Reaching the Speed of Light: A Thought Experiment on Halving Distance Traveled

[PeroK]

Quite right. I am having problems however understanding how 1 is reached. I appreciate you all trying to explain.

This is largely irrelevant to Zeno's paradox. Either it never reaches 1, in which case not even 1s can pass; or, in some sense it eventually reaches 1 in which case the universe lasts at most 1s.

Mathematically the finite sums never reach 1. But, we would like to identify 1 somehow as the limit of that sequence of sums. This can be done using the rigorous mathematics called real analysis, developed in the 19th century.

These infinite sums are extremely useful in physics but don't necessarily map to physical processes.

The real issue with Zeno's paradox, IMO, is that there is no reason to decompose time into smaller and smaller increments. It serves no purpose. Also, the conclusion is inherent in the approach. By only considering smaller and smaller increments, you can never consider any phenomena that take place outside that first second.

 

[jk_er_gamma]

That Zeno has done us no favours

What about giving calculus a day to save?

 

[sophiecentaur]

, IMO, is that there is no reason to decompose time into smaller and smaller increments. It serves no purpose.

The 'paradox' only goes half way with it so it falls over. Once Calculus, with its LIMITS, came along, it dealt with that problem and many others.
Perhaps Zeno and his paradox were just an early example of a Conspiracy Theory. Find an awkward question and then blame someone / something for it. Rationality is / was never very popular.
Edit: The function is actually not differentiable at the instant of impact, if you use a simple mathematical model. There will be a 'damped harmonic' type of motion as the ball and the wall collide. The actual time that the ball stops can be well defined that way - and the velocity approaches zero in a nice well behaved fashion. But Zeno can still muck about with the time of impact.

 

[sysprog]

The formulations that I've seen of Zeno's Paradox all begin with something like: before you can traverse the stated distance, you have to traverse half of it, and then you have to traverse half of the remainder, etc.. My reaction to that is that if that's so, you have to traverse the first half of the first half before you can traverse the second half of the first half, and before you can do that, you have to traverse the first half of the first half of the first half, etc., wherefore you could never even get started. This of course contradicts the postulate that you have a velocity, i.e., that you are in fact moving, with a speed. and in a direction, and so abuses the facts that all motions over finite distances may be described as inclusive of traversal of an infinite number of infinitesimal distances, and that all durations may be described as comprising an infinite number of infinitesimal moments or instants.

 

[scottdave]

I apologize that I haven't read all three pages of responses yet @ShaunM . In what context was this question posed to you? Is it possible that it was posed as a thought experiment? Another thing to think about - did they tell you that points A and B are fixed? What happens if B is also moving? Just thought I'd throw something out there to consider.