| Topic: |
Science > Physics |
| User: |
"Matthew" |
| Date: |
04 Jul 2007 11:01:21 AM |
| Object: |
Help in understanding relative velocity in 3D |
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
Thanks in advance,
Mattew
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| User: "Greg Neill" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 11:33:45 AM |
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"Matthew" <m_manni_m@hotmail.com> wrote in message
news:1183564881.139864.153310@c77g2000hse.googlegroups.com...
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
It sounds as though you are looking for the difference
between two vectors (i.e. vector subtraction is required).
That being the case, you might consider transforming the
vector pairs to Cartesian coordinates, taking the difference
(which is trivial in Cartesian coordinates) and transforming
the result back to spherical coordinates.
Your professor was correct in that two linearly independant
vectors define a unique plane, so that your 3D problem
reduces to the 2D case with an appropriate coordinate
transformation. But I'm not sure that it makes your life
any easier to use that approach.
.
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| User: "Matthew" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 12:00:42 PM |
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On Jul 4, 6:33 pm, "Greg Neill" <gneill...@OVEsympatico.ca> wrote:
"Matthew" <m_mann...@hotmail.com> wrote in message
news:1183564881.139864.153310@c77g2000hse.googlegroups.com...
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
It sounds as though you are looking for the difference
between two vectors (i.e. vector subtraction is required).
That being the case, you might consider transforming the
vector pairs to Cartesian coordinates, taking the difference
(which is trivial in Cartesian coordinates) and transforming
the result back to spherical coordinates.
Your professor was correct in that two linearly independant
vectors define a unique plane, so that your 3D problem
reduces to the 2D case with an appropriate coordinate
transformation. But I'm not sure that it makes your life
any easier to use that approach.
Actually it does m,ake it easier because I need to compute the
difference between two general vectors (I don't have, nor I need, the
magnitude or the values associated to them) as a proof of a theorem.
In the 2D dimension environment the final result express v(theta) with
a formula that contains the ratio between vectors's magnitude and some
square root of the cosine oh the (theta) angle difference between them
(computation is too long to be written clearly in here, but it's quite
straightforward). It then integrates this result over [0,2Pi] because
(I omitted that before) it needs to include all other objects moving
randomly in this framework.
In my case I've been told that I can compute v(theta) in the same way
and than integrate over theta and phi (double integral with limits
defined by polar and azimuthal, i.e: [0,Pi] and [0,2Pi], but it's the
idea below reducing to the 2D case is too thin for me:
You said:
Your professor was correct in that two linearly independant
vectors define a unique plane, so that your 3D problem
reduces to the 2D case with an appropriate coordinate
transformation.
In fact he told me something about fixing two of the axis and rotating
the third to get a unique plane for both vectors in order to express
relative velocity as a function of the only polar angle theta, but how
does it work exactly? As you can see I'm pretty confused about that
part...unfortunately I can't attach the article here, which probably
would better clarify you why I'm approaching the problem in this way !
Mat
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| User: "Greg Neill" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 12:55:30 PM |
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"Matthew" <m_manni_m@hotmail.com> wrote in message
news:1183568442.330366.122590@n2g2000hse.googlegroups.com...
On Jul 4, 6:33 pm, "Greg Neill" <gneill...@OVEsympatico.ca> wrote:
Your professor was correct in that two linearly independant
vectors define a unique plane, so that your 3D problem
reduces to the 2D case with an appropriate coordinate
transformation. But I'm not sure that it makes your life
any easier to use that approach.
Actually it does m,ake it easier because I need to compute the
difference between two general vectors (I don't have, nor I need, the
magnitude or the values associated to them) as a proof of a theorem.
In the 2D dimension environment the final result express v(theta) with
a formula that contains the ratio between vectors's magnitude and some
square root of the cosine oh the (theta) angle difference between them
(computation is too long to be written clearly in here, but it's quite
straightforward).
