What is Speed?

Discussion in 'Physics & Math' started by Johnny5, Jun 16, 2005.

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  1. Johnny5 Registered Senior Member

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    All this arguing about SR, and not getting anywhere got me thinking about something.

    Since SR involves multiple concepts of physics, why not have various individuals demonstrate their understanding of each concept one at a time.

    Then side by side, we can all see who understands what the best.

    To this end, I want this thread to be devoted to the concept of physics known as speed.

    It is extremely clear, that in order to understand the special theory of relativity, you have to understand the meaning of 'speed', since even to formulate the basic postulate of relativity, requires you to make reference to "the speed" of light.

    So what is speed?

    This thread is a chance for various members to elucidate this concept of physics in their own words.

    And remember, speed is not velocity.

    So pretend you are giving a lecture on 'speed' to other members of the forum.

    You can give links to other sites, you can do whatever you want.

    My goal is to decide who among us, understands the concept of 'speed' the best.

    You can differentiate between average speed, and instantaneous speed, talk about both, its up to you.
     
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  3. kevinalm Registered Senior Member

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    The absolute magnitude of the first derivitive of the position vector wrt time.
     
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  5. Johnny5 Registered Senior Member

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    This was hardly a lecture on the concept of speed, but at least you came foward with a response.

    One thing which bothers me about your response, is that you didn't use the concept of "moment in time" in your answer, which would help flesh out the meaning of instantaneous speed, which is what you attempted to define there.

    Also, you mention the term 'vector' which is not a necessary concept to already know, in order to understand speed.

    But you easily have the option of going into more detail, modifying your answer.

    Another thing, is that you didn't give enough information to show how to switch reference frames, and how speed changes in such a transformation.
     
    Last edited: Jun 16, 2005
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  7. Rogue Physicist Registered Senior Member

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    I would say traditionally 'speed' is something we relate to a more primary concept, 'motion'. 'motion' implies locality and substance (either mass, or nowadays energy*). From this we can see that there already problems when applying the concept of 'motion' to something like 'light'. We can imagine localized 'packets' of energy flying along a trajectory and arriving at a new location, but we can never prove such a concept: Since all 'photons' are identical, we have no way of discerning which 'photon' is which, in order to trace a path etc. So both motion and speed seem 'real' only in limited contexts where one can have the confidence to grant their applicability, such as with macroscopic (large scale) slow-moving objects having mass.
     
  8. Johnny5 Registered Senior Member

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    I'm going to give it a try

    Let it be agreed that any object has a center of mass, which is a single point, either inside the body, or outside of the body. But it is a single point.

    Let us begin the discussion of the concept of 'speed' by discussing the speed of the center of mass of an object.

    Now, speed is relative, since what is at rest in one person's frame of reference, is moving in another person's frame of reference.

    So speed is a frame dependent quantity.

    In any frame, the speed of the center of mass of an object can be precisely defined, in terms of the coordinates of the CM in the frame.

    Every location in a frame, can be specified by three coordinates, (x,y,z).

    A frame is a set of three mutually perpendicular infinitely long coordinate axes.

    Now, suppose that at some moment in time, the CM of some object has location (x1,y1,z1) in frame S.

    If at the very next moment in time, the location of the CM is still (x1,y1,z1) then we can say the object is at rest.

    Now, the object might be spinning, but the CM of the object is at rest, and when the CM of an object is at rest we will say the body is at rest.

    And we can define the rest frame of an object, as a frame of reference whose origin is permanently the center of mass of the object.

    We will treat rotation of rigid bodies separately.

    Now, if on the other hand, the center of mass of the object is no longer (x2,y2,z2) at the very next moment in time, then the center of mass of the object changed its location in frame S.

    We are now in a position to define the instantaneous speed of the center of mass of an object in an arbitrary reference frame.

    Let v denote the instantaneous speed of the center of mass of an object in some frame. v is mathematically defined as follows:

    v = distance traveled/time of travel = D/T

    distance traveled is found from the pythagorean theorem

    D = [(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2 ]^(1/2)

    And the time of travel is instantaneous.

    From the definition above, you can infer that speed is a strictly positive quantity.


    Ok so any comments on what i just wrote? (negative or positive are both welcome)

    PS: And don't think this was the best i can do.
     
  9. Johnny5 Registered Senior Member

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    The concept of speed is indeed linked to what started off being called motion. As for your comments on the photon, if a photon can be treated like a single coherent body, then you can imagine that the photon has a center of mass, and you can focus on the motion of the center of mass of the photon, and attempt to write equations of motion for the CM of a photon.

    I read your post first, and then wrote my previous post.
     
  10. chroot Crackpot killer Registered Senior Member

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    The answer given by kevinalm is all there is to know about velocity.

    You took offense because you needed to know what a vector is to understand what velocity is. That's funny, because your answer requires the same prerequisite. In fact, I would certainly prefer his answer over your crap-fest which involved not one but six independent concepts: (1) vectors, (2) the center of mass, (3) frames of reference, (4) instantaneous time, (5) rectangular coordinates, (6) the pythagorean theorem.

    - Warren
     
  11. Johnny5 Registered Senior Member

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    No, i didn't take any offense at all. Vectors don't bother me, nor do tensors, versors, scalars, or quaternions.

    But one thing at a time.

    Since relativity is totally wrong, the human race needs to be led by a superphysicist to a perfect understanding of time, because from my point of view no one undestands it but me.

