Prove that a particle will never pass through the centre of a sphere under a condition.

Question

Question: A particle was fired inside of a sphere. There was no gravity acting on the particle, no air resistance and each time it hit the inside of the sphere, it reflected without losing any velocity. If the particle doesn't pass through the centre of the sphere before the second bounce, show that it will never pass through the centre.

My attempts:

I considered representing the points where the particle reflects as a variable point and showing that the angle will never equal zero, no matter where the particle comes from (assuming it isn't coming from the centre).
I also considered looking for a vector representation of each reflection to see if there were any interesting results, although I personally couldn't find anything.
I also considered the possibility of a recurrence relation that related each angle of reflection, although this was also futile.

Note: Although a geometric proof would be helpful, I was looking for a more vector related proof. If there are no ways to do this, then I am happy to accept a geometric one. Vectors would be prefered, or at least some algebraic proof, but if nothing can be done, there's no issue.

Any help or guidance will be appreciated!

Answer 1

Let's assume the particle passes the origin at some iteration. It follows a straight line and hits the sphere. Notice that a straight line passing the origin and the sphere is a radius, and therefore normal to the sphere. Because of the normality to the surface, the particle returns in the same straight line (this can be proven by symmetry arguments)- and it must pass the origin again.

We conclude that if at some iteration we pass through the origin, we are forever confined to the same line, passing the origin at every iteration after the initial one.

If we use the same logic in reverse-time we conclude that it passed the origin at every iteration before the initial one.

We are told that the particle did not pass the origin at the first iteration, so we can conclude that it never will.

Answer 2

Suppose we solve this backwards, backtracking the path that needs to be taken from the center to the firing point.

Consider what needs to happen for the particle to reach the center for the first time. To reach the center, the particle must have traveled from the previous point where it bounced (or been launched from somewhere along the path that would have emanated from that bounce). Without loss of generality, we can call that the point $(0,0,1)$. At that point, the tangent is the plane $z=1$. Therefore, the particle hit at a perpendicular to the tangent and is reflected directly on its original path, meaning it goes straight towards the point $(0,0,-1)$. In doing so, it passes through the center once again. Therefore, in order for it to be the first time reaching the center, the particle must have been launched at some point $(0,0,a)$, for some $0<a<1$, with a launch angle either $\hat{z}$ (to reach after one bounce) or $-\hat{z}$ (to reach before the first bounce). The entire path taken by the particle (even after the first time it hits the center) lies on the z axis.