So the magnitudes do enter into it. It sounds suspiciously
like the essential mechanics of spherical to Cartesian
transformation are "built into" the method, but aren't
specifically called out.
It then integrates this result over [0,2Pi] because
(I omitted that before) it needs to include all other objects moving
randomly in this framework.
In my case I've been told that I can compute v(theta) in the same way
and than integrate over theta and phi (double integral with limits
defined by polar and azimuthal, i.e: [0,Pi] and [0,2Pi], but it's the
idea below reducing to the 2D case is too thin for me:
Perhaps I'm just thick today, but I don't understand
what the integration is all about. Two given vectors
have a fixed angle between them. In a given coordinate
system the angle that a given vector makes with the
axes is also fixed. So what is the angle being
integrated? How does it apply to all the disparate
pairs of vectors?
You said:
Your professor was correct in that two linearly independant
vectors define a unique plane, so that your 3D problem
reduces to the 2D case with an appropriate coordinate
transformation.
In fact he told me something about fixing two of the axis and rotating
the third to get a unique plane for both vectors in order to express
relative velocity as a function of the only polar angle theta, but how
does it work exactly? As you can see I'm pretty confused about that
part...unfortunately I can't attach the article here, which probably
would better clarify you why I'm approaching the problem in this way !
Perhaps you can put it up on a web page and provide
a link?
.
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| User: "Matthew" |
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| Title: Re: Help in understanding relative velocity in 3D |
05 Jul 2007 04:20:55 AM |
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On Jul 4, 7:55 pm, "Greg Neill" <gneill...@OVEsympatico.ca> wrote:
"Matthew" <m_mann...@hotmail.com> wrote in message
Perhaps you can put it up on a web page and provide
a link?
For whom has the patience to have a look, I found the link to the
article:
www.cs.uml.edu/~bliu/pub/mobihoc05Liu.pdf
the 2D proof of the theorem is the one in paragraph 4.2, and the
computation is showed at the beginning of page 7.
Basically the idea of cutting a plane where both of the vectors lie
helps me to keep the computation of v(theta) the same in 3D case (this
is what the prof said). The article is quite complicated, but
basically the integration is needed because our framework is composed
of many different sensors moving in a random fashion and a target: at
first we take one of the sensors (a general one) and the target to
compute the relative velocity...then we integrate over all the disk to
take into account all the other sensors...
My problem is not with the computation itself (I would use the x,y,z
coordinates in that case), but with the physical concepts behind this
idea of having the same computation as in pag.7 in the 3D case!
Thanks,
Mattew
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| User: "Androcles" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 12:45:24 PM |
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"Matthew" <m_manni_m@hotmail.com> wrote in message
news:1183564881.139864.153310@c77g2000hse.googlegroups.com...
: Hi everybody, I need your help in clarifying some phisysical concepts
: which are of great interest for my thesis...I'm working on a framework
: in which objects randomly move in a 3Dimensional space and at a
: certain point I have to deal with computing the relative velocity
: between two of them.
: My starting point is the same framework in 2D,
: where the vector of relative velocity v(theta) of the two objects is
: computed trigonometrically in simple way (Pitagora's theorem and so
: on), being Theta the angle [0,2Pi] which defines the direction of the
: objects (assume a disk as our movement environment).
: Now I have to transpose it to a 3D environment, with objects moving in
: a sphere, so I introduced the spherical coordinates (theta,phi)...this
: is the preface, now:
:
: I asked help to a physics University professor and he told me that the
: computation of relative velocity between two objects can be done
: exactly in the same way in 3D, because for relative velocity we can
: always find a plane cutting the sphere, a plane where both the
: velocity vectors lie. In this case I can use the same computation of
: the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
: he told me we're not dealing with position vectors (in that case we
: had to use both the spherical and azimuthal angles to define a
: position in 3D) but only with relative velocity vectors. To me it
: sounds like: "Velocity always lives in a 2Dimesional space", but I'm
: pretty confused about that...
