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Thu 4 Mar, 2004 10:28 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.
An infinitely tall tree would never completely descend below the mist. The point (line segment as the trunk approached horizontal) at which the trunk cut through the mist would simply progress outward as the tree fell.
BTW, an infinitely tall tree would have infnite mass and infinite inertia, so you could not push it hard enough to get it to fall, even if there was an impossibly uniform gravitational field also extending to infinity.
But it would have to. Take a look at the mathematical version. We can percieve the motion at the base of the ray. If the motion of the ray remains constant it will run into the other line. It is the extention of the ray that we can't percieve.
Why would it have to?
You could only perceive motion at the base if the motion of the "top" of the tree was not limited by the speed of light and could fall infinitely fast.
If you could get it to fall that far, a tree would be flexible enough that the trunk could lie flat on the ground as far as you could see even though the "top" was still above the cloud layer.
For imaginary rays where physics is not a constraint, you can either construct the ray parallel to the plane, or at an angle and rotate it arbitrarily close but never parallel.
truth
I would have to agree with Terry. This thought experiment DEFINES the tree as having an infinite hieght making it logically impossible for it ever to fall to a level equal to that of the ground. The tree would have to have a top for that.
Ooooooh, Terry, you're so profound!
I got smacked mid-coitus when I said that to a girl.
The point where the tree is touching the mist is
tan(f)*h , where f is the angle between the tree and the ground, and h is the height of the mist (10 feet).
If the tree is falling with uniform rotation, f goes from pi/2 to 0, and tan(f) goes from zero to infinity.
So the tree falls to the ground (f reaches zero), and the point of intersection of fog and tree goes to infinity.
Nevermind that the three reaches monstrous speeds while falling; since it is infinite, the speed of falling approaches infinity towards the infinite heights.
What was the question again?
The instant the angle between the tree and the ground is zero. Why make things more complicated than they are. The real question is who has to clean the damn thing up.
Computer ate my response this morning... but yeah, the infinite speed thing.
Infinity is a wonderful thing. I just can't get enough of it!
joefromchicago wrote:Craven de Kere wrote:I got smacked mid-coitus when I said that to a girl.
OK,
Craven, you need to
stop getting laid.
What could stop that kind of momentum?
So, Terry, you're saying if two rays are attached at the base, it is impossible to rotate one of them until it touches the other?
Second question, on the same subject: The angle between the tree and the ground remains the same as the angle between the tree and the fog as the first angle decreases to zero. So when it reaches zero and touches the ground, the angle between the tree and the fog will also reach zero, and the tree will lie upon the fog. What's the error in the math there?
::scratches head::, maybe I'm missing something, but won't the tree stop touching the fog when it's perfectly level with the ground? Don't see what's so complicated about this...
truth
WHen I was your age, Craven, coitus interruptus would have been an impossiblity. Now it is a persistent possibility. Even when I'm alone.
Scoates, the problem with trying to rotate an infinite ray is that since it is infinitely long, it the "end" is always infinitely far away from the base ray.
No matter how far or long it falls, there will still be some infinitely-distant part of the tree above the fog. But for the angle between the tree and ground to be zero, the tree must be parallel to the ground all the way to infinity and cannot extend through the fog at any point. This is only possible if the height of the tree is finite.
Are you familiar with Zeno's paradoxes?
Lets allow the damn tree to fall. || to the ground at an infinitesimal height above the groud.
|| to the fog, ten feet below (minus a hair.) Angle w/ fog 0°.
Angular motion. The thing starts at pi/2 radians, goes through pi/3, pi/4, pi/6, until it reaches zero. Zeno's paradoxes really aren't -- they're just exercises in faulty reasoning.
Zenophobia = fear of being divided in half.
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