timz
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Math/physics Problem: Elliptical Orbits

Thu Feb 21, 2013 9:37 pm

An object is in a circular orbit around a spherical planet with infinite mass.

If we instantly increase its speed by 10% the object's orbit becomes elliptical, with the ellipse tangent to the old circular orbit. If the increase is 20% the ellipse is larger and more elliptical. Given the amount of the speed increase, how do we calculate the eccentricity and size of the new orbit?

[Edited 2013-02-21 14:02:51]
 
BMI727
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RE: Math/physics Problem: Elliptical Orbits

Thu Feb 21, 2013 10:10 pm

Quoting timz (Thread starter):
Given the amount of the speed increase, how do we calculate the eccentricity and size of the new orbit?

Wouldn't it be impossible to have an orbit around an infinite mass? I think that would blow up the math.

But, ignoring that, wouldn't you just solve the vis-viva equation? You know your current velocity and current orbital distance so wouldn't you just rearrange the equation to solve for the semi-major axis? Then the periapse would be the distance of the initial circular orbit and apoapse would be twice the semi-major axis minus the periapse. At that point I think you can directly calculate the eccentricity.

I may be completely off on that, but I think that's basically how it should work.
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Mir
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RE: Math/physics Problem: Elliptical Orbits

Thu Feb 21, 2013 10:12 pm

Quoting BMI727 (Reply 1):
Wouldn't it be impossible to have an orbit around an infinite mass? I think that would blow up the math.

I suspect what the infinite mass thing is there for is to take away the possibility of an escape trajectory if you keep increasing the orbital velocity.

But yes, I did find that a bit confusing.

-Mir
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BMI727
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RE: Math/physics Problem: Elliptical Orbits

Thu Feb 21, 2013 10:16 pm

Quoting Mir (Reply 3):
I suspect what the infinite mass thing is there for is to take away the possibility of an escape trajectory if you keep increasing the orbital velocity.

I think it would also take away the possibility of any separation with the body. GM becomes infinite, which when plugged into the vis-viva equation makes it 0 (actually v^2/infinity) = (2/r - 1/a) so both the distance and semi-major axis would have to be infinite.
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vikkyvik
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RE: Math/physics Problem: Elliptical Orbits

Thu Feb 21, 2013 11:49 pm

Quoting BMI727 (Reply 1):

Wouldn't it be impossible to have an orbit around an infinite mass? I think that would blow up the math.

Yes, it would. Infinite mass = infinite gravitational pull = infinite velocity required for orbit.
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Tugger
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 12:01 am

You do realize that you are doing the OP's homework don't you?
 
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wingman
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 12:21 am

It's a sad day when you only understand the articles, verbs, and prepositions in the original question. What else could I be but a sales guy?
 
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 4:52 am

Quoting timz (Thread starter):
An object is in a circular orbit around a spherical planet with infinite mass.

At any distance from the planet with infinite mass, the "orbiting" object is subject to infinite gravitational force and falls at infinite speed into the planet. Infinite speed means that it occupies all points in the universe at once.

Ridiculous, yes. That's why we don't use infinity.
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comorin
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 5:01 am

Quoting DocLightning (Reply 8):
Infinite speed means that it occupies all points in the universe at once.

   We are now talking black hole trajectories.

I do confess that I would like to read up more to understand "the Infinite speed means that it occupies all points in the universe at once" - any reading suggestions?
 
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DocLightning
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 5:04 am

Quoting comorin (Reply 9):
I do confess that I would like to read up more to understand "the Infinite speed means that it occupies all points in the universe at once" - any reading suggestions?

If you are moving infinitely quickly, then this is the result. Or at least you would occupy all points between your origin and destination simultaneously.

Infinity is a tricky concept. It leads to a lot of conclusions that make no sense because there is no such thing as infinity in the real world.
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vikkyvik
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 6:52 am

Quoting comorin (Reply 9):
I do confess that I would like to read up more to understand "the Infinite speed means that it occupies all points in the universe at once" - any reading suggestions?

If you have infinite speed, you go infinite distance in zero time. Infinite distance in zero time means you occupy all points on your trajectory in that instant of time. But really, infinite distance means you have to hit every possible spot on every possible trajectory in the universe, because you can't leave a spot out, or you haven't gone an infinite distance.

