Cranberry sorting

Cranberries are a tart berry grown in the New England region of the United States. Fresh, high quality berries are fairly hard whereas rotten berries are squishy. To help sort the good berries from the bad, farmers developed a bounce board separator, in which the cranberries are dropped from a fixed height. Good ones bounce, rotten ones don't, and the machine is configured such that there is a wall the berries must bounce over to qualify as fresh. You can see a video of such a machine in operation here.

A rotten cranberry will lose at least \(90\%\) of its total energy during a bounce. If the cranberries are dropped from a height of \(10~\mbox{cm}\), what is the minimum height in cm of the wall so that no rotten cranberries could ever bounce over it?

Details and assumptions

  • Neglect any air resistance.
  • You may assume the path of the cranberries are almost vertical.

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