The following question is from 2016 $F=ma$ exam. My question is with regards to one step in the reasoning of the solution.
Original question: A small ball of mass $3m$ is at rest on the ground. A second small ball of mass $m$ is positioned above the ground by a vertical massless rod of length $L$ that is also attached to the ball on the ground. The original orientation of the rod is directly vertical, and the top ball is given a small horizontal nudge. There is no friction; assume that everything happens in a single plane. Determine the horizontal displacement $x$ of the second (originally top) ball just before it hits the ground.
The given answer is $3L/4$.
I understand that the C.M. does not move horizontally, so the horizontal displacements of the balls are always in $3:1$ ratio. However, how do we prove that the heavier ball never lifts off the ground? To me it seems plausible that while the C.M. is moving down and the rod is pivoting around the C.M., the heavier ball could lose contact with the ground. Hence, it is totally plausible that when the lighter ball hits the ground, the rod is not in a horizontal position. How do I eliminate this possibility?