A circular billiard table is given with a cue ball at the circumference. It is shot at an angle of θ to the line from the ball to the center of the table. For what angles θ will the ball return to this point, assuming the ball keeps going indefinitely?
I came across this question in a book written by a friend and I couldn't get a satisfactory answer for myself. All angles should satisfy this question, right? If not, Why? Can we get a generalized equation or range for all angles of θ that satisfies this question?
I would say the answer is no, not all angles will satisfy this question.
Let's start with the easiest to understand cases, where the ball does return to the start. If the angle is $\pi/n$ where n is a positive integer, then the ball returns in $n - 1$ bounces after going around the table once. Now consider what happens when $n$ is a rational fraction so that the $n=a/b$ where $a$ and $b$ are integers with $a>b$. Now the ball returns after $a-1$ bounces and has to go around the table $b$ times to return to the start. It should be clear that ball has to make an integer number of bounces to return to the start as we can't define a fractional bounce.
Now what if the angle is $\pi/m$ where m is an irrational number like $\sqrt{2}$? Now it is not possible to find an integer number of bounces that will return the ball to the start, because an irrational number multiplied by a positive integer is still irrational. Therefore, in this case the ball can bounce around the table indefinitely without ever returning to the start. That's kind of crazy, right and at the same time helps to give a mental image of the concept of infinite divisibility.
In case you are wondering why "an irrational number multiplied by a positive integer is still irrational" lets to the contrary, suppose there was a case where $n_1 \sqrt{2} = n_2$ where $n_1$ and $n_2$ are both positive integers and we know $\sqrt{2}$ is irrational, then we can rearrange the equation to $\sqrt{2} = n_2/n_1$ and $\sqrt{2}$ is now equal to the ratio of two integers, which is impossible by the definition of an irrational number.
In summary, the ball will return to the start for angles $\theta = \pi/r$ where r is a rational number and will not return to the start if r is irrational. There are many more irrational numbers than there are rational numbers, so if the angle is random, there is a great probability that the ball will never return exactly to the start.