comparing units of acceleration

[bwanaaa]

suppose i tell you that my velocity is increasing at 2 km/sec for every kilometer traveled. what is the formula describing my acceleration in km/sec^2?

Assume:
V velocity at end of time interval t
Vo initial velocity
D distance traveled
K constant (in this case 2 km/sec)

is the following equation correct?
V=Vo + K*D
(i dont like the above equation because K*D does not have units of velocity)

anyway, going forward
D~Vaverage*t and Vaverage=(V+Vo)/2
so D~(V+Vo)*t/2
so substituting into the original equation:
V = Vo + KVt/2 + KVot/2
V - KVt/2 = Vo + KVot/2
V (1 - Kt/2) = Vo (1 + Kt/2)
V = Vo (1 + Kt/2) / (1 - Kt/2)
one then has to differentiate the above expression against time to get acceleration.

of course the flaw is the part where i approximate D~Vavg*t. I really should use an integral there but it's been 20 years since i've done that. i am sure this has been done before but i cant find a link on the net. Can anyone here point me in the right direction?

 

[esun]

I like your thinking. The equation:

v = v0 + k*d

Where k = 2 km/sec/km (since the velocity is increasing at a rate of 2 km/sec/km) and d is in km. So k*d is in units of velocity.

Then we have to find d, which requires integrating the velocity up the current time. So d = int(v dt) from 0 to t, the point of time in question.

v = v0 + k * int(v dt)

If we differentiate both sides, we get:

dv/dt = k * v

Standard differential equation. Let's solve:

dv / v = k dt

ln(v) = k t + C

v = e^C e^(k t)

v = K e^(k t)

We know that v(t = 0) = v0 = K, meaning

v = v0 e^(k t)

a = dv/dt = k v0 e^(k t)

Anyway, I think that's right.

EDIT: Sorry, made some dumb math mistakes before.

 

[firewolfsm]

I lost you at differentiation...

Why can't

V = Vo (1 + Kt/2) / (1 - Kt/2)

work as the final equation?

 

[esun]

Well you said it yourself, the way you computed the average velocity is invalid. You must use calculus.

Here's a good way to see the solution I wrote (which I've fixed since you first looked at it probably) is correct. Just plot v(t) and x(t). See if v(t) increases by 2 when x(t) increases by 1. We can also check this analytically.

For example, let v0 = 1 for simplicity and k = 2 by our assumptions. We have:

v(t) = e^(2t)
x(t) = int(v(t) dt) = 1/2 e^(2t)

Let t1 and t2 be times between which x(t) increases by 1. Let's say we start at some x0 at time t1. That means at time t2, we should be at position x0 + 1. Let's solve for t1 and t2 in terms of x0.

x(t1) = x0 = 1/2 e^(2 t1)
t1 = 1/2 ln(2 x0)
x(t2) = x0 + 1 = 1/2 e^(2 t2)
t2 = 1/2 ln(2 x0 + 2)

Now let's plug in to v(t) and see if v(t2) - v(t1) = 2 (generally, it should equal k).

v(t2) = e^(2 t2) = e^[ln (2 x0 + 2)] = 2 x0 + 2
v(t1) = e^(2 t1) = e^[ln (2 x0)] = 2 x0
v(t2) - v(t1) = 2 x0 + 2 - 2 x0 = 2.

You can see that replacing 2 with k will produce the more general result.

If this isn't making sense, you'll need to brush up on your calculus. Check your own result and see if it satisfies this condition. It probably doesn't in general.

 

[bwanaaa]

thank you esun for that. you know; it was the thought of the calculus that was intimidating, not the actual doing of it-like going into a dark room that you havent been in for many years. The derivation of the naperian log should have conked me on the head when i saw that acceleration was proportional to velocity-but of course i wasnt seeing that-i was seeing velocity proportional to distance.

Now you might ask why I am asking about this. Well , it has nothing to do with real life and will not bring anyone any money. But I was reading about the hubble constant. You see in astronomy, the light has been very carefully dissected and observed. Instead of a smooth band of colors, each element when excited emits light at a specific set of frequencies, These 'spectral lines' have very specific frequencies based on what is emitting the light, the excited orbital and the ground state orbital that the electrons are jumping down to when they emit the photon. For example, when hydrogen burns, one of its spectral lines is light with a wavelength,656.2852 nm. If such a line is seen from a star very far away, we know it's hydrogen. but the measurements are off a little bit, the wavelengths are longer sometimes. This has been interpreted as a doppler shift resulting from the expansion of the universe. And expansion not in the sense of stars moving away from each other with momentum relative to each other, but rather stars separating because the entire universe is expanding-like raisins separating in a loaf of baking bread. NowI dont know how they came to this fine distinction, but they have measured the doppler shift for many stars and the farther ones seem to shift more. This has been interpreted to mean that the expanson of the universe is accelerating. This expansion rate has been measured at 72 plus/minus 8 km per sec per MegaParsec. A megaparsec is 3.088x10^19 km. you see where i am going with this.

