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Math/physics Problem: Elliptical Orbits  
User currently offlinetimz From United States of America, joined Sep 1999, 6903 posts, RR: 7
Posted (1 year 10 months 1 week 1 day 19 hours ago) and read 1499 times:

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]

30 replies: All unread, showing first 25:
 
User currently offlineBMI727 From United States of America, joined Feb 2009, 15841 posts, RR: 27
Reply 1, posted (1 year 10 months 1 week 1 day 19 hours ago) and read 1455 times:

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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User currently onlineAR385 From Mexico, joined Nov 2003, 6616 posts, RR: 35
Reply 2, posted (1 year 10 months 1 week 1 day 19 hours ago) and read 1456 times:
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http://www.mathopenref.com/ellipseeccentricity.html

User currently offlineMir From United States of America, joined Jan 2004, 21880 posts, RR: 55
Reply 3, posted (1 year 10 months 1 week 1 day 19 hours ago) and read 1450 times:

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



7 billion, one nation, imagination...it's a beautiful day
User currently offlineBMI727 From United States of America, joined Feb 2009, 15841 posts, RR: 27
Reply 4, posted (1 year 10 months 1 week 1 day 19 hours ago) and read 1446 times:

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.



Why do Aerospace Engineering students have to turn things in on time?
User currently offlinevikkyvik From United States of America, joined Jul 2003, 10350 posts, RR: 26
Reply 5, posted (1 year 10 months 1 week 1 day 17 hours ago) and read 1417 times:

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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User currently offlinetugger From United States of America, joined Apr 2006, 5797 posts, RR: 10
Reply 6, posted (1 year 10 months 1 week 1 day 17 hours ago) and read 1410 times:

You do realize that you are doing the OP's homework don't you?
 
Tugg



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User currently offlinewingman From Seychelles, joined May 1999, 2344 posts, RR: 5
Reply 7, posted (1 year 10 months 1 week 1 day 17 hours ago) and read 1404 times:

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?

User currently offlineDocLightning From United States of America, joined Nov 2005, 20365 posts, RR: 59
Reply 8, posted (1 year 10 months 1 week 1 day 12 hours ago) and read 1352 times:

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.


User currently offlinecomorin From United States of America, joined May 2005, 4903 posts, RR: 16
Reply 9, posted (1 year 10 months 1 week 1 day 12 hours ago) and read 1350 times:

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?


User currently offlineDocLightning From United States of America, joined Nov 2005, 20365 posts, RR: 59
Reply 10, posted (1 year 10 months 1 week 1 day 12 hours ago) and read 1348 times:

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.


User currently offlinevikkyvik From United States of America, joined Jul 2003, 10350 posts, RR: 26
Reply 11, posted (1 year 10 months 1 week 1 day 10 hours ago) and read 1335 times:

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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User currently offlinecomorin From United States of America, joined May 2005, 4903 posts, RR: 16
Reply 12, posted (1 year 10 months 1 week 1 day 2 hours ago) and read 1291 times:

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.


User currently offlineSmittyone From , joined Dec 1969, posts, RR:
Reply 13, posted (1 year 10 months 1 week 1 day 1 hour ago) and read 1281 times:

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.


User currently offlineDocLightning From United States of America, joined Nov 2005, 20365 posts, RR: 59
Reply 14, posted (1 year 10 months 1 week 1 day ago) and read 1270 times:

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.  


User currently offlineMir From United States of America, joined Jan 2004, 21880 posts, RR: 55
Reply 15, posted (1 year 10 months 1 week 1 day ago) and read 1262 times:

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.
http://usermeds.com/static/92b03f4f9295288e5f40c1586606fcb7.jpg

-Mir



7 billion, one nation, imagination...it's a beautiful day
User currently offlinevikkyvik From United States of America, joined Jul 2003, 10350 posts, RR: 26
Reply 16, posted (1 year 10 months 1 week 1 day ago) and read 1259 times:

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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User currently offlinetugger From United States of America, joined Apr 2006, 5797 posts, RR: 10
Reply 17, posted (1 year 10 months 1 week 1 day ago) and read 1256 times:

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....  

Tugg



I don’t know that I am unafraid to be myself, but it is hard to be somebody else. -W. Shatner
User currently offlineMir From United States of America, joined Jan 2004, 21880 posts, RR: 55
Reply 18, posted (1 year 10 months 1 week 1 day ago) and read 1255 times:

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



7 billion, one nation, imagination...it's a beautiful day
User currently offlineplanewasted From Sweden, joined Jan 2008, 541 posts, RR: 0
Reply 19, posted (1 year 10 months 1 week 23 hours ago) and read 1248 times:

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.


User currently offlinevikkyvik From United States of America, joined Jul 2003, 10350 posts, RR: 26
Reply 20, posted (1 year 10 months 1 week 22 hours ago) and read 1236 times:

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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User currently offlineMir From United States of America, joined Jan 2004, 21880 posts, RR: 55
Reply 21, posted (1 year 10 months 1 week 22 hours ago) and read 1233 times:

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



7 billion, one nation, imagination...it's a beautiful day
User currently offlinevikkyvik From United States of America, joined Jul 2003, 10350 posts, RR: 26
Reply 22, posted (1 year 10 months 1 week 13 hours ago) and read 1201 times:

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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User currently offlineEricR From United States of America, joined Jul 2010, 1904 posts, RR: 1
Reply 23, posted (1 year 10 months 6 days 13 hours ago) and read 1147 times:

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.


User currently offlinecomorin From United States of America, joined May 2005, 4903 posts, RR: 16
Reply 24, posted (1 year 10 months 6 days 11 hours ago) and read 1134 times:

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.


User currently offlineYYZatcboy From Canada, joined Apr 2005, 1098 posts, RR: 0
Reply 25, posted (1 year 10 months 5 days 21 hours ago) and read 1106 times:
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Don't forget some infinities are larger than other infinities...

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User currently offlineDocLightning From United States of America, joined Nov 2005, 20365 posts, RR: 59
Reply 26, posted (1 year 10 months 5 days 13 hours ago) and read 1083 times:

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.


User currently offlineNoWorries From United States of America, joined Oct 2006, 539 posts, RR: 1
Reply 27, posted (1 year 10 months 5 days 2 hours ago) and read 1051 times:

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.


User currently offlinevikkyvik From United States of America, joined Jul 2003, 10350 posts, RR: 26
Reply 28, posted (1 year 10 months 5 days 1 hour ago) and read 1043 times:

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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User currently offlineNoWorries From United States of America, joined Oct 2006, 539 posts, RR: 1
Reply 29, posted (1 year 10 months 5 days 1 hour ago) and read 1038 times:

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.


User currently offlinevikkyvik From United States of America, joined Jul 2003, 10350 posts, RR: 26
Reply 30, posted (1 year 10 months 4 days 23 hours ago) and read 1028 times:

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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