Suppose a large amount of particles are bouncing back and forth between x=0 and

Question

Suppose a large amount of particles are bouncing back and forth between x=0 and x=1, except that at each endpoint some escape. Let r be the fraction reflected each time; then (1-r) is the fraction escaping.Suppose particles start at x=0 and head toward x=1; eventually all particles escape. Write an infinite series for the fraction that escape at x=1 and similarly at x=0. Sum both the series. Which is the largest fraction of particles which can escape at x=0? (Remember r has to be between 0 and 1)

Any helpful hints as how to approach this and comments would be helpful. Thanks!

Explanation / Answer

LET Q BE THE NUMBER OF PARTICLES STARTING AT X=0 AT THE BEGINING TRAVEL 1 SO THEY GO TOWARDS X=1 WALL Q[1-R] ESCAPED AT X=1 Q*R GET REFLECTED AT X=1 THEY NOW GO TOWARDS WALL X=0 TRAVEL 2 QR[1-R] ESCAPED AT X=0 QR*R GET REFLECTED AT X=0.. TRAVEL 3 QR*R[1-R] ESCAPED AT X=1 QR*R*R GET REFLECTED AT X=1 TRAVEL 4 QR*R*R[1-R] ESCAPED AT X=0 QR*R*R*R*R GET REFLECTED AT X=0 THUS WE FIND AFTER MANY TRAVELS THE PARTICLES ESCAPING AT X=0 ARE Q[1-R][R+R^3+R^5+........+R^(2N-1)+.....] [USING FORMULA FOR SUM TO INFINITE TERMS OF A GEOMETRIC SERIES S=A/(1-R)......WHERE A IS THE I TERM AND R=COMMON RATIO

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