Bottle Throttle

Last week I got a question from my nephew Ezra: How does the bottle-flipping trick work? What's the best amount of water to use?

In case you're unfamiliar with the phenomenon, Ezra sent along a demo of one of his flips:

As a first approximation, I figured the water should stay fairly stable in the bottom of the bottle, and the main factor that dictates whether it lands upright is how much water is in it, and the range of impact angles that cause it to tip onto its base. I pictured the landing like this:

What matters here is the height of the center of mass, which we can calculate with

where mb and mw are the masses of the bottle and the water. The tipping point will be when the center of mass is over the contact point: Farther left, and it will fall on its side, farther right and it will stay upright. The maximum value of θ then is

We can plot this for different water levels to find the best height of water (assuming a 500 ml bottle, per this page):

This model gives the optimum water level as 12%, but I wasn't entirely confident in my simplified model. I wondered whether anyone had looked at this problem in detail, and lo and behold, an arXiv paper called Water Bottle Flipping Physics!

The paper looked at 3 cases: a rigid bottle, similar to the model I came up with; a can with a pair of tennis balls, which is a simpler model with mass moving around inside; and finally the water bottle:

Figure 3

The key finding in the paper was that the bottle's rotational momentum gets absorbed by the water. This happens in such a way that the bottle stops rotating with its base pointing down. In my model above, I didn't consider an existing rotational velocity on landing, which could easily tip the bottle. According to the paper, for the water bottle the ideal filling fraction is

where M is the ratio of the mass of water in a full bottle to the mass of the bottle itself. For the numbers I used, I get M = 56 and f = 12% again! It seems at least for the 500 ml bottle, the approximation works great, but for other sizes they'll diverge.