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(Solved) Imagine, if you will, a rigid-walled box (the system boundary) containing 10 billiard balls, each with a mass of 0.


Imagine, if you will, a rigid-walled box (the system boundary) containing 10 billiard balls,

each with a mass of 0.1kg. All collisions (ball-to-ball and ball-to-wall) are perfectly elastic

(no energy loss). At the beginning of time, each ball is traveling 10m. s

a) What is the total energy of the system (in Joules)?
Now we instantaneously add 5 more billiard balls, identical in every way to the first 10,

but each of these five has an initial speed of 5m. s

b) What is the total energy of the system (in Joules)?

Time passes. Recall that each ball-to-ball elastic collision conserves the net kinetic en- ergy, though balls with more energy deposit some into balls with less. No energy is lost from the system during this time lapse, and all the balls will equilibrate to the same speed.

c) What is the average speed of any given ball (in m)? s

Now consider a scary thought experiment: Say the “atmosphere” is made up of count- less identical 0.1kg billiard balls, all traveling in random directions with a speed of 5m.

s

If we let our box open in the initial condition found in part a, we would have 10 balls going 10m. Conversely, if we let our box open after the equilibrium speed from part c was

s

reached, we would release 15 balls going at the speed you calculated in part c. If we had a machine that could extract work from a single ball by colliding it with the balls flying around in the atmosphere (thereby reducing the velocity of each faster-moving ball to that of the “ambient” balls - meaning the energy extractable is proportional to the square of the velocity difference - the speed of the released balls relative to the ambient balls - it will NOT be the same as the change in the absolute energy of the balls):

d) How much work could be extracted from the 10 balls in part a?

e) How much work could be extracted from the 15 balls in part c?

f) Qualitative question: Through the process described in part a through part c, did the ability of the system to do work on its surroundings increase or decrease?


 


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