the constant k in the differential equation above is 72/3.08x10^19. i tried to graph that and you get pretty much a flat line. i has hoping in some sort of stupid simplicity that gravity could be explained this way but the numbers dont work. i was imagining that gravity is the result of our inertia in an expanding universe. but the acceleration of space expansion is not nearly enough. i dont think, but let's try.
a=gravity is 9.8 m/s^2
k=our constant is 72/3.08x10^19=2.33x10^-18 /sec
Vo i dont know
t = age of the universe=15 billion years is 4.7x10^16 seconds
K*t=.109
so e^K*t is a little greater than 1
.9.8=k*Vo*1.1
but k is an impossibly small number and Vo would therefore have to be impossibly large-way greater than the speed of light 2.9x10^9 m/s

the only way to get this to work is if our universe is a lot older than we think it is.



PS: btw, i didnt 'say it myself', that was firewolf. but i accept the rebuke as we are all human and responsible for each other.

 

[esun]

Heh, sorry, didn't realize it was someone different that responded. Glad I could help, though.

 

[PolymerTim]

I'm not sure I understand your application. Were you trying to see if the motion of the earth could be responsible for gravity? The problem with that is that the motion has a direction, so there would be opposite gravity on the opposite side of the earth and not in any way related to the center of mass. Oh yeah, and gravity on the moon would fluctuate widely as it slingshots around the earth. There's a lot of simple observations that make this not possible. Maybe I misunderstood where you're going with it though.

 

[bwanaaa]

indeed if the the universe were expanding in the classical sense-with all the planets and stars flying away from each other-then it makes no sense. but that is not the expansion we are talking about. the universe is expanding because every subatomic particle is getting bigger. one of the websites had an interesting movie:
http://brahms.phy.vanderbilt.e.../astromovies/dots.html

andthis shows why there is cosmological redshift:
http://brahms.phy.vanderbilt.e...stromovies/cosmoz2.avi


the analogy of raisins in baking bread has also been used. but i have a hard time believing that space is expanding without the stuff in it expanding also. to extend the analogy, as the bread bakes, i expect the raisins to swell as well. if you are standing on a planet and you are both expanding, then i would expect your inertia to resist that change. no matter where you stand, you feel the expansion as gravity. and if you are in the sky, your expansion and the earth's expansion will eventually cause you to collide-which you perceive as falling.

since the expansion of the universe is observable as redshift, i am trying to correlate the gravitational constant with the observed and documented accelerating expansion of the universe. now go write it up and win a nobel.

and btw, if the universe is expanding, doesnt that address the galactic rotation curve anomaly?
http://en.wikipedia.org/wiki/Galaxy_rotation_curve
in other words, Keplers laws say that the farther out you get from the center of the galaxy, the slower you should orbit(assuming most of the mass is at the center) But what if the universe is expanding-doesnt the expansion velocity add into the observed galactic rotation velocity and so you get a flatter curve.

 

[firewolfsm]

bwanaa...what the hell?

I've NEVER heard that idea before, no one has even proposed that all particles are expanding. We can prove the universe is expanding in the classical sense because galaxies are all moving away from each other. The size of the galaxies relative to the distance between them is changing. I don't think your idea works at all.

And also, the further you are from a rotating galaxy, the faster you move, because you have to cover more distance (larger elipse) in the same amount of time. There were paradoxes that if you got far enough, you'd travel faster than light but it was shown that those bodies broke off of the orbit before they hit that speed.

To get back on topic, he was just trying to get a formula for acceleration per distance traveled, as opposed to time.

 

[bwanaaa]

firewolf, please check out the links above. the universe is known to be expanding because of redshift. this doppler effect can result from many possible scenarios, not just the classical motion of emitter and detector separating. the wavelength of light can increase if the space between 2 bodies increases without them moving in their own reference frame. raisins in bread that's baking-the raisins have zero velocity relative to the bread but are moving apart.

regarding your comments about galctic rotation, please look up Vera Rubin. I find it interesting that given the awesome scarcity of women in science in the past, how many of the greatest discoveries of the physical world were made by them-Madame Curie, Rosalind Franklin, Lise Meitner,etc. I also have a logical proof that God is a woman but that is a different thread.

