Excerpt from Infinite Loops in Finite Time Some Observations The second difficulty is that a number of physical theories actually predict the occurrence of clustered variation. These theories are idealizations, but they are reasonable and useful idealizations, that can only be avoided at substantial cost. The best known of these (discussed in [fleck, is the bouncing ball (figure The simplest and most straightforward model of a ball bouncing on the ground is that the ball is a rigid object, and that at each bounce, the ball ...
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Excerpt from Infinite Loops in Finite Time Some Observations The second difficulty is that a number of physical theories actually predict the occurrence of clustered variation. These theories are idealizations, but they are reasonable and useful idealizations, that can only be avoided at substantial cost. The best known of these (discussed in [fleck, is the bouncing ball (figure The simplest and most straightforward model of a ball bouncing on the ground is that the ball is a rigid object, and that at each bounce, the ball bounces up with a speed that is a fixed fraction p of the speed with which it hit.4 Since the time between bounces is proportional to the speed leaving the ground, it follows that the time between bounces decreases in a geometric series, and that therefore the ball attains perfect rest in a finite time, having carried out infinitely many bounces. (note that this is different from a standard damped harmonic oscillator, in which the frequency is constant, and the system therefore in principle never reaches a state of absolute rest.) Admittedly, this is an idealization; with an actual ball, beyond a certain point, the system will be dominated by the internal vibrations of the ball, which have a fixed frequency. However, to take this into account would require modelling the ball as an elastic object, which greatly complicates reasoning. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at ... This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
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