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Thread: Another special relativity problem

  1. #1 Another special relativity problem 
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    Here's another special relativity problem, this one's a bit harder to wrap your head around than the last one:

    A car approaches a ditch at relativistic speed. When both at rest, the car is large enough to drive over the ditch without falling in. However, from the ditch's reference frame, the moving car is length contracted such that it is small enough to fall into the ditch. But remember that from the car's reference frame, the ditch, which was already too small, is length contracted, which means its even smaller, so it would definitely be too small for the car to fall in.

    The car is rigid, which means it cannot bend. Also, assume that if it were to fall into the ditch, it would stop moving the instant it fell.

    Will the car drive over the ditch or will it fall in? Justify your answer.


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  3. #2  
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    This is a trick question. The friction on the tires will cause the car to instantly ignite upon reaching relativistic speed, and the ball of fire will travel only as long as the momentum allows, stopping just a single foot short of the now normal sized ditch.


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  4. #3  
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    Haha, touche,

    Lets assume the car is a hovercraft, so we neglect friction and surface contact. But the hovercraft will still fall if there is no ground below it.
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  5. #4 Re: Another special relativity problem 
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    Quote Originally Posted by Arvand
    Here's another special relativity problem, this one's a bit harder to wrap your head around than the last one:

    A car approaches a ditch at relativistic speed. When both at rest, the car is large enough to drive over the ditch without falling in. However, from the ditch's reference frame, the moving car is length contracted such that it is small enough to fall into the ditch. But remember that from the car's reference frame, the ditch, which was already too small, is length contracted, which means its even smaller, so it would definitely be too small for the car to fall in.

    The car is rigid, which means it cannot bend. Also, assume that if it were to fall into the ditch, it would stop moving the instant it fell.

    Will the car drive over the ditch or will it fall in? Justify your answer.
    This is a standard special relativity question, the resolution of which involves being very careful with "simultaneity". See for instance the problem stated in terms of trapping a train in a tunnel in Wolfgang Rindler's Introduction to Special Relativity

    There are other variations on this theme.

    http://en.wikipedia.org/wiki/Ladder_paradox
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  6. #5 Re: Another special relativity problem 
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    Dr. Rocket is of course correct.

    Quote Originally Posted by Arvand
    Also, assume that if it were to fall into the ditch, it would stop moving the instant it fell.
    This assumption is of course a problematic one because there is an issue of simultaneity involved. You assume that it would fall the instant that there is no support for either front or back wheels at the same time. But of course we know that simultaneity depends on which inertial frame we are looking in. If there is a special inertial frame then it is the rest frame of those that are moving - the rest frame of the vehicle, but of course in their frame there is never a time when there is no support for both front wheels and back wheels at the same time and thus there is no time at which "it would stop moving".

    Those in the rest frame of the ditch do see a time in which it seems to be the case that there is no support for both front wheels and back wheels but clocks on the front and the back of the vehicle would reveal that they are seeing these parts of the car at different times in their own frame. This is in fact a good explanation for lorentz contraction. They are simply seeing the back end of the car at a later time when the car has traveled farther in the direction of motion.
    See my physics of spaceflight simulator at http://www.relspace.astahost.com

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  7. #6  
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    The car would drive over it! b/c you said, at rest the ditch is small enough to drive over. Just because the perspective or view changes(to the viewer) does not mean the object has changed. when you look through a magnifying glass the words appear larger, but in reality, are they??!
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  8. #7 Re: Another special relativity problem 
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    Quote Originally Posted by mitchellmckain
    Dr. Rocket is of course correct.
    Of course!
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  9. #8  
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    If an object is in front of a object moving at relativistic speed would they even see length contraction of the object in front of them? I thought that was something that happened to objects moving along side the object. I heard that at such speed the view in front of you is distorted in that the center of your view is blue shifted and the sides of your view may contain objects you have passed and they are red shifted. Also the view takes on a tunnel vision like look. So would objects in front of you look shorter?
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  10. #9  
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    Quote Originally Posted by Wildstar
    If an object is in front of a object moving at relativistic speed would they even see length contraction of the object in front of them? I thought that was something that happened to objects moving along side the object. I heard that at such speed the view in front of you is distorted in that the center of your view is blue shifted and the sides of your view may contain objects you have passed and they are red shifted. Also the view takes on a tunnel vision like look. So would objects in front of you look shorter?
    You have to be careful with the verb "to see" in relativity.

    Most often, when you see a statement as to what an observer "sees" that statement relates to what he would physically measure in his own reference frame -- so you notions of time dilation and length contraction as expressed quantitatively with the Lorentz transformation.

    On the other hand, it appears that what you are addressing in your post is what an observer would infer from the pattern of photons that impact on his retina. That is a different kettle of fish altogether.

    Penrose reported on some calculations that he did with regard to what an observer would actually see with is eyes n(I think this may be in his book the Road to Reality. Based on the Lorentz contraction, you would expect that a circlular hoop passing near you at high speed would appear to be an oval, due to length contraction in the direction of motion. A physical measurement, say using a ruler, would indeed show it to be an oval. But, according to Penrose, the visually perceived shape of the hoop would be circular, due to the finite speed of light and differences in path length between different pieces of the hoop,
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