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Relativity Paradox

  1. Nov 27, 2003 #1
    Lets say a ship, travelling 98% the speed of light approaches another ship travelling the opposite direction at the same speed. Both at a constant speed, so both have grounds for being at rest, at least from my understanding, but then wouldnt each record the other's speed to be 198% the speed of light? Relativity declares the universal "speed limit" to be light speed. Im probably missing something, so dont laugh at my naivety, but I cant figure it out.
  2. jcsd
  3. Nov 27, 2003 #2

    sorry, my math was wrong there, it would be 196% light speed
  4. Nov 27, 2003 #3


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    The thing you are missing is that velocities don't add the way we thought they did under Newtonian physics.

    To add two velocities such as in your example you need to use the formula:


    In which case you get an answer of


    Since the advent of Relativity we have learned that this is the formula to use when adding velocities. It is just that in normal situations where u and v are small compared to c, the answer comes out to be so close to


    that we can use the second formula and get an answer that is within reasonable limits of error.
  5. Nov 28, 2003 #4
    thank you Janus, that makes sense to me now. I thought it might be something like that, but I've never seen that equation before. Thank you again.
  6. Dec 3, 2003 #5
    So, can anyone tell me WHY this is how it is? and not just because thats what einstein said. Are there actual physical reasons this equation exists? And how do we know this, if we've never accelerated 2 objects to 99% light speed?
  7. Dec 3, 2003 #6


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    It is a consequence of the fact that light always goes c in every frame of reference. The question of why light always goes c in every frame of reference doesn't really have an answer -- it just happens this universe operates that way.

    - Warren
  8. Dec 4, 2003 #7
    Sure we do!

    In particle accelerators particles are routinely accelerated to velocities >0.99 c.
  9. Dec 4, 2003 #8
    I know this, but have 2 particles been accelerated towards each other, with some way of "viewing" one from the other?

    Also, c always remains the same in all references, but c is not beign measured here, the speed of a ship in comparison with another ship is. Are you saying that the LIGHT from the ship can only appear to go this fast?
  10. Dec 4, 2003 #9


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    "Collider" accelerators are exactly that, and the LEP (Large Electron Positron) collider is perhaps the most powerful, at least until the LHC (Large Hadron Collider) comes on stream at CERN.

    One way to assess what one particle 'sees' of the other that it collides with is to measure the energies of the particles before they collide head on, and the total energy of the 'rubble' after the collision.

    In short, each of the two particles 'sees' the other moving towards it at exactly the speed given by the equation in Janus' post.
  11. Dec 5, 2003 #10
    Another good check is to use particles with quite short lifetimes, such as a muon (lifetime ~ 10 us). If you accelerate such a muon to velocities ~0.99 c then due to the time dilation the lifetime seems to increase in the lab-reference and you will get reasonable probabilities to still detect the muon after a few minutes.
  12. Dec 5, 2003 #11
    But according to the equation above, accelerating a particle to 99% light speed, and having it smash into a stationary particle, this collision will produce more energy that if 2 particles were accelerated towards each other at 99% light speed, or it would be really close, am i correct? This may be how it is, but it just doesnt seem logical to me
  13. Dec 5, 2003 #12
    The two-accelerated-particle case has a higher total energy (invariant mass/energy + kinetic energy) than the one-accelerated particle case, so if the two particles were to annihilate on contact, for example, the EM radiation produce in the first case would have higher energy (higher frequency) than in the second case.

    That's not worded very well - I appear to have lost any ability I ever had to form sentences, but I hope you understand what I mean...

  14. Dec 5, 2003 #13

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    Let me just add that this is not just something weird about light---it's something weird about the structure of the universe. Anything traveling at the speed of light would be measured to have the same speed in any frame.
  15. Dec 5, 2003 #14
    I think I understand it, but that doesnt explain there relative speeds beign almost identical, just that one has higher energy. How can accelerating one particle to 99% c with respect to earth, say, and another towards the first yeaild the same relative speed as a particle hurled at a stationary object? And how can the energy's be different if ones speed is the same?
  16. Dec 6, 2003 #15
    "And how can the energy's be different if ones speed is the same?"

    Because E = mc^2, and m = m0/sqrt.(1 - v^2/c^2), so at relativistic speeds (more of) the energy you put in hoping to increase v instead increases the mass. But the energy is still there, and when it hits something else it's expended (or some of it is).

    So if you took a (charged!) pea, and accelerated it in some stupendous accelerator to near light speed, it might eventually "weigh" (have the same mass as) a double-decker bus, but would actually be smaller than the original pea from the viewpoint of the experimenter! I don't need to tell you the effect of a a double-decker bus hitting something at near light speed (or two double-decker buses, ie: fast peas, hitting each other at near light speed ;-)
  17. Dec 7, 2003 #16
    Obvious this would be a new Theory maybe called:The Big-Bus-Bang?

    The question you then have to ask, what are the chances of two buses arriving at the same time?:wink:
  18. Dec 8, 2003 #17
    No, it would be the theory of the small pea crunch. Alternatively if the peas were energetic enough, they might create a brane, then that would most likely end the universe as we know it, and could be called the "Pea-Brane" Theory of the little crunch.

    I thought that's what they were supposed to do, rather than evenly seperated in time (whatever that is), which would be more useful.

    BTW, what the hell's a "Cave-Ranyart", and the "Moorglade"?
    Last edited: Dec 8, 2003
  19. Dec 8, 2003 #18
    Last edited: May 2, 2004
  20. Dec 8, 2003 #19
    One thing I don't understand is:
    The fact that light always goes the same speed independant of the reference. How does this give an equation such as:
    (u+v)/(1-u*v/c^2) which at low speeds is approx. u+v
    instead of
    (u+v)/(1-(u^2)*(v^2)/c^2) which at low speeds is approx. u+v

    I'm pretty sure it has to do with the hypotenuse (sp) of some triangle light makes... but please explain... thanks
  21. Dec 8, 2003 #20


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    The equation

    [tex]u = \frac{v+u^\prime}{\sqrt{1+v^2 u^{\prime 2} / c^2}}[/tex]

    would not be dimensionally consistent. Inside the square root, you have a dimensionless quantity (1) added to a quantity with units of velocity squared, which doesn't make sense. The total equation would not be dimensionally consistent.

    - Warren
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