    Since speed involves time, you need to discuss moments in time in the definition. What you call a crap-fest, was merely me focusing on the speed of the CM of an object, in some arbitrary frame of reference.

    Speed is thus defined using the coordinates of the CM in a frame, exactly as I did.

    The same definition can then be extended to particles.

    Lastly, I fail to see how I used vectors in what i described. Perhaps you can show this to me?


    PS: Also, i do not wish to exchange insults, that isn't my way, and it's totally counterproductive. Perhaps this will help you understand where I am going with this thread...

    In order to define speed, you need two moments in time.

    In order to define acceleration, you need three moments in time.
     
  12. chroot Crackpot killer Registered Senior Member

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    See that (x,y,z) thing you used in your definition? That's a vector.

    Given a position vector r, the speed of r is |d/dt (r)|.

    You don't need to involve coordinate systems or centers of mass, and doing so just muddies the water.

    - Warren
     
  13. Johnny5 Registered Senior Member

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    No, that was supposed to denote a point.

    Coordinates of point P in frame S: (x,y,z)

    Position vector from origin of frame S to point P in frame S:
    xi+yj+zk
     
  14. chroot Crackpot killer Registered Senior Member

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    A point and a vector are the same thing; the only difference is that a vector is permitted to act as an operator.

    - Warren
     
  15. Johnny5 Registered Senior Member

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    A point and a vector aren't the same thing. A vector has magnitude and direction. A point is just a location, either in some reference frame, or in fixed in absolute space.

    As for your comment that a vector is permitted to act as an operator, what are you referring to?
     
  16. chroot Crackpot killer Registered Senior Member

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    You sure do lack higher math skills for being such an Advanced Reasoning Agent.

    A point is defined with respect to a known origin, yes? There is a one-to-one correspondence between a point and the vector between the origin and that point.

    A vector can act as an operator; for example, a vector can multiply a scalar, while a point cannot.

    - Warren
     
  17. MetaKron Registered Senior Member

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    It seems like Johnny5 came closer to providing the whole picture. It had the needed detail. Technically he should have called "velocity" "speed" in one place, but it's a prevalent habit, not as bad as the one I've seen of treating singularities and black holes as interchangeable. Kevinalm's "position vector" needs to be explained for anyone to be able to use it.

    Johnny5's explanation is good for a vector that is absolutely linear. If it curves, strictly speaking it is only a vector from a given point along the curve. In other words, it's a constantly changing vector that follows a curve in 3 space or 4 space. This is where I have to admit that I am not an expert, although I'm pretty good. I can somewhat follow curves in these spaces if I reconstruct the equations using cruder methods of integration, but I'm having trouble learning how to read the equations that describe geodesics. What on earth is a "tensor", anyway?

    Johnny5 does write some pretty decent walk-throughs. He may be good enough to write a book about this, but it takes a lot of solid work to get from nebulous concepts to a product that will impress those who understand it. I've been on something for five years and have just the last few weeks reached enough clarity to be able to put forth something useful. I have had the habit of arguing instead of working, and that has slowed me down.

    I have recently become aware also that understanding mathematics takes time and even a genius like myself has to take that time and absorb concepts and put them together in their proper framework. Make no mistake, I have a pretty big ego and am likely to go on my own little tangents, but I am trying to understand and control this, and I want to understand what people are trying to say when they think they are on to something. I have other habits I need to control, too, like playing video games instead of working and indulging in lengthy self-analysis during a discussion on mathematics.

    Johnny5, the statement "the time of travel is instantaneous" needs clarification. I'm not good with terms that are used ambiguously, if that is what you did that time. Sometimes I'm the guy who has to repeat something back before he understands what was said to him. I no longer care if that makes me look like a dummy. I just want to understand what people are saying.
     
  18. Johnny5 Registered Senior Member

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    Mmmm, let me ask you something.

    Given any two arbitrary points in a reference frame, how many vectors are there from one point to the other?
     
  19. chroot Crackpot killer Registered Senior Member

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    A "position vector" is the vector from some origin to some point of interest.

    The speed of the point of interest is the absolute magnitude of the first time derivative of its position vector.

    There is no better way to describe it.

    - Warren
     
  20. chroot Crackpot killer Registered Senior Member

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    There is exactly one vector between any two points.

    - Warren
     
  21. Johnny5 Registered Senior Member

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    Yep, it sure does.

    If you go back and look at kevin's post, which i liked by the way, you will see that somewhere in there, he also uses the idea of "instantaneous time of travel"

    Neither he nor I have yet gone into the meaning of that, but it's simple enough.

    Let A, B denote two moments in time, and let it be the case that A before B. If it is not the case that there is a moment in time C, such that A before C and C before B, then A,B are consecutive moments in time, or better yet adjacent moments in time.

    Suppose that we formulate the laws of physics using the variable t for time.

    Let us use the natural number system, to indicate that two moments in time are adjacent as follows:

    t1

    t2

    Which implies that t2 after t1.

    Definition: Instantaneous change in the time coordinate t, is represented by dt, the differential of the variable t, and is defined in terms of moments in time as follows

    dt = t2 - t1
     
  22. Johnny5 Registered Senior Member

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    So there is no such thing as a curved vector?

    Like a vector arc along the surface of a sphere?

    Also, vector from P to P` is not the vector from P` to P, they have opposite directions.
     
  23. chroot Crackpot killer Registered Senior Member

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    Yikes. You gotta love the lazy calc teachers who lead students to misconceptions like this!

    - Warren
     
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