 

[firewolfsm]

redshift means the wave length is getting lower, simply because the emitter is moving away from us, the signal is stretched. The raisins are the same size as ever.

 

[PolymerTim]

Wow, that is an interesting mind experiment. But have you really tested that theory against basic observations?

First let me see if I understand you correctly. You think that the force we feel as gravity is really caused by the expansion of all matter, even at subatomic levels so that essentially, the earth and my body are expanding against each other in an accelerating fashion and I feel the force due to inertia.

If that was true, then how do you explain gravitational effects through space without a physical connection? For instance, the moon orbiting the earth. If we assume a symmetric expansion of all mass such that the center of mass of any object is not changing, then why wouldn't the moon and earth expand into each other? I guess you could say that this same expansion causes the moon to spiral in an outward direction maintaining a constant apparent distance, but this still wouldn't account for the measurable forces between the earth and the moon such as tidal forces.

It just doesn't seem to fit well.

 

[firewolfsm]

oh...one more thing, how did you get from

V - KVt/2 = Vo + KVot/2

to

V (1 - Kt/2) = Vo (1 + Kt/2)

?

edit: also, I read your first explanation, you assume the expansion is accelerating, well...from what I read, it is, but the force driving expansion isn't growing or anything.

Think about the bread rising, the raisins in the middle move away from each other because the space between them expands, but the raisins on the outside move away from each other faster because of the distance between them.

 

[Mday]

Originally posted by: firewolfsm
oh...one more thing, how did you get from

V - KVt/2 = Vo + KVot/2

to

V (1 - Kt/2) = Vo (1 + Kt/2)

?

By factoring out V on the left, and Vo on the right

 

[firewolfsm]

oh yes....

that would seem obvious...

 

[bwanaaa]

Originally posted by: PolymerTim
...

I guess you could say that this same expansion causes the moon to spiral in an outward direction maintaining a constant apparent distance, but this still wouldn't account for the measurable forces between the earth and the moon such as tidal forces.

It just doesn't seem to fit well.

help me understand what you mean by tidal forces. as i interpret this concept , it is a weaker force exerted on the earth by the far side of the moon than the near side. likewise the earth, so that the water on earth facing the moon is pulled towards the moon more than the earth beneath it. And that sea bed beneath that water is pulled toward the moon more than the far side of the earth. This results in the 'bulges' we perceive as tides.

now, going back to the original observation of the expansion of the universe as measured by the cosmological redshift-that is a measurement of velocity (72km/sec) that increases the more 2 bodies separate (per megaparsec). so, the velocity of the far side of the moon relative to the far side of the earth is greater than the near side of the moon to the near side of the earth. this difference in velocities should result in tides or even ripping a planet apart -and that's what happens sometimes when tidal forces are too strong relative to the mass of the object..

the appeal of this model is that it is entirely geometric. gravitational mass and inertial mass are related through motion. the problem with this model is that it is only geometric. if jupiter were magically and instantaneously converted from a gas giant to something much denser, then this theory would not explain why the gravity pulling on jupiter's moons would be different. You would have to complicate the model by making a second assumption: that the inertia of an object increases as the presence of inertial mass near it increases.

@firewolf
i dont believe that the raisins do not enlarge as the bread rises because the raisins and the bread represent components of the universe that are actually made of the same stuff. do you really believe that the universe is expanding by radial motion of everything away from each other through space? why then does redshift increase at greater distances? would you posit that the movement is accelerating? if that is the case then redshift should be different when we look in one direction or another-look 'outwards' where the velocities would be expected to be ever increasing gives what we see-but 'looking inwards' should give smaller redshifts. and looking sideways should give no redshift. but i dont think those things have been obseved.

 

[PolymerTim]

Originally posted by: bwanaaa
help me understand what you mean by tidal forces. as i interpret this concept , it is a weaker force exerted on the earth by the far side of the moon than the near side. likewise the earth, so that the water on earth facing the moon is pulled towards the moon more than the earth beneath it. And that sea bed beneath that water is pulled toward the moon more than the far side of the earth. This results in the 'bulges' we perceive as tides.

Well, thats close but not quite how I read tidal forces. Looking at the effect of the moon on the earth, you can consider the moon as a point source of mass. If you accept traditional gravitational theory, the earth and moon are orbiting each other rotating about the center of mass of the system. The water on the earth that is closer to the moon is closer to the center of this rotation and thus feels much less acceleration relative to the far side of the earth. This low acceleration is not cancelled by the gravity of the moon (this would occur at the center of mass of the earth) and so water is attracted to this point closer to the moon.

