**Abstract**

The study was started to analyze the widely known ‘bottle flip’ phenomenon. The bottle flip is when a half-filled water bottle stands upright when thrown. It is difficult to analyze the motion because the distribution of fluids in a bottle is constantly changing. Thus, a parabolic motion model was constructed to analyze the motion of the bottle. This is due to the law of conservation of angular momentum being established because the resistance of air and resistance other than gravity can be ignored. Through the statistical position mean, the center of mass and moment of inertia were induced and the model was constructed. As a result of the experiment, the correlation between the amount of oil inside the bottle and the distribution of location was identified.

**Contents**

**1. Introduction**

**2. Theory**

2.1 Projectile Motion

2.2 Conservation of Angular Momentum

2.3 Model based on fluids

2.3.1 Tennis Ball Approach

2.3.2 Center of Mass Approach

**3. Experiment**

3.1 Angular Momentum-Center of Mass Experiment

3.2 Projectile Motion based off Center of Mass

3.3 Experiment Model based off fluid mass

**4. Experiment Result**

4.1 Angular Momentum-Center of Mass Experiment

4.2 Projectile Motion based off Center of Mass

4.3 Experiment Model based off fluid mass

**5. Conclusion**

**6. Reference List**

**Motion analysis of the water bottle flip **

**1. Introduction**

* Figure 1*

When you throw a bottle as you spin it, it seems very simple to land upright again. But there are several difficulties in making an accurate model of the bottle flip. This study focuses on approaching the bottle flip not as a simple parabolic motion but also considering the distribution of fluids by dividing them into tiny particles in order to find the center of mass. The center of mass is used for the function of position versus time. This study will complement these points and propose a model for a reasonable analysis of the motion of the water bottle.

In addition to the qualitative description of simply analyzing the distribution of water in more detail in analyzing this phenomenon, it is also important to argue on a quantitative basis how much water changes motion through which movement. If a precise quantitative analysis of the relationship between the two becomes possible, it is of great significance to build a valid model for a phenomenon that is not yet well known.

While analyzing the papers that have so far dealt with this phenomenon, prior studies found common problems. In most preceding studies, the distribution of water from a bottle of water is concluded by considering it as a simplified rigid shape, which is a critical problem that makes the distribution of real water approximately one-dimensional. Instead of simplifying and analyzing the water bottle itself, the researchers wanted to describe it as an approach that would more directly track movement changes in the water distribution. The study also focused on what physical factors would affect the water bottle after it hit the ground and landed safely without falling.

Based on the problems identified in the preceding study, the following study problems were established:

- What specific exercise does a bottle do from when it is released to when it hits the ground?
- How does the movement of the water bottle change depend on the type of liquid in the bottle (water, glycerol, cooking oil, etc.) and its amount?
- What are the specific conditions under which a bottle does not fall when it lands on the ground?

**2. Theory**

A previous study of the study found that most of the time, water bottles were developed based on the premise that they were rigid. However, in this situation, the researcher determined that the motion of the water bottle was not applicable to a simple parabolic motion, and thus tried to approach the problem in a more theoretical way : with very little water distribution. As a result, the magnitude of the force acting on the water bottle and its point of use were given as a quantitative measure, so the motion equation was obtained.

**2.1 Projectile Motion**

*Figure 2*

Projectile Motion is a two dimensional motion with the object only experiencing gravitational acceleration. Thus, the motion of the water bottle appears such as figure 3.

If the initial velocity is defined as , the x-component and the y-component can be defined as , respectively. There is no reason to consider the – directional vector component and directional vector component since they do not affect each other. Then, given Newton\’s second law, *F = ma *(where ‘*F*’ is the force acting on a mass ‘*m*’ by which ‘*a*’ acceleration is produced), the only gravity *mg *(where ‘*m*’ is mass and ‘*g*’ is gravitational acceleration*)* in the direction of -. The position equation of the x-axis is given, and the position equation of the y-axis is obtained by.

(1)

(2)

In the equations (1) and (2), ‘*t*’ is the time elapsed. The reason why you can see the parabolic motion of a water bottle to the extent that there is no force other than gravity or to the point of neglect. Air resistance has confirmed that it does not affect the motion change of the bottle, and the resistance between the fluids inside the bottle does not affect the center of mass throughout the bottle. So, it can be concluded that a parabola motion model of a water bottle can be set up.

