Undergrad Measurement precision question

[calinvass]

Supposing we have 2 circles in a 2 d space that can bounce off each other like balls. These circles are made of an infinite number of points. We put the centre of each circle on a line and send them towards each other along the line. Are they going to bounce off keeping their trajectory on the same line? The derivative on the point of impact is zero and I expect they will return on the same line. However, if we want to place each circle on the same line, we will need to measure if it is in a right position with a certain accuracy. Say one circle starts from the left and the other from the right. If the accuracy is a finite number, I suppose the left hand circle will always bounce in 50% of the cases up and 50 %down. How can we tell the difference between precision of measurement and the world intrinsic randomness, as QM would suggest?

 

[phinds]

You seem to be talking about mathematical circles, in which case there is no imprecision involved. By the way "finite number" includes zero, which you seem to have overlooked. Also, if there IS imprecision (a real world scenario) I don't see why you would think "the left hand circle will always bounce in 50% of the cases up and 50 %down". If the center is off one way, the bounce will always be in the same direction, depending on which way the center's position is off from the axis.

I find your whole scenario imprecise and vaguely stated.

 

[calinvass]

Yes, my scenario requires a better description. The 50-50 chances are clear by the way of defining precision. If you try to put a point on a line, you place it by continuously measuring its position and then you leave the point where you think it is on the line.
"If the center is off way ..." The idea is that you try to put the center on the line. I should've add that once the circles are on the line you always give them an vector velocity perfectly parallel with the line.

Anyway, I've realized that my scenario implies a hidden variable, which QM seems to demonstrate it doesn't exist in many situations. My example is a case of a deterministic system that still behaves like a non deterministic one.

 

[Nugatory]

This isn't a quantum mechanics question (and the concern about precision that it raises is unrelated to the uncertainty principle of quantum mechanics). There are many classical systems whose behavior is exquisitely sensitive to the initial conditions. If we do not completely specify the initial conditions with sufficient accuracy, the behavior will be for all practical purposes random. The example that most people will be most familiar with is a tossed coin, which lands heads or tails apparently at random. The OP's example (calculating the rebound abgletwo identical perfect disks or perfect spheres colliding) will be familiar to everyone who has been through a first-year college-level classical mechanics class and is relatively easy to analyze mathematically.