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Thu 4 Mar, 2004 10:42 pm
Imagine you are on an absolutely flat plane. The plane extends infinitely in all of it's directions. Ten feet above you there is a thin layer of fog, a blanket of white extending infinitely in all directions, at a constant of ten feet above the ground (initial plane). Nearby is a tree. It is a thin tree with no branches and of infinite height. It goes up through the fog, and off into forever. You have an axe, and you chop down the tree. It falls down through the mist and hits into th ground. So it is now parallel to the ground plane, and extends forever off in one direction (so the tree serves as a ray). The question: at which point would the tree descend below the mist? Keep in mind, initially the tree was at a 90 degree angle with the ground plane and mist plane. When the tree has fallen so far that it is at a 30 degree angle with the ground, it will also be at a thirty degree angle with the plane of fog (simple rule of a transversal crossing parallel lines). When the tree is at 10 degrees with the ground it will be at ten degrees to the fog, and when it hits the ground it is obviously at zero degrees with the ground. So when is the jump when the tree no longer extends above the fog, and lays entirely along the ground? (Mathematically, and with one less demension, take two parallel lines, and attach a ray to the bottom one, extending through the top. Rotate the ray from it's point on the bottom line, until it is parallel and lies upon the bottom line. At which point as it no longer crossing the top line?)
This is kind of more of a riddle. I hope everyone has the patience to visualize the motion.
Only when the included angle between the tree and ground reaches zero.
Not a moment before.
That's the way I'm seeing it too, Adrian. Since it's of infinate length, even an angle of 0.00001 would still have it sticking above the fog.
Infinity is a wonderful thing. I just can't get enough of it.
You can't chop down the tree. Period. If the tree were to rotate, even at an infinitesimal rate, the other 'end' of the tree would be moving at infinite velocity.
Even if the tree could fall, wouldn't the (infinately distant) top-end therefore take infinately long to drop below the fog?
There's a problem with that answer, adrian, because now matter how infinitely far away it is, the point where the tree crosses the fog is exactly ten feet above the ground. So how could the tree "jump" from being ten feet above the ground, to touching it and just "skip" the motion between the ten feet?
No, Grand Duke. The velocity of the far end is determined by the velocity of the base of the trunk. Mathematically it's based on rotating a ray, which is intirely possible... I think?
SCoates wrote:No, Grand Duke. The velocity of the far end is determined by the velocity of the base of the trunk. Mathematically it's based on rotating a ray, which is intirely possible... I think?
I have to hold my hands up in ignorance at this point! (I'm an accountant by training and a railway engineer by profession).
I was working on the assumption that the longer the tree was, the longer the end would take to fall below the fog, for a constant velocity of falling. I'm curious to know if the good people of A2K can come up with a consensus of opinion on this problem.
I understand most of the answers, but not the question. Down is toward the center of mass. Is the mass infinite and the distance to the mass center infinite? If not, the ground can be flat, perhaps, but the layer of fog will form a segment of a sphere with the center at the center of mass. Thus the fog will be a dome which touches the ground in a circle, radius 10 miles?) with the center not necessarily near the base of the tree. Infinity produces lots of problems. Neil
Assuming significant gravity over a few hundred miles of tree height; the ground would be crushed by the enormous mass of the tree, and begin sinking vertically into the plane. The fact that the tree base was moving would be propagated up the tree trunk at hundreds of miles per hour. Long before the first hour elapsed the lower part of the tree would be stretched beyond it's allowable elasticity and the tree would begin to break into pieces.
If the plane is made of unobtainium and resists the enormous mass, the leaning of the lower part of the tree (after chopping it) would propagate up the tree at hundreds of miles per hour causing the tree to bend in a segment of an ellipse (hyperbola?)until the tree began to break into pieces, unless this tree material was able to endure being bent while heavily loaded at an angle.
