in vacuum and separated by a distance, d. They are isolated from all forces except their mutual gravity. Find the time elapsed beginning when the separation is d/2 and ending when the separation is d/3.

I set up the plunge orbit problem by starting with the Vis Viva equation, solved for the speed in orbit:

v = √[2G(M+m)(1/r−1/d)]

Where d is the apoapsis distance and r is the current distance. Since all motion in a plunge orbit is radial, we can replace v by the derivative form.

dr/dt = √[2G(M+m)(1/r−1/d)]

That's an ordinary non-linear differential equation with variables separable. We then set up the integration.

∫ dt = ∫ dr / √[2G(M+m)(1/r−1/d)]

The two key substitutions to make in this instance are

a² = −1/d
x² = −1/r

Notice that

2x dx = dr/r²
dr = (2/x³) dx

Our integral becomes

∫ dt = √{2/[G(M+m)} ∫ dx / [x³√(a²−x²)]

I recognize the form of the integral, and I know that the solution is

∫ dx / [x³√(a²−x²)] = −√(a²−x²)/(2a²x²) − [1/(2a³)] ln{[a + √(a²−x²)]/x}

So,

√[G(M+m)/2] ∫ dt = −√(a²−x²)/(2a²x²) − [1/(2a³)] ln{[a + √(a²−x²)]/x}

And I note that

a²−x² = 1/r−1/d
1/(a²x²) = rd
x = i/√r
a = i/√d
1/a³ = i d√d

Substitution and factorization gives

∫ dt = √{d/[2G(M+m)]} { √(rd−r²) − id ln{[√(r/d) − i √(1−r/d)]} }

Euler's Theorem describes a complex relationship between the trigonometric functions and the natural logarithms. In particular,

arctan{√[(1−x)/x]} = i ln[√x − i √(1−x)]

Which means

t(r) = tᵣ−t₀

t(r) = {d/[2G(M+m)]} { √(rd−r²) + d arctan[√(d/r−1)] }

That's the exact classical solution for the time to fall from rest in a plunge orbit.

Putting in the limits,

t₁−t₀ = t(d/2)
t(d/2) = (1/2 + π/4) √{d³/[2G(M+m)]}

t₂−t₀ = t(d/3)
t(d/3) = {√(2)/3 + arctan[√(2)]} √{d³/[2G(M+m)]}

t₂−t₁ = t(d/2)−t(d/3) = (t₂−t₀)−(t₁−t₀)
t(d/2)−t(d/3) = {√(2)/3 + arctan[√(2)] − 1/2 − π/4} √{d³/[2G(M+m)]}

t₂−t₁ ≈ 0.141322975518 √{d³/[2G(M+m)]}

Suppose that God (or whatever) stopped the moon's transverse orbital motion, causing Earth and the moon to be mutually at rest and separated by a distance, d, for an instant. How long would it take the moon to fall from d/2 to d/3?

G = 6.67384e-11 m³ kg⁻¹ sec⁻²
M = 5.9722e24 kg = Earth's mass
m = 7.3477e22 kg = Moon's mass
d = 3.84402e8 m = radius of moon's orbit

Then, using the equation above, t₂−t₁ = 37494.35 seconds

Let's check that by using a time-step simulation with the acceleration calculated for each small interval of time. The program that I wrote to do that is in Python 3.4, shown below.

GM = 4.034788e14 # Gravitational parameter for the total mass
dt = 0.01 # one-hundredth second increments in time
d = 3.84402e8
t = 0
v = 0
r = d
while r > d/2:
____t = t + dt
____a = −GM/(r*r)
____r = r + v*dt + a*dt*dt/2
____v = v + a*dt
t1 = t
t = 0
v = 0
r = d
while r > d/3:
____t = t + dt
____a = −GM/(r*r)
____r = r + v*dt + a*dt*dt/2
____v = v + a*dt
t2 = t
T = t2 − t1

When we run that code, we get

T = 37494.35

Just as before.