**2.2 Conservation of Angular Momentum**

The angular momentum is defined as the cross product of the position vector and the linear momentum.

: Angular momentum [ *kg /s*]

: linear momentum [*kg m/s*]

The amount of angular momentum defined as is also induced in relation to torque, it is important in this study that the amount of angular momentum does not change without the torque being applied.

(4)

(5)

: torque [ N m]

: Force applied [N]

If air resistance is ignored, the torque applied to the rotating water bottle is zero because gravity is the only force applied. The applied torque is zero because the becomes zero if the center of mass of the entire bottle is defined as the axis of rotation.

Therefore, the total amount of movement should be constant after the bottle leaves the hand and hits the ground. If it is possible to calculate the moment of inertia using the center of mass, it can be induced to the angular velocity of the inertia moment, or vice versa, by means of a formula of constant angular momentum (I: moment of inertia, ω: angular velocity). Thus, the law of angular momentum conservation can be used to predict angular velocity through the distribution of liquids in the water bottle.

**2.3 Model based on fluids**

This study approached the stable landing of water bottles in two ways. The first approach is a tennis ball approach that simplifies water bottles using two tennis balls, and the second approach is a center of mass approach that shows the center of mass and position distribution of water bottles as the center of mass and the distribution of liquid.

**2.3.1 Tennis Ball Approach**

In order for a bottle to land successfully, one must understand the conservation of angular momentum and decreasing angular velocity [1]. The only force acting on a water bottle with regard to angular motion is gravity and therefore no resulting torque is produced. The main point of the tennis ball approach is that, as a result, L around the center of mass should be preserved when defining the center of mass as a rotating axis. Namely, L=I ω is a constant, where I is the moment of inertia. Although it may be thought ω should be kept constant, the redistribution of liquids in water bottles identifies the change in the center of mass.

*Figure 3*

As shown in Figure 3, if you were to simplify the distribution of fluid to two tennis balls, and try to find the momentum and center of mass, the bottle was considered as a cylinder. If the cylinder radius is defined as R, bottle height H, and bottle mass , the centre of mass is defined as H/2. The tennis balls are modelled by hollow spheres of radius R and mass .

The lower ball stays at the bottom of the cylinder, while the top of the upper ball is located at the position h. The location of the center of mass according to the position of the two tennis balls and bottle is derived and varies during the experiment, as it is a function of the position of h of the second ball:

(6)

However, in this study, the fluid model was approached, rather than simplifying it with a rigid one.

We first determine the center of mass of the combined system of the bottle (mass ) and the water (mass ). The center of mass can be found by taking the weighted average of the center of mass of the bottle, located at H/2, and of the distributed water, located at h/2. With this, the center of mass position can be found as :

(7)

This expression has been employed for obtaining the position of the center of mass in the water bottle experiments. The next step is to determine the moment of inertia of the system I, measured with respect to the center of mass . In analogy to the tennis bottle, the moments of inertia of the bottle and of the water were determined separately, which leads to the total moment of inertia . Using the parallel axis theorem, the bottle’s moment of inertia is found to be:

Here is the moment of inertia of the bottle with respect to its own center of mass (located approximately at H/2), while the second term accounts for the shift to the system’s center of mass at . Since a simplified one-dimensional description is being considered,from now on shall be used. In similar fashion, the moment of inertia of the one-dimensional water column can be expressed as:

(9)

The total moment of inertia then reads:

(10)

where it is understood that is given by (7)

Therefore, if f is defined as a value of the center of mass of the fluid divided by the value of the center of mass of the entire bottle, as namely , the ratio of inertia moment and angular velocity can be compared.

Define G (f) as

. (11)

(12)

and is attained when the water is maximally distributed, i.e for h = H

G (f) is the ratio of the maximum moment of inertia, the largest spread of water, and the earliest moment of inertia when the water bottle leaves the hand. Since the angular velocity ω is inversely proportional to the moment of inertia, G (f) should be small if the angular velocity is minimal.