: I whish I've been clear enough, can anyone helps me in understanding
: this concept?
You seem to be more confused by polar coordinates than anything else.
These are not "physical concepts" but mathematical ones.
I suggest you use Google Sketchup, you can download it for free
and it has a protractor and a ruler. Create an exact cube and
then measure the distance across the long diagonal of the cube
and the angles it makes to the ground plane.
Play with it until you understand.
.
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| User: "Sam Wormley" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 11:41:50 AM |
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Matthew wrote:
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
Thanks in advance,
Mattew
If you know the xyz components of velocity for each of the two
particles, then the resultant velocity is the square root of the
sum of the three velocity differences squared.
The introductory chapter(s) of most classical mechanics textbooks
review vector algebra. See equation (1)
http://mathworld.wolfram.com/Vector.html
.
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| User: "Puppet_Sock" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 05:13:12 PM |
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On Jul 4, 12:41 pm, Sam Wormley <sworml...@mchsi.com> wrote:
[snip]
If you know the xyz components of velocity for each of the two
particles, then the resultant velocity is the square root of the
sum of the three velocity differences squared.
Um. What? A velocity is a square root of a sum of squares?
There are three "velocity differences"?
Maybe what you meant was, the magnitude of the relative
velocity is the square root of the sum of the squares of
the three components in Cartesian coordinates of the
relative velocity, and that the relative velocity is the difference
between the two velocities.
The introductory chapter(s) of most classical mechanics textbooks
review vector algebra. See equation (1)
http://mathworld.wolfram.com/Vector.html
Pretty sure that reference isn't going to be useful.
Socks
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| User: "Sam Wormley" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 05:27:49 PM |
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Sam Wormley wrote:
Matthew wrote:
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
Thanks in advance,
Mattew
If you know the xyz components of velocity for each of the two
particles, then the resultant velocity is the square root of the
sum of the three velocity differences squared.
The introductory chapter(s) of most classical mechanics textbooks
review vector algebra. See equation (1)
http://mathworld.wolfram.com/Vector.html
Oop... sorry for the above post!
.
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| User: "" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 04:58:33 PM |
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On 4 juil, 12:01, Matthew <m_mann...@hotmail.com> wrote:
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie.
He was right about this, but maybe not specific enough.
What you do is cut a plane on which the origins of both
vectors lie. Then, you mentally "pull over" one of the vectors so
that both origins coincide. Then you cut the plane on which
both vectors lie. The latter is the plane he was talking about.
Andr=E9 Michaud
In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
Thanks in advance,
Mattew
.
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| User: "PD" |
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| Title: Re: Help in understanding relative velocity in 3D |
05 Jul 2007 03:56:03 PM |
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On Jul 4, 11:01 am, Matthew <m_mann...@hotmail.com> wrote:
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
Thanks in advance,
Mattew
There's a few ways you can look at this. Here's two.
1. Vector addition and subtraction (which is what you're doing here to
get relative velocities) always admits the freedom to slide the
vectors around as you wish. This is what allows you to use the "head-
to-tail" method of adding vectors -- sliding the second vector so that
the tail of one sits on the head of the other. When you do that, you
find that you've defined three points: the tail of the first vector,
the head of the first vector (coincident with the tail of the second
vector), and the head of the second vector. Three points always define
a plane. [If you don't like sliding the vectors that way, then slide
them so that the two vectors both have their tails at the coordinate
system origin, and you then still have three points: the origin and
the heads of the two vectors.]
2. The vector cross-product of two velocity vectors is another vector,
perpendicular to both vectors. This third vector is of course
orthogonal to some plane (as is any vector). The plane is the one
containing the two velocity vectors. So if you want to find the plane,
take the cross-product, and then define the plane to be the one where
the projection of the cross-product onto that plane vanishes.
