Optimal curve for a marble drop I'm doing a marble roller coaster project for my physics class at school. The first part of our roller coaster involves a marble falling downwards into a curve that will drop then go up 12 inches. What is the optimal shape of the curve so that the ball preserves the most momentum possible (so we can minimize the height that we drop it from)?


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*Maximum initial height of 36 inches (although I'm trying to minimize that)

*Initial speed from drop of 0

*Time taken is irrelevant

*We're assuming a level surface

*PVC insulator is pretty rough but will still allow a steel ball to roll down it

*The PVC pipe is pliable (since we cut it in half) so that we can pretty much shape it in whatever way we want

*The steel ball is around 7/16 inch diameter


Thanks
Edit: Feel free to close the question as Norbert has already sufficiently explained it both in his comments and in my answer (a restatement of what he said)
 A: You want to minimize energy lost by the marble since you want the lowest height that still allows you to climb 12 inches.
A marble starting from rest will lose energy due to slipping before rolling - this means the initial slope should not be too steep; after that you want the curvature to be gradual but the more rapidly you curve back up, the shorter the total distance - which ought to minimize the total energy lost due to friction.
A bit of math: we can calculate the maximum slope allowed for a ball to roll down the slope without sliding. If the coefficient of friction is $\mu$, then the maximum slope is given by
$$\tan\theta = \frac72 \mu$$
for a solid sphere - the derivation of this can be found in an earlier answer I wrote, which also includes some interesting calculations on the amount of power dissipated due to sliding. If you have a sphere of arbitrary moment of inertia $I = k m r^2$, then the general equation is
$$\tan\theta < \frac{\mu}{1+k}$$
The derivation of this can be found at this link - although my definition of $k$ is different than the one in that link, the math should be easy to follow.
Since the stated goal here is to lose the least amount of energy (so you can start the marble from the lowest height) it seems obvious to me that you want the initial slope to be just less than the critical slope: this will minimize the distance travelled which is favorable to minimize rolling friction (but I am pretty sure sliding friction would be much more significant).
Note that the rising slope (the 12" climb at the end) also needs to be less than the critical slope - otherwise the ball will not be able to convert its rotational energy back into potential energy (it will slow down without stopping spinning).
Incidentally, if you wanted a setup that was both "fast" and "lose little energy", you might find that dropping the marble vertically off a (slightly angled) steel anvil so it bounces back to the required height might be the most efficient mechanism - although that will depend on the coefficient of restitution of the marble-anvil system... worth a little side experiment.
A: Thanks to @Norbert Schuch for this answer
While time taken for the ball to travel the curve can be minimized through the use of a Brachistochrone curve, the shape of the curve doesn't really affect the upwards distance that the ball travess when you drop it from a certain height (unless there are oddities like sharp bends in the "track")