If so, the cut end of the tree might move horizontally at an accelerating speed, to partially straighten the curve in the lower part of the tree. The perfectly flat, not crushed plane would offer little or no friction to impede the cut end from moving/scraping horizontally. I suppose the cut end would not be abraded by the scraping if there is no friction. Any friction would convert the tree's cut end to flaming saw dust at hunreds of miles per hour as it scraped horizontally forever if the tree did not break in pieces. Neil
IMO
IF The tree was infinitely long it could not fall as at some point the top of the tree would have to exceed the speed of light and this is not allowed.
If the tree broke up then it would not be infinitely long and then it could be measured and we may be able to do something with it. :wink:
Infinity is kind of hard to do much with.
Think about it conceptually, with parallel lines and rays. The problem has nothing to do with physics. For example, it geometry (which of course is very basic) the concept of rotating a line is taught. Are we concluding that it is impossible to rotate a line, since it's length is infinite?
I suppose we can rotate a line of infinite length if we ignore the fact that the far end would be moving at light speed, even if we rotate the line at one second of arc per eon. Relativistic effects would dominate and the line would not be straight while rotating relative to any observer. If we assume Isaac Newton was correct and modern physicists are wrong, then, I suppose we can rotate a line of infinite length. Neil
You're thinking about this horribly.
The concept of a line is merely a point of reference. It is not subject to gravity, or other confines of physics, like maximum speed. Concept can travel far faster than the speed of light. Any third-grader can multiply the speed of light if you give them a few minutes. It may not be useful to physics, but I think I have stressed enough times that we are not talking about physics. If anyone else has comments about breaking the speed of light or gravity, or inertia, please post them somewhere else. We are getting terribly off subject. Conceptual lines are not subject to any of these things.
Sorry, the use of a "tree" tends to make me think "Mechanically
) and a tree implies rigidity. (limited flexibility)
Poetic license may allow an "infinitely tall" tree but to make one that has some of the characteristics of a jellyfish may mean that we can't use that analogy at all.
Or perhaps we are barking up the wrong tree?
If the line has no mass, it will not fall or rotate if you cut it near the surface of the plane.
We can rotate the line consceptually about a fulcrum, at the plane until the far end is at an altitude of 10 feet. If the far end continues toward the the plane at one foot per second, we will have 3 parallel lines 10 seconds later. Total rotation time is infinity plus 10 seconds. Neil
Your thinking is to concrete. I should never have included a story problem involving physical items. In any case, "parallel lines" suffers conflicting terms. In relation to infinity any finite number functions as 0. The reason we cannot percieve the moment the tree crosses the layer of fog, is because as the distance down the top line approaches infinity, the distance between the two line approaches 0. The tree will only touch the bottom plane once it has touched every point along the infinitely long plane of fog, and with respect to that infinity the distance between the "parallel lines" is now neglible. So at that point, since we must factor in "infinity," there is only one line. This is supported by the fact that as the angle fromt he ground plane to the tree approaches zero, so too does the angle between the tree and the top plane, so when the tree touches the bottom plane, it does indeed also lie upon the plane of fog, for at that point the distance between them is dismissed.
Hi SCoates,
I believe the paradox you have designed here derives from the fact that you are trying to relate two different conceptual units, ie zero and infinity.
Zero has a finite value (zero), but Infinity does not have a finite value. Infinity is not simply a "very large number", it in-finite. The two concepts do not mix, and one cannot merge into the other through any transition.
When trying to imagine where the infinite line descends below the fog to traverse that last ten feet of space before becoming parallel to the ground, you are trying to define a "transition" which can never occur.
Which, rosborne, is why I feel that "parallel lines" is an incorrect term. It tries to apply both finite and infinite concepts. Unless of course the distance between the lines were also infinite, in which case it seems to me the lines would not exist, but that is probably just because I cannot comprehend infinity. My proposal is that from now on, whenever someone says "parallel lines" we correct them and say "You mean line segments." And we roll our eyes, as if we shouldn't even have to explain ourselves.
SCoates wrote:My proposal is that from now on, whenever someone says "parallel lines" we correct them and say "You mean line segments."
Unless of course, the lines are defined as being parallel to begin with, then they can be infinite in length and not segments. It's only the transition from a non parallel condition to a parallel condition in which the paradox arises.
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