Obviously, in the real world, you can't do anything in zero time, nor can you attain infinite speed, or go an infinite distance. Or, as Douglas Adams so eloquently put it:

One of the problems has to do with the speed of light and the difficulties involved in trying to exceed it. You can't. Nothing travels faster than the speed of light with the possible exception of bad news, which obeys its own special laws. The Hingefreel people of Arkintoofle Minor did try to build spaceships that were powered by bad news but they didn't work particularly well and were so extremely unwelcome whenever they arrived anywhere that there wasn't really any point in being there.
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comorin
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 2:50 pm

Quoting DocLightning (Reply 10):
Quoting vikkyvik (Reply 11):

Much appreciate the explanations! Vikkyvik, thanks for the academic citation  

I just finished reading 'Origins' by Neal De Grasse Tyson - great book - and looking to expand my horizons.
 
smittyone
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 3:34 pm

Quoting DocLightning (Reply 10):
Infinity is a tricky concept. It leads to a lot of conclusions that make no sense because there is no such thing as infinity in the real world.

Same for every movie plot involving time travel.
 
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DocLightning
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 4:38 pm

Quoting vikkyvik (Reply 11):
If you have infinite speed, you go infinite distance in zero time.

Quick point. You would move a finite distance in zero time. The solution to "infinity divided by zero" is basically the set of all finite numbers, which is why it is undefined.

However, in any finite amount of time, no matter how infinitesimal, you would move an infinite distance and the only way to do that would be to occupy all points in an infinite universe at once.

Here's another result of infinity:

Given: You (and the planet Earth up to the present moment) have a finite, non-zero chance of existing.
Given: The universe is infinite in extent (and this is debatable, but may well be true).

Then: There MUST be an infinite number of perfectly identical copies of Earth in the Universe that have had the exact same history up to this very day. Thus, there are an infinite number of copies of you in the universe.

Furthermore: There MUST be a far larger (but still infinite) number of near-copies of Earth in which history has varied from Earth's history to some extent.

"Impossible!" No. Actually, mathematically absolutely necessarily true. Go wrap your brain around that one.  
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Mir
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 4:58 pm

Quoting DocLightning (Reply 14):
Then: There MUST be an infinite number of perfectly identical copies of Earth in the Universe that have had the exact same history up to this very day. Thus, there are an infinite number of copies of you in the universe.

Furthermore: There MUST be a far larger (but still infinite) number of near-copies of Earth in which history has varied from Earth's history to some extent.
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vikkyvik
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 5:04 pm

Quoting DocLightning (Reply 14):
Quick point. You would move a finite distance in zero time. The solution to "infinity divided by zero" is basically the set of all finite numbers, which is why it is undefined.

Infinity is, by definition, larger than any finite number. To go at infinite speed, you'd have to move an infinite distance in literally no time. The limit of "infinitesimal" is zero. Infinite speed takes you all the way to that limit.

Any number divided by zero yields infinity. Infinity divided by zero would be the same thing.

Quoting DocLightning (Reply 14):
However, in any finite amount of time, no matter how infinitesimal, you would move an infinite distance

In any finite time, you'd actually move MORE than infinite distance (because for any finite time, there is a finite time that is shorter...and anyway, it's not possible to travel more than infinite distance).
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Tugger
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 5:11 pm

Quoting DocLightning (Reply 14):
"Impossible!" No. Actually, mathematically absolutely necessarily true. Go wrap your brain around that one.

Wait.... but where is God in all of that?.....    I mean if God is infinite and the universe is infinite then..... Wait....  

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Mir
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 5:12 pm

Quoting vikkyvik (Reply 16):
To go at infinite speed, you'd have to move an infinite distance in literally no time.

You can't go anywhere if there is no time. You can be at all points in the universe at once at any particular moment in time (as you would if you were moving at infinite speed), but in order for there to be motion there must be a period of time for that motion to occur in (even if it's only a fraction of a nanosecond).

Quoting vikkyvik (Reply 16):
Any number divided by zero yields infinity.

Any number divided by zero yields a "does not compute". Infinity is just the area between the limit of what we can calculate and a pure division by zero (i.e. a division by an incredibly small number that is still not zero).

-Mir
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planewasted
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 5:49 pm

Quoting DocLightning (Reply 8):
At any distance from the planet with infinite mass, the "orbiting" object is subject to infinite gravitational force and falls at infinite speed into the planet. Infinite speed means that it occupies all points in the universe at once.

