# Problem: Marble and tube

My family recently visited Moon Marble Company, where my daughter picked up a bunch of marbles.  We’ve been having fun concocting ways to entertain ourselves, one of which was to roll the marbles down a cardboard tube.

A marble is rolled down a cardboard tube. What can we calculate?

To calculate how far a marble will roll along the floor after being dropped down a tube leaves the comfortable world of school physics and enters the real world of rolling friction on cardboard and carpet.  If the tube is close to vertical, the marble will be going quite fast as it exits the tube but won’t go very far horizontally.  If the tube is close to horizontal, the marble will be going quite slowly as it exits the tube.  Our intuition tells us that neither will work as well as somewhere in the middle, so we can search for the angle $\theta$ that maximizes the horizontal distance of the marble along the floor.  This is a case for which the rolling resistances in the tube and on the floor will partly “cancel”, and my daughter’s play tests show that we can get pretty good results just by tracking the proportionality of the velocity to a function of the angle $\theta$.

What physics models do you have to use to solve this problem?