A 1.0 -mm-diameter sphere bounces back and forth between two...

A 1.0 -mm-diameter sphere bounces back and forth between two walls at $x=0 \mathrm{mm}$ and $x=100 \mathrm{mm} .$ The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of speed. At a random instant of time, what is the probability that the center of the sphere is a. At exactly $x=50.0 \mathrm{mm} ?$ b. Between $x=49.0 \mathrm{mm}$ and $x=51.0 \mathrm{mm} ?$ c. At $x \geq 75$ mm?

A 1.0 -mm-diameter sphere bounces back and forth between two walls at $x=0 \mathrm{mm}$ and $x=100 \mathrm{mm} .$ The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of speed. At a random instant of time, what is the probability that the center of the sphere is
a. At exactly $x=50.0 \mathrm{mm} ?$
b. Between $x=49.0 \mathrm{mm}$ and $x=51.0 \mathrm{mm} ?$
c. At $x \geq 75$ mm?

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Key Concepts

Measure Zero in Continuous Distributions

A key concept in probability involving continuous measures is that the probability assigned to any single point is zero because a point has no length (or measure). In contrast, the probability of finding the object within an interval is proportional to the measure (length) of that interval.

Continuous Probability Distribution

In continuous probability theory, the set of possible outcomes is uncountably infinite, which means that the probability of the occurrence of any single, precisely defined point is zero. Instead, probabilities are assigned to intervals or regions, with the probability calculated as the integral of the probability density over that region.

Uniform Motion

When an object moves at a constant speed along a path, its position at any random time is uniformly distributed over the accessible range. This implies that the fraction of time the object spends in any particular interval is directly proportional to the length of that interval, assuming no variations in speed.

Elastic Collisions

Elastic collisions are interactions in which both momentum and kinetic energy are conserved. In the context of a sphere bouncing between two walls, this means that the sphere's speed remains constant throughout its motion, leading to a periodic and time-symmetric trajectory.

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