We start by describing how to visualize the dynamics during a bottle flip and how to analyze the resulting motion. The experiment is designed with the classical approach to the dynamics of extended bodies:The motion is decomposed into aof the center of mass and aaround the center of mass. This decomposition is natural since the only external force is gravity, which exerts no torque around the center of mass. As such, the water bottle flip serves as a prime example of conservation of angular momentum.

In addition to the water bottle and the tennis bottle shown in Fig., we will also consider a “rigid bottle” that contains an immobilized mass. The rigid bottle serves two purposes: to verify that we recover the usual rigid body rotation and to highlight the importance of the movable mass for a successful bottle flip.

A. Experimental setup and analysis

The experimental setup used in this study consists of a black background, a lamp for illumination, and a digital camera α-6000. Each experimental run (or bottle flip) takes roughly 1 s. Here, we recorded the films at 50 frames per second, a shutter time of 1/1600 s, and a resolution of 2 megapixels. Our recommendation is to use a minimum of 20 frames per second to gather enough data points, a maximum shutter time of 1/200 s to avoid blur in the moving bottle, and a minimum resolution of 1 megapixel (most smartphone-cameras satisfy such requirements nowadays). We typically ran 10 successful flips per bottle type with the same fillings and select the cleanest landings among them for analysis.

ω = dθ/dt. This quantity can be measured by tracing the top and bottom of the bottle on the videos. Another key ingredient of the analysis is to determine the motion of the center of mass of the total system. For the rigid bottle, the center of mass obviously remains at a fixed position along the bottle for all times. However, it is rather difficult to accurately determine the center of mass of the sloshing water—from the images, one cannot infer the precise distribution of water inside the bottle. Here, we simply proceed by an approximate analysis that is detailed in Sec. 1(a) 1(b) The rotational motion is quantified by the angular velocity. This quantity can be measured by tracing the top and bottom of the bottle on the videos. Another key ingredient of the analysis is to determine the motion of the center of mass of the total system. For the rigid bottle, the center of mass obviously remains at a fixed position along the bottle for all times. However, it is rather difficult to accurately determine the center of mass of the sloshing water—from the images, one cannot infer the precise distribution of water inside the bottle. Here, we simply proceed by an approximate analysis that is detailed in Sec. III B , based on the maximum height of the water mass along the bottle. This complexity of the water bottle [Fig.] is our prime motivation for introducing the tennis bottle [Fig.]. Namely, the exact positions of the tennis balls are easily determined. Subsequently, the center of mass is obtained by taking the mass-weighted average of the positions of the two balls and of the bottle's center.

h on each frame (see Fig. 2 10 NIH Image to ImageJ: 25 years of image analysis ,” Nat. Methods 9, 671– 675 (2012). 10. C. A. Schneider, W. S. Rasband, and K. W. Eliceiri, “,” Nat. Methods, 671–(2012). https://doi.org/10.1038/nmeth.2089 11 11. Mathworks, MATLAB 2015b ( The MathWorks , Natick , 2015). In summary, the experimental measurements consist of tracking the top and bottom of the bottle in each frame and of tracking the following additional points to determine the center of mass: (a) Water bottle: the maximum height of the sloshing wateron each frame (see Fig.). (b) Tennis bottle: the position of the tennis balls for each frame during the flip. The acquired digital images were imported into a computer, and the tracking was performed manually using ImageJ,simply using the point tool with auto-measure. The data were then processed using MATLAB.All manually tracked sets of data were filtered using smoothing splines (with a low smoothing factor of 0.99) to reduce the user-induced bias and noise.