Not chaos
The gas of non-interacting molecules actually provides a good example with which to discuss chaotic vs. non-chaotic systems. An "easier", but still similar system to our gas is billiard balls on a table. This system is 2-d, rather than 3-d, and is much easier to visualize.
Our billiard table is square with corners at (-2,2), (2,2), (2,-2), and (-2,-2), where I measure everything in meters. It's a wonderful billiard table as the balls have no friction with the table and bounce off elastically from the walls. Hence if I start a ball rolling, it rolls forever.
A simple problem first: I start a ball rolling at the origin with velocity (0,1) m/s. How long does the ball take to return to the origin in seconds ?
You can read this Introductory note - What is Chaos anyway?
The answer is 4.
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1 solution
In the problem, it is said that the table is a square with vertices as ( − 2 , 2 ) , ( 2 , 2 ) , ( 2 , − 2 ) and ( − 2 , − 2 ) . This shows that the side length of the square table is 2 m. Now let us consider the origin as point O on the table where the ball to be rolled is placed at beginning.
The velocity of the ball is ( 0 , 1 ) m/s. This means that the ball starts to move along positive direction of y-axis with a velocity of 1 m/s.
Now, when the ball is rolled, it moves parallel to the y-axis. So, we can observe that since the vel. is 1 m/s along (+ve) y-axis, we can track its movement like this ---->
After a total of 1 sec, the ball is at ( 0 , 1 )
After a total of 2 sec, the ball is at ( 0 , 2 )
Now, since the table is bounded by given coordinates, the ball elastically bounces from the walls as said in the problem and now moves in the antiparallel direction to the initial movement, i.e., the ball is now moving parallel to (-ve) direction of y-axis with the same vel as earlier since this was an elastic collision. So -->
After a total of 3 sec, the ball is again at ( 0 , 1 )
After a total of 4 sec, the ball is finally again at ( 0 , 0 ) .
So, total time taken = 4 sec