PD
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| User: "Matthew" |
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| Title: Re: Help in understanding relative velocity in 3D |
06 Jul 2007 06:07:52 AM |
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On Jul 5, 10:56 pm, PD <TheDraperFam...@gmail.com> wrote:
On Jul 4, 11:01 am, Matthew <m_mann...@hotmail.com> wrote:
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
Thanks in advance,
Mattew
There's a few ways you can look at this. Here's two.
1. Vector addition and subtraction (which is what you're doing here to
get relative velocities) always admits the freedom to slide the
vectors around as you wish. This is what allows you to use the "head-
to-tail" method of adding vectors -- sliding the second vector so that
the tail of one sits on the head of the other. When you do that, you
find that you've defined three points: the tail of the first vector,
the head of the first vector (coincident with the tail of the second
vector), and the head of the second vector. Three points always define
a plane. [If you don't like sliding the vectors that way, then slide
them so that the two vectors both have their tails at the coordinate
system origin, and you then still have three points: the origin and
the heads of the two vectors.]
2. The vector cross-product of two velocity vectors is another vector,
perpendicular to both vectors. This third vector is of course
orthogonal to some plane (as is any vector). The plane is the one
containing the two velocity vectors. So if you want to find the plane,
take the cross-product, and then define the plane to be the one where
the projection of the cross-product onto that plane vanishes.
PD
Your explanation and Andre's one are somehow answering my question: I
understood why it si possible to reduce the vector velocity problem
from 3D to a 2Denvironment, and you suggested me the way to define the
plane the vectors belong to. Now I only need to "connect" this
concepts to the computation I made in order to relate the angle that
appears in it to the plane...
.
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| User: "Puppet_Sock" |
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| Title: Re: Help in understanding relative velocity in 3D |
04 Jul 2007 05:24:37 PM |
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On Jul 4, 12:01 pm, Matthew <m_mann...@hotmail.com> wrote:
[snips]
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie.
While your prof's explanation is technically correct, it is not
very helpful.
Look down this page
http://en.wikipedia.org/wiki/Curvilinear_coordinates
till you come to spherical coordinates. That should
let you convert between spherical coords (r, theta, phi)
and Cartesians (x,y,z). In Cartesians, the relative
velocity of two particles is just the difference between
their individual velocity vectors, and that's just the
difference between the coordinate values.
(x1, y1, z1) - (x2, y2, z2) = (x1 - x2, y1 - y2, z1 - z2)
And then you can convert the result back to spherical
coords if you need to.
To do what your prof suggests, you can note that two
vectors that are not parallel to eachother can be used
to define a plane. If you do that, then you can make
your coordinate system such that two of the coordinates
are in the plane, and the third is perpendicular. In that
set of coordinates, call the "z" coordinate the one
perpendicular to the plane. And then the z component
of each velocity vector in those coordinates is zero.
In other words, you are back to 2-d.
But the easiest way to *find* that plane is to use Cartesians.
It *can* be done in spherical coords, but it's painful.
Socks
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| User: "" |
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| Title: Re: Help in understanding relative velocity in 3D |
05 Jul 2007 04:25:07 AM |
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On Jul 5, 2:01 am, Matthew <m_mann...@hotmail.com> wrote:
Hi everybody, I need your help in clarifying some phisysical concepts
which are of great interest for my thesis...I'm working on a framework
in which objects randomly move in a 3Dimensional space and at a
certain point I have to deal with computing the relative velocity
between two of them. My starting point is the same framework in 2D,
where the vector of relative velocity v(theta) of the two objects is
computed trigonometrically in simple way (Pitagora's theorem and so
on), being Theta the angle [0,2Pi] which defines the direction of the
objects (assume a disk as our movement environment).
Now I have to transpose it to a 3D environment, with objects moving in
a sphere, so I introduced the spherical coordinates (theta,phi)...this
is the preface, now:
I asked help to a physics University professor and he told me that the
computation of relative velocity between two objects can be done
exactly in the same way in 3D, because for relative velocity we can
always find a plane cutting the sphere, a plane where both the
velocity vectors lie. In this case I can use the same computation of
the 2D case to find v(theta), being theta the polar angle in [0,Pi]:
he told me we're not dealing with position vectors (in that case we
had to use both the spherical and azimuthal angles to define a
position in 3D) but only with relative velocity vectors. To me it
sounds like: "Velocity always lives in a 2Dimesional space", but I'm
pretty confused about that...
