Water Fountains and Bottles

Have you ever tried to refill one of those disposable sport water bottles at a drinking fountain? The Evian or Poland Spring one liter bottles with the pop tops? Those things are pretty long and skinny, so unless your drinking fountain has really high water pressure there's little chance you can hold the bottle upright. Amazingly, even when the bottle is at an extreme angle, you can still fill the bottle almost to the brim. So this got me thinking, what is the fullness of the bottle as a function of the height of the water fountain stream?

First, I made a model of the bottle sitting on my desk. I approximated the dimensions like this:

The problem is formalized as so. We have a bottle whose lower lip is height h above the ground, and the angle of the bottle is defined as the angle between the ground and the bottle's vertical axis. So completely upside down is -90 degrees, and fully vertical is 90 degrees.


We're going to use MATLAB to solve this problem. The 3D model of the bottle is like this:
We're going to solve the problem by integrating a series of slices, each of which is a disc subtended by a chord. The slices of the bottle are integrated using a trapezoidal approximation, and each slice is added to the final volume. Like this:


If we plot the percentage of volume as a function of theta, we get this:
See!!?? The huge jump in volume right around 0 degrees, when the bottle is flat. This means the bottle is about half full for a tiny tilt, and if the bottle is only 20 degrees up, we're already 80-90% full.

Still when we fill a water bottle we aren't really thinking about the angle of the bottle. We're thinking more of the height of the bottle's mouth above the ground. So let's plot first the bottle's angle versus lip height.

There's that funny bump because for lip heights lower than 1" the bottle is hinging on the front rather than the back. We then plot the final relation between height and volume.
See? The bottle doesn't have to be tipped up at all. The full height is 7.75", but it's 70% full for just half the height. And that's the problem solved.