Suitability of Torricelli’s Theorem Formulation in Cases of Leaking Reservoirs with Video Analysis Tracker

This study aims to empirically prove the Torricelli equation formula in the case of leaky reservoirs with the help of video tracker analysis. The method used in this research is quantitative descriptive. The experiment was carried out with a simple tool: a 19-liter gallon of water filled with water and dyed, and then three holes were made vertically with different heights. The gallon is filled with water with a constant water level. Next, take a video of each leaking hole. Video is analyzed with Tracker software. Variables observed were the velocity of water exiting from the leak point (v), the time it took for water to gush from the leak point to the bottom (t), and the horizontal distance from the leak point position to the bottom (x). The results obtained based on video analysis with the tracker are that the farther the distance from the surface of the water to the leak point, the farther the horizontal distance of the resulting jet of water will be. This study concludes that theoretical data and experimental data have significant value, so the video analysis tracker software is feasible to use in dynamic and static fluid learning.


Introduction
Torricelli's theorem is the kinematic equation for a fluid falling from a height regardless of resistance.Torricelli's theorem can also be referred to as Torricelli's Law, Torricelli's Equation, or Torricelli's Principle, coined by Evangelista Torricelli (15 th October 1608 -25 th October 1647).Further explanation of Torricelli's theorem is water gushing out of a hole/leak in a water tank.The amount of kinetic energy of water that spurts out of the water tank hole equals the amount of potential energy [1].Therefore, the speed of the water spraying at the hole is the same as water falling freely from the water level limit.Because the greater the difference between the height of the hole and the water level limit, the faster the water spray will be [2].
Toricelli's theorem is actually a special application of Bernoulli's law.However, this principle was discovered by Toricelli one century before Bernoulli's law was formulated, so the name Toricelli's principle has been commonly used [3].Bernoulli's law, the fundamental equation of an incompressible fluid that flows laminarly, has been applied to various things [4].
Several variables that can be determined in Torricelli's theorem include the velocity of fluid flowing out of the leak hole (v), the time it takes for water to fall from the leak hole (t), and the horizontal distance from the leak point to the falling water point (x) [5].Torricelli's theorem states that the velocity of the fluid (v) that comes out of the leaky hole in the tank is proportional to the square root of twice the vertical distance or height, which is measured from the vertical distance of the water surface to the point of leakage [6].The higher the position of the hole where the fluid comes out, the greater the speed at which the fluid will spray.At the same time, the time of falling water in the reservoir (t) is proportional to the square root of twice the vertical distance between the leaking point and the soil surface [7].The farthest horizontal distance from the leaking point to the falling water point/spout (x) is directly proportional to the magnitude of v and t, meaning that x depends on the variables v and t [8].Based on the application of Bernoulli's law, the higher the vertical distance between the water surface and the leak point, the farther the water jets.However, in applying Torricelli's theorem found in physics lessons in class, the equations that refer to Torricelli's theorem, when applied to several positions of inline vertical leak holes, have a different visualization.Therefore, an experiment is needed to empirically prove the formulation of the Toricelli theorem with precision and accuracy.One way to determine precision and accuracy is to use Tracker video analysis.It works by slowing down the video of the physical phenomenon to be observed so that the researcher can analyze the desired variable with the slowed movement.The analysis was also carried out using the tools provided by the Tracker software.

Method
The method used in this research is descriptive quantitative.Before the experiment, theoretical analysis of leaky reservoir simulations with Torricelli's theorem was first carried out for conditions similar to those during the experiment.The analysis was carried out to theoretically determine the use of the Torricelli theorem equation in the case of leaking reservoirs and contextual cases related to the conditions during the experiment to obtain a visualization of water jets.Furthermore, an experiment was carried out using a 19-liter gallon of water, and holes were made with three variations.The experiment was carried out by making three small holes in a gallon of water vertically in line with the following position variations (see Table 1).where h1 is the vertical distance from the water surface to the ground, h2 is the distance from the leak hole to the ground, and Δh is the distance from the water surface to the leak point.Gallons are filled with colored water, and water is conditioned to come out of each hole to take video and analyze with tracker software.Video of the experimental results was analyzed with Tracer software directly after the experiment.The results obtained are then described [9].Tracker Video Analysis and Modeling Tool is a free video analysis and modeling tool built on the Open-Source Physics (OSP) Java framework.It is designed to be used in physics education and allows students to model and analyze the motion of objects in videos.Some of the advantages of Tracker Video Analysis software are: (1) Free: Tracker is a free software that can be downloaded and installed on Windows, Mac OS X, and Linux operating systems [10]; (2) Easy to use: Tracker is designed to be userfriendly and easy to use, even for beginners.It has a simple interface and provides step-by-step instructions for video analysis; (3) Accurate: Tracker is a powerful tool that can accurately track the motion of objects in videos.It can rescale each frame, move the origin, rotate the video, and more [11]; (4) Versatile: Tracker can open various types of videos and can be used for a wide range of applications, including physics education, sports analysis, and motion tracking; (5) Clean: Tracker has been independently tested by Softpedia and found to be 100% clean.In summary, Tracker Video Analysis and Modeling Tool is a free, easy-to-use, accurate, versatile, and clean software that can be used for video analysis and modeling.It is designed for physics education but can also be used for other applications [12].

Results and Discussion
The results and discussion section will be divided into three main sections.The first part is a theoretical analysis of reservoir leaks using Torricelli's theorem, and the second part is a theoretical analysis of leaking reservoir simulations for experimental cases.In the second part, a case calculation simulation analysis will be carried out in actual conditions using Torricelli's theorem.The third part is a leaky reservoir experiment analyzed with tracker software.