Ridiculous, yes. That's why we don't use infinity.

But if the orbiting object is at an infinite distance from the planet the force will not be infinite, and the speed not infinite? Or?

And if the force was infinite the objects speed would be the speed of light.
 
vikkyvik
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 7:01 pm

Quoting Mir (Reply 18):
You can't go anywhere if there is no time. You can be at all points in the universe at once at any particular moment in time (as you would if you were moving at infinite speed), but in order for there to be motion there must be a period of time for that motion to occur in (even if it's only a fraction of a nanosecond).

But you also can't have infinite speed, nor move an infinite distance, so saying you can't go anywhere in zero time is only 1/3 of that problem.

Quoting Mir (Reply 18):
Any number divided by zero yields a "does not compute". Infinity is just the area between the limit of what we can calculate and a pure division by zero (i.e. a division by an incredibly small number that is still not zero).

I know it's a does not compute. But what I meant is that the limit as you approach division by zero is infinity (or rather, there is no limit). So it stands to reason that if you were actually able to divide by zero, the answer would be infinity.

By the same token, as you approach infinite speed, you approach zero time to cover any distance.

Quoting DocLightning (Reply 14):
"Impossible!" No. Actually, mathematically absolutely necessarily true. Go wrap your brain around that one.

That one actually doesn't bother me at all. I happily admit that possibility.

Quoting Mir (Reply 18):
Infinity is just the area between the limit of what we can calculate and a pure division by zero (i.e. a division by an incredibly small number that is still not zero).

But if you're dividing by an incredibly small number that is not zero, then you can divide by a smaller number that is still not zero. So you haven't hit infinity yet.
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Mir
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RE: Math/physics Problem: Elliptical Orbits

Fri Feb 22, 2013 7:08 pm

Quoting vikkyvik (Reply 20):
By the same token, as you approach infinite speed, you approach zero time to cover any distance.

You approach it, yes. But you do not reach it. That's what a limit is all about. If you were to ever reach zero time, then you're not moving.

-Mir
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vikkyvik
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RE: Math/physics Problem: Elliptical Orbits

Sat Feb 23, 2013 3:39 am

Quoting Mir (Reply 21):
You approach it, yes. But you do not reach it. That's what a limit is all about. If you were to ever reach zero time, then you're not moving.

True. But if you're AT infinite speed (which is what I was originally addressing), then it's a different story. It isn't possible, nor is moving an infinite distance in no time.
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EricR
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RE: Math/physics Problem: Elliptical Orbits

Sun Feb 24, 2013 4:24 am

Quoting DocLightning (Reply 14):
Given: You (and the planet Earth up to the present moment) have a finite, non-zero chance of existing.
Given: The universe is infinite in extent (and this is debatable, but may well be true).

Then: There MUST be an infinite number of perfectly identical copies of Earth in the Universe that have had the exact same history up to this very day. Thus, there are an infinite number of copies of you in the universe.

Furthermore: There MUST be a far larger (but still infinite) number of near-copies of Earth in which history has varied from Earth's history to some extent.
.



Most recent consensus suggests the universe is finite and expanding as opposed to an infinite space. Nevertheless, assuming it is infinite, then your "furthermore" statement makes no sense. Infinite is just that - infinite. There cannot be a "larger" infinite number of examples.
 
comorin
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RE: Math/physics Problem: Elliptical Orbits

Sun Feb 24, 2013 6:02 am

Quoting EricR (Reply 23):

Most recent consensus suggests the universe is finite and expanding as opposed to an infinite space.

Correct - if the Universe is of finite age (13.7 billion years old) as per cosmic microwave emissions, then it cannot be infinite. The only room for infinity in that context is positing multiverses.

IMO, Infinity and zero are useful algebraic (and later, calculus) based concepts, not real numbers.
 
YYZatcboy
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RE: Math/physics Problem: Elliptical Orbits

Sun Feb 24, 2013 8:30 pm

Don't forget some infinities are larger than other infinities...

http://www.youtube.com/watch?v=elvOZm0d4H0

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DocLightning
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RE: Math/physics Problem: Elliptical Orbits

Mon Feb 25, 2013 3:36 am

Quoting EricR (Reply 23):
Most recent consensus suggests the universe is finite and expanding as opposed to an infinite space. Nevertheless, assuming it is infinite, then your "furthermore" statement makes no sense. Infinite is just that - infinite. There cannot be a "larger" infinite number of examples.