**2.3.2 Center of Mass Approach**

The positions of colored dots move in different forms from the shape of the parabola movement. The center of mass can be obtained through the distribution of colored dots and water. To do this, reference [2] has calculated the dot (i.e. white, yellow, green, red, blue, and top red) calculation model. The model uses and to represent the parabolic motion and the uniform acceleration motion of the y-axis respectively. This represents the distribution of water through and as a function of time, and it is difficult to directly explain why center of mass varies with the distribution of water. When a bucket filled with water rotates horizontally, the water is pressed downward. You can think of the force acting on a bucket by centrifugal force, but the centrifugal force by this type of acceleration is spinning. The higher or lower the speed, the stronger the centrifugal force. With the influence of a non-inertial reference coordinate system, the centrifugal force can be thought of as changing the fluid in the bottle. When a bottle leaves the hand, the center of mass changes, which is the rotating center of the bottle and center of mass rises because the water is moving away from the initial rotational axis.

*Figure 4. CM Approach*

The center of mass is obtained using a dot float. Depending on the distribution of water, the distribution in the bottle changes with several dots. At this time, the center of mass can be obtained by measuring the location of each dot and the amount of water. This is called the Least Square Minimum Method,

(13)

(14)

where the center of mass is defined as this minimum [(t)]. Since the tracker program was used in this study, a function of calculating the position’s mean was used to determine the center of mass. Using the statistical nature that the sum of squares of deviations is minimal in the mean, the search was conducted by obtaining a center of mass and then accepting the parabolic model obtained from.

**3. Method**

An experiment was conducted earlier to verify our water bottle model developed in theoretical exploration. As for the progress and planning of the experiment, first of all, the necessary results should be to analyze the movement changes in water bottles that vary when each variable is different. He decided to rotate the bottle around and throw it at a certain amount of force and analyze the movement through a program called Tracker. The planning of the experiment will follow the factors described above. First, the experiment was conducted by varying the volume of liquid when the type of liquid in the water bottle was unchanged, increasing the volume of liquid by 20 ml from 100 to 300 (ml). The results were more reliable by doing an experiment 10 times in total. The second experiment is to vary the type of liquid when the volume of the liquid is fixed. This experiment was conducted in the same way as the first one. The test was conducted taking into account the density and viscosity of the liquid, and the results were more reliable by doing each 10 times. The water bottle is 25 grams in mass and 23 centimeters in height.

**3.1 Projectile Motion based off Center of Mass**

*Figure 5. Water Bottle Used in Experiment*

As a model using the previously and while exercising the water bottle, the experiment was the same as in, but it focused on the location of changes in the center of mass and the actual center of mass of the water bottle distribution. The high-speed camera is measured each frame per 0.02 seconds, and MATLAB is used to compare the models in the simulation with those derived from the actual experiment.

*Fig 6. Bottle flip and MATLAB Program Analysis*

**3.2 Experiment Model based off fluid mass**

The volume of fluid was adjusted to the desired amount of volume across several times on the bottle, and the distribution of the existing fluid to the bottle was set constant (ex, stationary floor, etc.). It developed an experiment model by taking pictures by time interval using a camera. In particular, during the exercise of the water bottle, the amount of fluid moving should not change. The amount of fluid was tested with a change of 20 ml from 100 ml to 300 ml, and the type of fluid was also explored taking into account.

**4. Discussion**

To scientifically explain the stable landing of a water bottle, the following research issues have been established:

**4.1. What specific exercise does a bottle do from when it is released to when it hits the ground? **

The distribution of water in the bottle was expected to change the rotational inertia of the water, causing erratic rotation of the bottle.

*Figure 7. x and y position with time. The computed CM (curve fit) from the model and the experimental CM (from Tracker)*

Using the previously referenced model using and while exercising the water bottle, the location of the actual center of mass and the location of the center of mass changes along with the distribution of the fluid and water bottles were not significant. The conclusion is that the model in the simulation is accurate when comparing the model derived from the actual experiment. Thus, the parabolic model using the current center of mass is valid and can be reliably concluded on the basis of this result whether the water bottle has a stable landing.

The center of mass of x-axis and y-axis position graphs in Figure 7. show the data types that can be described by parabolic motion models. Thus, if the initial angular velocity and linear velocity are measured, the location of the center of mass of the water bottle or moment of inertia can be obtained over a given time. Chi-square test of product of the actual center of mass was used to compare the value of the theoretical center of mass with that of the actual center of mass. This makes sense because the sample size is more than 10 items and two independent variables are used.