Theoretical analysis of leaky reservoir simulation using Torricelli's theorem
The following is a theoretical analysis to generate a general leaky reservoir case formulation using Torricelli's theorem [13] (see Figure 2).where h 1 is the vertical distance from the water surface to the ground, h2 is the distance from the leak hole to the ground, and Δh or (h1 -h2) is the distance from the water surface to the leak point.Apply Bernoulli's law at location 1 and location 2, namely at the water level in gallons and at the leak point.
Focus at location 1 and location 2; water is pushed by an outside air pressure of 1 atm.So given, P 1 = P2 = P0 = 1 atm [14].Because the cross-sectional area at location 1 is much larger than the crosssectional area at location 2, the rate of descent of the water level in the tub is very small and can be assumed to be zero.So, we take v1 ≈ 0. Finally, Bernoulli's law can be approximated by: Equation ( 2) is known as Toricelli's Theorem.Look carefully at equation ( 2); the speed of the fluid coming out of the hole is the same as the rate of a free-falling object at a height of h2 when released from a height of h 1 [15].There is a variable (g) is the acceleration due to gravity, whose constant is set at 9.8 m/s 2 [16,17].Furthermore, another parameter that is analyzed is the time the water falls from the leak point to the ground (t), referring to the free fall equation, namely: The next variable is the horizontal distance (x), which is the horizontal distance from the leak point to the water point falling to the ground.The analysis of this x variable is dependent on the v and t variables.Thus obtained: Based on the results of the derivation in equations ( 2), (3), and ( 4), an analysis of cases of leaking reservoirs at certain height values (special cases) can be carried out [18].

Theoretical analysis of leaky reservoir simulation for experimental cases
Next is a simulation analysis using Torricelli's theorem formulation in leaky reservoirs for cases with certain height values, as in the experiments.Table 2 shows data from the simulation results by calculating Torricelli's theorem.Based on these data, the visualization of the simulation data of water jets from the leaking holes can be expressed in the figure: This visualization is a form of mathematical analysis of the Torricelli equation, which, of course, can change if the state of the size of the determining variable also changes.So, this form of visualization must be studied more deeply with a method to produce high accuracy.In this experiment, the method of video analysis software tracker was used.This method can make slow-motion movements easier to analyze the desired variables more accurately and precisely directly [19].

The leaking reservoir experiment was analyzed with tracker software
The third part is a video analysis of the experiments carried out.The experiment was conducted in the Unesa Physics Laboratory with a 19-liter gallon of water with three small holes.The hole is made in a vertical and straight line with a predetermined distance (see Figure 2).Video capture is done for each hole.Video results are directly analyzed using a software tracker.The following is the result of velocity data tabulation (v), the time the water falls from the leak point to the ground (t), and the horizontal distance from the leak point to the ground (x) using the tracker.The theoretical data is obtained through Bernouli's equation, which is kept constant with several variables such as acceleration of gravity, velocity of fluid flow, height of leaking tank, and height of tank hole.The theoretical data Toricelli needs is speed, the time it takes to touch the ground, and the distance traveled by the airflow [20].In a tank with a three-hole state, it is necessary to identify the value of the velocity of the fluid flow through the seller Bernoulli, which is kept constant with the initial velocity or surface velocity of the fluid being zero due to fluid flow, which refers to free-fall motion or because the direction of fluid motion is in the y direction [21], on the time variable used with reference to the height of the hole to the ground surface and due to the influence of the earth's gravitational acceleration.Furthermore, the distance variable obtained in the x-axis direction is obtained from the time and velocity of liquid fluid in the x-axis direction [22].
The experimental data was obtained through an experiment with 19 liters of gallon material and water with three holes opened simultaneously, which were then recorded using a camera to process the data using tracker software.In using the tracker for the Toricelli experiment, are as follows: (1) import the video from the file manager; (2) perform calibration and continue with a meter with a length of 1 meter; (3) track with a mass point to record the airflow coming out of the hole; (4) look at the tracking process, activate the speed, time, and distance column in the direction of the x-axis according to the number of frames that appear, then do the average process for each of these variables [23].The following (Table 4) is comparative data from three types of analysis (theoretical analysis, analysis with special cases/ real situation simulation, and analysis with Tracker video analysis for special cases/ real situation simulation).Based on the data collection results, it is known that data obtained through theory and processed through the Toricelli theorem equation has a significant value to the data obtained from direct experimental activities.The experimental data is then processed using the Tracker Video Analysis and Modeling Tool software to determine the value of the velocity of the fluid flow, the time it takes to touch the ground, and the farthest distance that the liquid can travel in the x-axis direction.

Conclusion
Based on theoretical and experimental data, it can be concluded that Torricelli's theorem is in the case of three holes with a significant division of the heights of each hole.The theoretical data determined are the variable speed, time to touch the ground, and the furthest distance traveled through the equation of significant value to the experiment applied to a 19-liter gallon with the position of the farthest distance, namely the second hole, then the first hole, and finally the third hole.Theoretical and experimental data have significant value, so the video analysis tracker software is feasible for dynamic and static fluid learning.

Figure 1 .
Figure 1.Illustration of several variables x, v, and t in Torricelli's theorem.

Figure 4 .
Figure 4. Visualization of the simulation results of water jets from leaky reservoirs.

Figure 5 .
Figure 5. Video analyzed using software tracker

Table 1 .
Position of each Hole in the Experiment

Table 2 .
Real-state simulation data during experiments with Torricelli's theorem

Table 3 .
Leaking reservoir data during the experiment were analyzed with Tracker software

Table 4 .
Comparative data from three types of analysis