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Probability in Classical Mechanics: The Bouncing Ball

The discussion in the previous two sections has concentrated on use of the reciprocal velocity function to calculate time intervals. The same function can be used to calculate probabilities. All that is required to convert reciprocal velocity to probability density is to divide by the time to go from one turning point to the other. In probability terms, this 'normalizes' the function: it makes the probability of finding the particle somewhere between the turning points 100%. To convert probability density to probability, we have to integrate it between the limits we are interested in. If we take the bouncing ball as our example, we might ask the question "where is it most likely to be found?" The answer is simple: where it is moving slowest: near y = h. We can use the probability density to quantify the answer. If we divide the region between y = 0 and y = h into eight regions of equal width, we can calculate the probability of finding it in any particular region. The probabilities, in order of increasing y, are .0646, .0694, .0756, .0835, .0947, .112, .146, and .354. Hence it is over 5 times more probable to find the ball in the top region than in the bottom region, and the probability of finding the ball somewhere in the top two regions is 50%. Probability in quantum mechanics is the subject of the next section.

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