**4.2 . How does the movement of the water bottle change depending on the type of liquid in the bottle (water, glycerol, cooking oil, etc.) and its amount?**

The changing motion of the water bottle was difficult to obtain because its initial linear velocity and angular velocity were wide of the error range when subjected to constant force. Thus, the results were drawn by the changes in motion depending on the amount of liquid. The f represents the ratio of the fluid volume, G (f) represents the ratio of the moment of inertia, i.e. the ratio of angular velocity based on the theoretical search. As the amount of fluid in the water bottle changes the ratio of G. The simultaneous presentation of the two graphs will result in a reduction in the appropriate volume of fluid, or greater than or equal to, making it harder for the water bottle to land reliably. Therefore, the proper amount of water and predictable models of water bottles can be built.

*Figure 8. the G(f) and height ratio of the bottle versus Mass ratio of water*

*Figure 9. G(f) and the height and mass ratio of the bottle*

*Figure 9. G(f) and the height and mass ratio of the bottle*

Figure 8 and Figure 9 show the graph with the same variables, but the minimum values and graph shapes are different than those in the quadratic function, which is convex below. This is because Figure 9 shows a theoretical model that is not based on experiments and does not take into account the movement of the water bottle and the distribution of the position of the liquid when using the tennis ball approach. Figure 8 defines the appropriate conditions in 0.3 and 0.52, whereas the preceding studies assume 0.2 and 0.4. Therefore, it can be concluded that a stable landing is possible when the amount of oil is greater than that of the previous study.

**4.3. What are the specific conditions under which a bottle does not fall when it lands on the ground?**

The study found that the water bottle would draw a parabolic trajectory in a 2-dimensional motion, and then it would measure the center of mass to create a new object.

*Figure 10. The position of CM versus Angular Velocity compared to the axis of rotation*

Figure 10 shows the location of the center of mass of the water bottle in a ground collision, expressed in cm the distance from the axis of rotation, to quantify how inclined the bottle is and how it is. When comparing the location of the angle of contact and the center of mass in a bottle crash, it was judged that the inverse relationship of the type of linear function was appropriate, and the change in the direction of motion and the location of the center of mass of the bottle was related to each other. Thus, the initial condition of angular velocity can be used to confirm the success of a stable landing after the impact. The angular velocity at the critical point was measured at 3.8 rad/s. Critical points are those points where, from any point on, the angular speed does not land reliably, regardless of the location of the center of mass. The minimum and maximum angular velocity of success were obtained when the water bottle was thrown, given that the frequency of landing failures during several experiments was more than 3.8 rad/s.

**5. Conclusion**

First, the parabolic motion model formed in the water bottle has been verified theoretically and experimentally that the travel distance and angular velocity are proportional to the initial angular velocity and linear velocity. Based on this, the object is related to the length change and quadratic function. By unraveling the concentrational equation for the mass variation and location of fluids in the bottle, the location distribution of fluids in the bottle was predicted. Tests have shown that the effect of fluid on motion is significantly less than the amount of fluid. The results of the experiment showed that the larger the radius of the water bottle, the smaller the amount of pull in the bottle, and the slower the angle, the more successfully the bottle landed. It was also concluded that the reason why the water bottle did not fall when it landed on the ground was because the fluid in the bottle was acting on a torque opposite to the landing and reduced angular velocity.

**6. References**

[1] *Pim J. Dekker, Lumen A.G. Eek, Water Bottle Flipping Physics* *arXiv:1712.08271 [physics.flu-dyn] (2017)*

*[2] Paulo Simeão Carvalho and Marcelo José Rodrigues Phys. Educ. 52 045020 (2017)*

*[3] Carvalho, P.S., Rodrigues, M., The centre of mass of a “flying” body revealed by a computational model, Eur. J. Phys., 38(1), 015002. (2017)*

*[4] Walker, Jearl, David Halliday, and Robert Resnick. Fundamentals of Physics. Hoboken, NJ: Wiley, Print. (2011)*

### About the Author

Tae Hyun is a high school senior in South Korea. He likes to read science fiction, listen to 1940\’s swing, and play the oboe. Tae Hyun is lives in a small apartment with his mother, father, and a dog called Apple.