Is This a Valid Solution to Newton's Equation for Gravitational Attraction?

[Lucw]

Hello.

Sorry for my English.
I have found on the net a solution of Newton equation. And I just want to have your view.
The speed between two mobiles moving towards each other is: (only with gravitation)

dx/dt = - square root (2.G.(m1+m2).(1/X0 -1/X))

Where X0 is the distance between the two mobiles when they are released without initial speed.
And X the distance where we measure the speed. (X0>X).

A solution is: (SR is square root)
t=SR (X0³/(2.G.(m1+m2))).(X/X0.SR(X0/X-1)+arctg(SR(X0/X-1)))

The author of the solution poses: u = SR(X0/X-1).
Thank you for your opinion.

Lucw

 

[nomadreid]

What kind of comments would you like?
It would help if you would supply the link to which you are referring.
You can look at post #8 on
https://www.physicsforums.com/threads/remarkably-difficult-Newtonian-problem.360987/#post-2497264
where v appears in the course of the solution.

 

[Lucw]

Oh. Thank you very much.
I just come from a french forum. But i will read carefully the reference you give.
It is the first time that i see a solution of Newton's equation in the form:
t = f(m1, m2, X0 and X).
And if X tends towards 0, it is the law of Kepler, to a whole factor close.
Best Regard

Lucw

 

[berkeman]

Oh. Thank you very much.
I just come from a french forum. But i will read carefully the reference you give.

Welcome to the PF.

You can find a tutorial for LaTeX math equations here: https://www.physicsforums.com/help/latexhelp/

Using LaTeX makes it much easier to post and read mathematical equations.

 

[Lucw]

Hello.

I have made some calculations with the formula.
For distances from 10 meters to 2 meters, the two following mass take as duration:
1 and 10 kg: 345.5 hours
2 and 10 kg: 330.8 hours.
it seems to me, therefore, that a heavy body falls faster than a light body towards a body of the same mass. In terms of duration.
This contradicts Galileo's reasoning. And gives reason to Aristotle.
But, for the same distances.
1 and 9 kg: 362.4 hours
2 and 8 kg: 362.4 hours
5 and 5 kg: 362.4 hours.
Galileo's reasoning is correct, provided that the object that falls comes from the Earth ...

Thanks for your opinion.

Luxw

 

[nomadreid]

First, I presume you are talking about the time which two masses, starting at rest (i.e. both having the same velocity) at 10 meters from one another, without other influences (e.g., off in space away from heavy bodies etc), take to gravitate towards each other until they are 2 meters away from one another. Yes, they will have different times according to the distribution of mass, which shouldn't be surprising: an object falling on the moon and an object falling from the same height on the Earth will take two different times to get to the ground. Galileo and Aristotle were dealing with pairs of objects, one of which was the very large Earth and comparatively small objects: a more precise statement than "both will take the same time (ignoring air resistance) to get to the earth" would be "the differences in the times which the two objects will get to the Earth will not differ significantly (again, ignoring air resistance)." That is, the terms m₁ + mass₂ in your equation will be almost the same in m_Earth+mass_apple versus m_Earth+mass_cannonball, but the difference between 10kg+1kg and 10kg+2kg is significant. By the way, a nice way to calculate it is given in the first answer in https://physics.stackexchange.com/q...-masses-will-collide-due-to-Newtonian-gravity. There, the integration is done from r₀ to 0, but when you do the integration from r₀ to r₁ , you just get r₀^3/2-r₁^3/2 instead of just r₀^3/2 in the numerator of the result.

 

[Lucw]

Hello Nomadreid.

I have used the formule you give to me.
This does not give the same results as with the formule i have given.
I have made numérical approximation. With computer...
Divide the distance into 10000 parts. Calculate the speed in each point. Calculate the average speed for each part. Calculate the duration for each part ... And make the sum ...