Originally posted by: bwanaaa
now, going back to the original observation of the expansion of the universe as measured by the cosmological redshift-that is a measurement of velocity (72km/sec) that increases the more 2 bodies separate (per megaparsec). so, the velocity of the far side of the moon relative to the far side of the earth is greater than the near side of the moon to the near side of the earth. this difference in velocities should result in tides or even ripping a planet apart -and that's what happens sometimes when tidal forces are too strong relative to the mass of the object..

I fail to see why the different velocities of two different objects that are not mechanically coupled results in any kind of force. Additionally, I don't see how your theory explains objects in orbit at all. It takes a force to cause an object in linear motion to be accelerated into a circular path. Your attempt to relate gravity to a radial acceleration leaves no room for these types of forces between objects that are not mechanically coupled.

Originally posted by: bwanaaa
@firewolf
i dont believe that the raisins do not enlarge as the bread rises because the raisins and the bread represent components of the universe that are actually made of the same stuff. do you really believe that the universe is expanding by radial motion of everything away from each other through space? why then does redshift increase at greater distances? would you posit that the movement is accelerating? if that is the case then redshift should be different when we look in one direction or another-look 'outwards' where the velocities would be expected to be ever increasing gives what we see-but 'looking inwards' should give smaller redshifts. and looking sideways should give no redshift. but i dont think those things have been obseved.

Your own links from vanderbilt explained this very well. In a uniform, isotropic expansion, any point you choose as your reference point will see all other points moving away at the same rate, no matter what direction you look in. Also, points farther from you will appear to be moving faster (more redshift) simply because they are farther from you and are affected by the uniform expansion over a larger distance (again with no effect in direction). If I understand correctly, the redshift does not tell us about acceleration, only relative velocity between the two points.

Originally posted by: bwanaaa
indeed if the the universe were expanding in the classical sense-with all the planets and stars flying away from each other-then it makes no sense. but that is not the expansion we are talking about. the universe is expanding because every subatomic particle is getting bigger...
... the analogy of raisins in baking bread has also been used. but i have a hard time believing that space is expanding without the stuff in it expanding also.

It is quite common in nature to findle multiple, overlapping forces that have different magnitudes at different length scales. Just because galaxies within the universe have been shown to be expanding away from each other in an accelerated fashion, does not mean that the responsible force is dominant on all length scales. I see no reason to assume that just because the galaxies are expanding from each other means that planets are expanding out of their solar system, or my body or cells or atoms are expanding at all, much less at the same rate. This seams to be the primary basis for your new theory and I'm not sure where you came up with it.

 

[bwanaaa]

Originally posted by: PolymerTim
I fail to see why the different velocities of two different objects that are not mechanically coupled results in any kind of force. Additionally, I don't see how your theory explains objects in orbit at all. It takes a force to cause an object in linear motion to be accelerated into a circular path. Your attempt to relate gravity to a radial acceleration leaves no room for these types of forces between objects that are not mechanically coupled.

Consider a person who wants to stay 100 miles above the earth's surface. According to the crazy model, because everything is expanding, that person will perceive he is falling towards earth. To maintain his altitude, he will have to move away from earth as he and the earth expand. A motive force has to be applied. If the spaceman is to remain stationary, then the force is directed away from the earth-the spaceman is moving away at the same speed that the universe is expanding. If the spaceman is to orbit, the force is slanted away from the perpendicular-and the spaceman's motion spirals outward. His altitude relative to the earth remains constant as the universe expands.

The problem i have is the following: consider a person in a box. If the box is expanding in all directions, the wall thickness should be increasing. as these objects are concentric, wont they eventually collide? I mean the inside of the box should be getting smaller and the person is getting bigger right?

It is quite common in nature to findle multiple, overlapping forces that have different magnitudes at different length scales. Just because galaxies within the universe have been shown to be expanding away from each other in an accelerated fashion, does not mean that the responsible force is dominant on all length scales. I see no reason to assume that just because the galaxies are expanding from each other means that planets are expanding out of their solar system, or my body or cells or atoms are expanding at all, much less at the same rate. This seams to be the primary basis for your new theory and I'm not sure where you came up with it.

you are right in deducing how i came up with this idea. if we have data that says the universe is expanding, then it must be valid for all points in the universe. Otherwise we would see inhomogeneities. i was trying to make the numbers fit--correlate gravity which is the derivative of velocity with respect to time ( D(v)/D(t) ) with the cosmological constant which is the derivative of velocity with respect to distance ( D(v)/ D(d) ). Before you shoot me, please recognize that i am not speaking of gravitational force (newtons) but rather of gravity (units of acceleration).

