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Q. 26

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
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Short Answer

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

b. Between and ?

c. At ?

a) The Probability is 0

b) The Probability is

.c) The Probability is

See the step by step solution

Step by Step Solution

Step 1 Given Information : 1.0-mm-diameter sphere bounces back and forth between two walls at x=0 mm and x=100 mm. 

Step 2 Calculation

We know that the Probability of finding a particle within the range exactly at point x is determined by multiplying the probability density P(x) by x

Prob (in x at x) = P(x) x

Notice that the center of the ball will be confined in the region from to , and that is because the radius of the ball is which is the distance between the center of the ball and the walls when the ball hits any of them.

.a) We are asked to find the probability that the center of the sphere being located at an exact x, meaning that . Hence the probability is

Prob (in 0mm at 50mm ) = P(x) 0 = 0

.b) Since the center of the sphere has an equal possibility to be anywhere in the given range, then the probability of the center being located within (  to x = 51.0mm ) is given by dividing the length of this region by the length of the total region that the ball is bouncing in

.c) Following the same procedures as in part (b), the probability of the center being located in the range between x = 75.0 mm and x = 99.5 mm is given by dividing the length of this region by the length of the total region that the ball is bouncing in

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