But i come back with the raisoning of Galileo.
He takes a small piece, m1, on the ground (from Earth). And let it fall (from Pisa tower).(Without atmosphere).
It lasts X seconds.
Do not forget that the Earth's mass is now: M-m1.
Than he takes a big piece, m2, and let it fall in the same conditions. (Earth's mass is now: M-m2...)
It lasts also X seconds. But Galileo do not know that the durations are the same. He has not Newton's laws...
And in his day, it was thought that heavy bodies would fall faster ...
Galileo, in his mind, glues the two objects, m1 and m2.
And he reasons thus:
The mass m1,if it falls less quickly,must slow down the mass m2. Mass m2 which must accelerate the mass m1.
We must have a duration between that of the mass m1 and that of the mass m2.
But the whole m1 + m2 is heavier than m2 and should fall faster than m2. His conclusion is that all bodies fall in the same way ...
But the mass m1 + m2 has a duration of fall also of X seconds, since it comes from the ground ...

You have spoken of the moon.
Imagine that Galileo take a part of the Earth the same mass that the moon is.
How many time it takes to fall from Pisa tower (58 m...): 3.365 seconds...
And if Galileo take the moon on Pisa tower: 3.344 seconds...

These are of course theoretical calculations.
So we have the following paradox:
Aristotle and Galileo are rigth.

Thanks for your opinion.

Lucw

 

[nomadreid]

I am presuming that the difference between the calculation from your numerical method and the result of the integral used in the formula in the link I provided is more significant that allowed for this numerical method. The way you wrote your formula is a little ambiguous: from your manner of writing it, I think you wrote

If this is wrong, please correct it. It would help if you showed your formulas upon which you are basing your calculations in a readable way, because if you are getting two different answers, then obviously something is wrong., then rather than make the calculations too tedious to follow, you can do one of three things:
(1) write them in Latex
(2) scan them and upload them using the "upload" button, or
(3) provide a link to the site where you got the formulas from. (Note: I am fluent in French, so there is no problem if you provide a link to a French website.) [However, please don't provide your calculations in a computer program, as I am less fluent in programming languages.]
Going to more subjective considerations: yes, Galileo was correct in what he was saying (and Aristotle was wrong), but he was handling a different situation than the one that you posit. Both your formula and the newer one contains the sum of the masses. Galileo was using the sum of the two masses to be constant, so that the result for (taking one of the two formulas, say

for convenience until we check what is going on with your formula) will be the same in both cases. However if the sum of the masses is different, as in the situation you stated, obviously the answers will also be different (in either formula).

 

[Lucw]

Hello.

Oups. This is the rigth formule...
You can find here how the calculation is made
Thank you for your attention.

Lucw.

 

[nomadreid]

Thank you. I shall work through it later. In the meantime, for anyone else who is reading this, the appropriate link is
http://forums.futura-sciences.com/physique/814010-galilee-experience-de-pensee-chute-corps-7.html
where you have to scroll down to the post by Archi3 at 17/03/2018 at 16h33. (Unlike Physics Forums, this forum doesn't have the good idea to number the posts )

 

[nomadreid]

OK, back for a few minutes, so a quick observation: since

that is, the formula from http://forums.futura-sciences.com/physique/814010-galilee-experience-de-pensee-chute-corps-7.html matches the one from https://physics.stackexchange.com/q...-masses-will-collide-due-to-Newtonian-gravity (and https://www.physicsforums.com/threads/remarkably-difficult-Newtonian-problem.360987/#post-2497264 ) for the case when x=0, and since I don't see anything wrong with the derivation, I will make the leap of faith that the former formula is also valid for x≠0. But without going into calculations, my comments still hold that when the sum of the two masses is the same, the time will be the same, and when the sum is varied to an insignificant amount (e.g., apple versus cannonball on the earth), the time will vary also only by an insignificant amount (especially if one is doing numerical integration), but when the sum changes significantly, then the time will correspondingly change significantly. What I find odd in the French forum is that one comment was about proving both Galileo and Aristotle right, whereas they are two opposing positions. To quote from Wikipedia, "Galileo supposedly discovered that the objects fell with the same acceleration, proving his prediction true, while at the same time disproving Aristotle's theory of gravity (which states that objects fall at speed proportional to their mass)." https://en.wikipedia.org/wiki/History_of_gravitational_theory (or, if you prefer, https://fr.wikipedia.org/wiki/Histoire_de_la_gravitation)

 

[Lucw]

Hello Nomadreid.