 

[PolymerTim]

I can see how you're trying to make a connection between two things that have the same units. But your theory tries to do a lot more than that.

Why should there be a perceived falling? If all matter is expanding homogeneously as you put it, then the distance between the spaceman and the earth will be growing just as fast as the two are expanding. By definition the two should cancel each other out and there will be no perceived change in position. This is the same reasoning that gets you in trouble with the man in the box problem. You recognize that the box walls are getting thicker and that the man is getting larger, but fail to remember that dimensions of the box are also growing at the same rate. Again, there would be know perceived change in size. I'm getting lost in this myself, but it seems to me that every time you come up with a force, it is due to the lack of one something expanding (which should be expanding by your theory).

What do you mean by a stationary spaceman? While an object can be stationary relative to the surface of the earth (as it rotates) I don't believe it is possible for an object to sit in orbit without "orbiting". It would simply fall to the ground.

Originally posted by: bwanaaa
if we have data that says the universe is expanding, then it must be valid for all points in the universe.

This is something I will have to disagree with, at least if you mean at all length scales. Lets make an example. I'm going to make a rubber sheet and embed metal spheres in it. Now I'm going to stretch the sheet biaxially and watch the spheres expand relative to each other. But are the spheres getting any larger? Of course not. Why? The force of expansion is being transferred to the metal spheres through the rubber sheet, but there is another force holding the atoms of the sphere together more tightly than the expansion force can overcome and thus there is insignificant deformation of the metal spheres. My point is that there is no reason to assume that the expansion is homogenous. And I would argue that we do see inhomogeneities at different length scales.

 

[bwanaaa]

Originally posted by: PolymerTim
I can see how you're trying to make a connection between two things that have the same units. But your theory tries to do a lot more than that.

Why should there be a perceived falling? If all matter is expanding homogeneously as you put it, then the distance between the spaceman and the earth will be growing just as fast as the two are expanding. By definition the two should cancel each other out and there will be no perceived change in position.

you're right. i am not high or drunk, just blinded by an attempt to make an equivalence that probably doesnt exist. we know that space is expanding at 72 km/sec per megaparsec. i am trying to work that fact into the observations that we call gravity. when 2 things are close, then the space they are in should be expanding relatively homogeneously-so, yes, everything cancels out to the accuracy that we can observe in our neighborhood. but i suppose over great distances, we can start to see differences that are not measurable locally.

...What do you mean by a stationary spaceman? While an object can be stationary relative to the surface of the earth (as it rotates) I don't believe it is possible for an object to sit in orbit without "orbiting". It would simply fall to the ground.

if a rocket took off from the equator, it is imparted with 2 velocities, 1 radial -out and away from the earth, and the other the speed that it had from the earth's rotation-horizontal. if you fix your position by taking a star sight and then fire retro rockets to stop the orbital velocity you are left with a unidirectional motion away from the earth. to stay at that altitude, you need to continue to apply some thrust to keep from falling back to earth. In the crazy model, the thrust is need to keep "running away from an expanding earth". One could try an cheat and say that empty space isnt expanding because there is no stuff to expand. The apparent distance between two stationary bodies in space should then decrease because the bodies are larger. But the distance between their centers of mass would actually increase because they are growing.

... And I would argue that we do see inhomogeneities at different length scales.

yes i recognized that above. so there are 2 points that the crazy model fails at-
1) different densities are not accounted for - a planet of uranium and an identically sized planet of cork exert different gravitational fields
2) topologic inconsistency exists. if matter expansion is used to account for things falling closer, and that is observable to us as an apple falls from a tree, then all empty space should be obliterated in our neighborhood-an apple in abox will expand to fill it, the branches of a tree will merge, the pores of a sponge should shrink, etc,


but you know, i still havent converted the cosmological constant of expansion to units of acceleration. off to the notebook to do my homework. if i had crunched the numbers before i opened my mouth, i might have discovered that the contribution of the universe's expansion to gravity measurements is trivial.

 

[firewolfsm]

Look...you're an intelligent guy, so stop wasting your time on this rediculous theory, you KNOW it's flawed and it can't work.

You, one guy, are actually trying to say that everything every physicist since Einstein has believed is wrong.