Detecting ground distance through rubber
It may be easier to suggest a solution if you were to "describe the goal" of your project.  

If this is just a science experiment, I like your idea of accelerometers.  What you need then is to take the next step and understand dead-reckoning in order to determine the height of the ball.

In other words, the integral of acceleration is velocity.  And the integral of velocity is position.  That means the integral of the integral of acceleration is position. As you may or may not know, for each integral you add a constant. The current speed for the first integral operation and the current position for the second.  But we can assume we can detect hitting the ground.  And, at that point, our position is X = 0 and our velocity is also 0.

That said, dead reckoning is difficult to use mostly because of the accumulated errors.  However, you may have more luck as you get to "reset" your values each time you detect impacting the ground.