In Zeno’s paradox
How did he even manage to think of the theory behind the paradox when it’s so obvious that if you throw a rock at a tree, it will actually hit the tree.
I love paradoxes but they are totally mind boggling. 
The same holds true for not only the last bit of the rock’s journey, but every part of it.
As far as we know, the universe isn’t actually infinitely divisible, but is granular. I don’t think it’s this that solves the problem, but rather that, as the number of divisions approaches infinity, their size approaches zero.
That’s the whole point of a paradox: good premises, good reasoning, and a completely unacceptable conclusion. And on the surface, the reasoning seems reasonable, doesn’t it?
Anyway, the resolution is simple. When the distance is halved, the time it takes to cover it is halved as well.
Only if the rock were traveling at a constant velocity, which is unlikely.
If you want to get really picky about it, sure. But the deceleration is due to air resistance, which for a small rock is going to be damn near zero.
This is the same thing as placing trip wires at 4’, 2’, 1’, 6", 3", etc., you will place an infinite number of trip wires without reaching the tree. Then imagine throwing a ball so that it trips all the wires. It will trip each wire and still hit the tree.
The paradox is based on the assumption that an infinite number of non-zero steps must require an infinite amount of time. In Xeno’s time this was the conventional wisdom.
But we now know that the sum of an infinite series can be finite. Ultrafilter is quite correct - as the distance is halved, so is the time. The fact that we can imagine an infinite number of steps in no way implies that these cannot all be accomplished.
It works for shoes as well. The number of shoes my wife owns is finite but the number she needs to own is infinite.
There is a fallacy in the reasoning. The reasoner assumes that the endpoint is the tree. (Or, in the case of dropping a hammer, the ground.) This is wrong. Since the rock has enough velocity to go past the tree, it’s “target” is actually a point beyond the tree. In the case of the hammer, the “target” is the centre of the Earth. In both cases, something gets in the way of the projectile on its way to the target. Therefore, the rock hits the tree and the hammer hits the ground.
Holy crap, I think we’re married to the same woman!
What a paradox! :smack:
if you got that from a website, id be interested in the link.
thanks.
I don’t think the problem has anything to do with loss of velocity, but rather the conceptual impossibility of passing an infinite number of points in a finite space of time.
Make it a man walking between two trees, then.
This is one of Zeno’s four paradoxes of motion. You can read more about the interesting ones here.
The fourth paradox of motion, FYI, is only difficult to understand if you’ve never encountered the idea of relative velocity. Back then, no one had, but nowadays, the opposite is true.
My point was that the rock does hit the tree and the hammer does hit the ground, because the tree and the ground get in the way before the projectile reaches its natural destination.
Yes, but in doing so, it still traverses a distance that is conceptually infinitely divisible, the same is also true if someone leaps in and catches it halfway, or the thrower swings it and never releases - it is still moving along a path that is conceptually infinitely divisible.
I understand what you’re saying, but I don’t agree with it. If Zeno drops a hammer, he is assuming that the hammer’s destination is the ground. As you say, this is a finite distance that is infinitely divisible so the hammer must never readh the ground. I contend that the hammer’s destination is not the ground, but the centre of the Earth because gravity does not stop at the ground. Since the destination of the hammer is thousands of miles away, the ground five feet from the hammer’s starting point is well within the first half-distance that must be covered.
And how does it cover the first 2.5 feet of that?
The point of the paradox is that whatever distance the rock has to cover, it first has to cover half of that. And in order to cover that distance, it has to cover the first half of that. And so on and so forth, and you never get anywhere. Like I said, think of it as a man walking from Athens to Sparta.
It doesn’t matter. The destination is about 12,000 km away. In order to reach the centre of the Earth, the hammer must first traverse 6,000 km. two-thirds of a meter is much less than 6,000 kilometers. So the hammer never reaches the first half-distance because it does hit the ground before it can even get near it. By saying “how does it cover the first 2.5 feet of that?” the assumption is that gravity stops at the ground, and that the if you dug a hole, then the hammer would still stop at the original ground level. But since gravity does not stop at ground level, then the hammer must continue to fall and the ground gets in the way before it can reach the first half-distance.
I understand the paradox. But using a dropped hammer is a bad example of it. Walking from Athens to Sparta comes closer to it.