I whish I've been clear enough, can anyone helps me in understanding
this concept?
Thanks in advance,
Mattew
The easiest way to deal with this sort of situation is to define an
initial coordinate system called world space. Define your objects in
terms of world space
x,y,z position
vx,vy,vz velocity vector
For each time unit dt, move the objects
x = x + vx*dt
y = y + vy*dt
z = z + vz*dt
Now, to do a coordinate transformation from to object O you do
foreach (object in world)
object.x = object.x - O.x
object.y = object.y - O.y
object.z = object.z - O.z
If your objects have a concept of direction, then your transformation
has to include a rotation at the end.
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| User: "Matthew" |
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| Title: Re: Help in understanding relative velocity in 3D |
05 Jul 2007 05:19:37 AM |
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Again, I already proceeded in calculating the formula for the proof in
3D, and I based it on the concept of calculating the relative velocity
(as shown in the article linked above) as in the 2Dimensional case,
because the professor told me I could do so. My problem is not
calculting anything now, but in understanding why I can reduce the
problem to 2-dimensions using this idea of cutting the sphere with
planes...I can't figure it out which plane is related to which angle
and so on...mainly I'm looking for a physical explanation for that,
not for an alternative way to compute te relative velocity!
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| User: "" |
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| Title: Re: Help in understanding relative velocity in 3D |
05 Jul 2007 08:01:11 AM |
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On 5 juil, 06:19, Matthew <m_mann...@hotmail.com> wrote:
Again, I already proceeded in calculating the formula for the proof in
3D, and I based it on the concept of calculating the relative velocity
(as shown in the article linked above) as in the 2Dimensional case,
because the professor told me I could do so. My problem is not
calculting anything now, but in understanding why I can reduce the
problem to 2-dimensions using this idea of cutting the sphere with
planes...I can't figure it out which plane is related to which angle
and so on...mainly I'm looking for a physical explanation for that,
not for an alternative way to compute te relative velocity!
You said you started with a theta angle in 2D between the
vectors (or projections of your vectors), so if you transpose
to 3D you should have the same angle between one vector
and the projection of the other in your 3D setup.
But if you start with 2 random vectors in the 3D space, then you
need a reference frame external to your vectors to which you can
relate them if you are to find theta.
I suggested one way to find the plane, but there are more than
one way to address this. For example, cut a plane on which
one of the vectors and the origin of the other one lie. You then
project the other vector so that both origins coincide. You then
cut the final plane on which the first vector and the projection
of the other lie.
There you have your theta from your initial 2D set up or
alternately, the theta that you calculate from the external
reference frame you mapped your 3D events into.
Andr=E9 Michaud
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| User: "" |
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| Title: Re: Help in understanding relative velocity in 3D |
05 Jul 2007 09:34:21 AM |
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On Jul 5, 8:19 pm, Matthew <m_mann...@hotmail.com> wrote:
Again, I already proceeded in calculating the formula for the proof in
3D, and I based it on the concept of calculating the relative velocity
(as shown in the article linked above) as in the 2Dimensional case,
because the professor told me I could do so. My problem is not
calculting anything now, but in understanding why I can reduce the
problem to 2-dimensions using this idea of cutting the sphere with
planes...I can't figure it out which plane is related to which angle
and so on...mainly I'm looking for a physical explanation for that,
not for an alternative way to compute te relative velocity!
I don't understand what you've done, but suppose objects 1 and 2 are
moving in 3D space. Relative to some coordinate system, suppose object
1 is moving along the x-axis, and object 2 is moving along the y-axis.
You can just scrap the z-axis in that case, and consider 2 objects
moving in 2D space.
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