There is ample mathematical reasoning to show that although all infinities are infinite, there are different sizes of infinities. It depends on exactly how you are bounding sets.

For example, a 1x1cm square has an infinite number of points inside it. A 2x2cm square is 4x larger, but also has an infinite number of points inside. One could view that as an example of different sizes of infinities. That's one of the strange things about infinities. They are equally infinite, and yet one is larger.

The set of real numbers between 0 and 1 is infinite. The set of real numbers between 0 and 324,254,890,124 is also infinite, but much larger.

I'm not a mathematician, but I remember this stuff from my mathematics courses in college. Infinity does very strange things.
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NoWorries
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RE: Math/physics Problem: Elliptical Orbits

Mon Feb 25, 2013 2:39 pm

Quoting DocLightning (Reply 26):
For example, a 1x1cm square has an infinite number of points inside it. A 2x2cm square is 4x larger, but also has an infinite number of points inside. One could view that as an example of different sizes of infinities. That's one of the strange things about infinities. They are equally infinite, and yet one is larger.

The set of real numbers between 0 and 1 is infinite. The set of real numbers between 0 and 324,254,890,124 is also infinite, but much larger.

While it is true there are different sizes of infinities, those are not examples. If a one-to-one relationship can be established between the points in two different sets, the sets are considered to be the same size. For the squares, the relation is simply x2 = 2 * x1 and y2 = 2 *y1. The same basic reasoning applies to the second example.

An example of different sized infinities would be the rational numbers vs the real numbers -- even though both are infinite, the set of real numbers is larger than the set of rational numbers because it is not possible to establish a correspondence between them.
 
vikkyvik
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RE: Math/physics Problem: Elliptical Orbits

Mon Feb 25, 2013 3:54 pm

Quoting NoWorries (Reply 27):
While it is true there are different sizes of infinities, those are not examples. If a one-to-one relationship can be established between the points in two different sets, the sets are considered to be the same size. For the squares, the relation is simply x2 = 2 * x1 and y2 = 2 *y1. The same basic reasoning applies to the second example.

Sorry, I don't understand what you're explaining here. is "x2" x-squared? 2*x? Or just "x number 2"?
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NoWorries
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RE: Math/physics Problem: Elliptical Orbits

Mon Feb 25, 2013 4:27 pm

Quoting vikkyvik (Reply 28):
Sorry, I don't understand what you're explaining here. is "x2" x-squared? 2*x? Or just "x number 2"?

I don't know how to subscripts on this forum -- is there a way?

Any how -- let x1 mean x subscript 1, a point along the x axis in the 1 cm square, similarly y1 for a point along the y axis. Let x2 mean x subscript 2, a point along the x axis in the 2 cm square, similarly y2 for a point along the y axis. Use * to denote multiplication (I was FORTRAN programmer 30 years ago). So x2 = x1 * 2 means pick any point along the x axis in the 1cm square, multiply by 2 to get the corresponding point along the x axis in the 2 cm square. Same for y. So for any point (x1, y1) in one square there is a corresponding point (x2, y2) in the other square. Seems a bit counter-intuitive, but it's a basic principle in set theory.
 
vikkyvik
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RE: Math/physics Problem: Elliptical Orbits

Mon Feb 25, 2013 5:53 pm

Quoting NoWorries (Reply 29):
I don't know how to subscripts on this forum -- is there a way?

No idea. I usually use x_2 or something like that.

Quoting NoWorries (Reply 29):
(I was FORTRAN programmer 30 years ago).

They actually taught us FORTRAN my freshman year of college (in 2000). Don't remember any of it, which probably shows how involved it's been in my life since then.

Quoting NoWorries (Reply 29):
So x2 = x1 * 2 means pick any point along the x axis in the 1cm square, multiply by 2 to get the corresponding point along the x axis in the 2 cm square. Same for y. So for any point (x1, y1) in one square there is a corresponding point (x2, y2) in the other square. Seems a bit counter-intuitive, but it's a basic principle in set theory.

Gotcha, makes perfect sense. Thanks.
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