Thank you for your answer.
We have an equation ...
First remark.
If we set X0, X and mb. And if we increase m, the duration decreases. Aristotle is therefore right. The heavy ones fall faster ...
Second remark.
If we set X0, X and the sum (ma + mb). And if we increase ma (mb decreases ...) the duration is the same. And Galileo is right. All bodies fall in the same way.

This is not the first time I have tackled this problem. But I still see this same "human" resistance. To changes.
Think of the initial acceleration of the masses in both configurations; the distance traveled by the mass a in both configurations. It's the magic of Newton's laws ...
Lucw

 

[nomadreid]

If we set X0, X and mb.

I think you mean, if we fix X0, X and mb.
Yes, Aristotle is right in one context, and Galileo is right in another. However, to say that both are right... this reminds me of

or, a bit more fully,

 

[Lucw]

Hey.

Aristote is rigth if the falling objects do not come from Earth.
And Galileo is rigth if the falling objects come from Earth.
But don't mind.
It will still take some years before mind of people change....

Lucw.

 

[nomadreid]

Given the reluctance of the general population to study physics as well as falling standards in many countries' educational programs, I would not hold my breath until people change their basic prejudices.

 

[Lucw]

Euh.
So we are like in 1600. When Galileo said, the Earth turns around the sun ...
It's funny. I find.

About the study of physics ...
In 1824 steam locomotives began their commercial career. And for the "scientists", heat was always a fluid without mass that passed hot bodies to cold bodies. We have come a long way since ...
And that will continue. Maybe not in "our world" where we fall in comfort and abundance. Other where I think.

Candy. I will ask a question about the gravitational deviation. But I will (again) be ejected. It's weird. There are topics that can not be addressed in the forums. In 1600, we prepared a small pile of wood. And the one who did not agree with the "universal thought", we put it on and we lit the wood (Giordano Bruno). It was simple and definitive ........

On it, have a good day.

Lucw

 

[nomadreid]

I will ask a question about the gravitational deviation. But I will (again) be ejected. It's weird. There are topics that can not be addressed in the forums.

I am not sure what topics you have had problems posting. In general, the guidelines are outlined here
https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/
(that is, at the top, under "Info"> "Terms and Rules")
but most topics are allowed as long as they are not offensive , or are irrelevant to mathematics or physics, or trying to promote a philosophical or political agenda, or are repeating your own post just to get another listing; if you put it in the wrong rubrik, a mentor will usually kindly move it. Of course, a danger is that simply no one will answer. I myself have a very limited grasp of Einstein's Field Equations, but there are lots of good physicists on this site. So, if you have a specific question about the curvature of spacetime, go ahead and post it, either in this thread or in a new one. (You could post it here and see if I could answer it -- although probably not today -- and I could advise you whether you need to start a new thread.)

 

[rcgldr]

In a two body system, consisting of two objects with mass m₁ and m₂, both objects accelerate towards a common center of mass for the two objects, and each objects acceleration towards the common center of mass is due to the gravitational field of the "other" object. Assume the common center of mass is in the positive direction with respect to object₁ and negative with respect to object₂ . The rate at which object₁ accelerates towards the common center of mass = G m₂ / r² and the rate at which object₂ accelerates towards the common center of mass = -G m₁ / r² .

 

[rcgldr]

There are a few old threads about this. Note that these threads are considering the time it takes for two objects to collide (both objects accelerate towards each other), but it seems they have similar intermediate formula for velocity, M for the one object case, (m1+m2) for the two object case. For the 2 object case, if m1 >> m2, then the sum (m1+m2) ~= m1, so let M = m1. Some of these consider point objects others consider objects with non-zero radius.

See post #11 for a Kepler's law approach and post #19 for other approaches:

Non constant accelleration equation(s)

Using a different substitution here, and also allowing for non-zero radius objects:

Rectilinear motion of two attracting masses

See post #66, #68, and #73 for examples of both substitutions:

time-of-a-falling-object-when-the-force-of-gravity-isnt-constant

 

[Lucw]

Thank you Rcgldr