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    Modeling Scenario
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    147

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    9-030-WaterHammer-ModelingScenario
    We develop and apply a numerical algorithm that solves a system of two nonlinear partial differential equations (PDEs) that describes the time evolution of the water hammer phenomenon.
    Stock and Flow modelhyperbolic equationsfinite differenceswater hammerhydrodynamicscompressibilitywave celerity
    Potential Scenario
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    68

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    2000-deOlivera-Delfino-Costa-Leite-Pinhole Water Flow From Cylindrical Bottles
    The water flowing from the first bottle feeds the second one located below it. The same concepts of mass and energy conservation now lead to a non-trivial differential equation for the lowest bottle dynamics.
    Torricelli's LawConservation of EnergytConservation of MassBernoulli’s Theorem
    Potential Scenario
    275

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    83

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    2009-Munz-EtAl-When Zombies Attack-Mathematical modelling of an Outbreak of Zombie Infection
    We introduce a basic model for zombie infection, determine equilibria and their stability, and illustrate the outcome with numerical solutions.
    zombiesinfectionepidemicwater treatmentlatent infectionquarantineimpulsive eradication
    Modeling Scenario
    446

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    357

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    1-150-CancerTherapy-ModelingScenario
    This activity builds upon elementary models on population growth. In particular, we compare two different treatment models of cancer therapy where in one, surgery happens before therapy and in the other, surgery happens after therapy.
    population dynamicsphage therapyexponential growthcancerwater treatmentdecaysurgery
    Potential Scenario
    161

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    74

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    2015-Joshi-EtAl-Optimal control of an SIR model with changing behavior through an education campaign
    We study stability analysis and use optimal control theory on the system of differential equations to achieve the goal of minimizing the infected population (while minimizing the cost).
    SIR modelsvaccinepopulationwater treatmentsusceptibleeducation campaignminimizing costinfected
    Potential Scenario
    153

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    75

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    2017-David_Morin-Oscillations
    So needless to say, an understanding of oscillations is required for an understanding of waves.
    dampingfrequency responseresonanceoscillationbeatsunderdampingharmonic oscillatorsover dampingcritical damping
    Modeling Scenario
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    181

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    1-064-TorricelliBox-ModelingScenario
    The time it takes a column of water to empty and the time it takes the same volume of column of water with a box (various sizes) submerged in the column of water are compared through modeling with Torricelli's Law.
    groundwater flowcolumn of waterTorricelli's Lawwater displacementsubmerged box
    Potential Scenario
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    70

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    2015-Goodnow-EtAl-Mathematical Models of Water Clocks
    This is a historical tour of water clocks, known as clepsydra a Greek word meaning water thief. These devices were for telling time.
    Torricelli's Lawwater clockBernoulli's Lawclepsydra
    Modeling Scenario
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    138

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    6-030-SaltAndTorricelli-ModelingScenario
    We build on a model for the height of a falling column of water with a small hole in the container at the bottom of the column of water. We use data from one video of a falling column of water.
    whole picturetankmixingflowcolumn of watersalt mixinggraduated cylinderapertureTorricelli's Law
    Modeling Scenario
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    1-090-EmptySphericalTank-ModelingScenario
    We model the emptying of water from a spherical tank. First, we pump out water at a constant rate. Second, we allow the water to exit through a small hole in the bottom of the tank. We seek to determine how fast the water level is falling in both...
    VolumesphereTorricelli's Lawtimingspherical tankfluidempty
    Potential Scenario
    154

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    58

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    2006-G_Ashline-J_Ellis-Monaghan-How high-fast-long-Modeling water rocket flight with calculus
    We describe an easy and fun project using water rockets to demonstrate applications of single variable calculus concepts.
    calculusFlightprojectilewater rocket
    Modeling Scenario
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    277

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    1-029-ConeToCubeFlow-ModelingScenario
    An inverted right circular cone with a hole at the bottom is suspended above an open-topped cube which also has a hole in the center of the bottom. The cone is filled with water and we model water flow from cone to cube and out the bottom of the...
    waterflowTorricelli's Lawconecube
    Potential Scenario
    140

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    45

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    2008-Huangi-Lu-Modeling long-term longitudinal HIV dynamics with applications to an AIDS clinical study 
    To better understand the factors responsible for the virological failure, this paper develops the mechanism-based nonlinear differential equation models for characterizing long-term viral dynamics with ARV therapy.
    longitudinal dataAIDSantiretroviral drug therapyBayesian nonlinear mixed-effectstime-varying drug efficacyHIV dynamics
    Modeling Scenario
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    185

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    1-057-FiguringFluidFlow-ModelingScenario
    We propose three differential equations models for the height of a column of falling water as the water exits a small bore hole at the bottom of the cylinder and ask students to determine which model is the best of the three.
    dischargetankTorricelli's LawConservation of Energyfluid flowheightliquid
    Modeling Scenario
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    1054

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    1-015-Torricelli-ModelingScenario
    We help students develop a model (Torricelli's Law) for the height of a falling column of water with a small hole in the container at the bottom of the column of water through which water exits the column.
    dataSpanishtankcolumn of waterapertureTorricelli's LawConservation of EnergytSolverwater flowExcel Solver
    Potential Scenario
    148

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    45

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    1999-Leonid_Hanin-Which tank empties first
    Suppose that water towers like those shown in the opening are initially filled with water and have the same volume, height, and cross-sectional outlet area. Which one empties first?
    tankTorricelli's Lawspherical tankemptyfrustrum of conecylinder
    Modeling Scenario
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    206

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    1-058-WaterClocks-ModelingScenario
    We apply Torricelli's Law to the task of building a water clock in which the height of the water in a container falls at a constant rate when the container has a hole in the bottom to let the water flow out.
    flowTorricelli's Lawspigotswater clock
    Modeling Scenario
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    1-063-ThreeHoleColumnOfWater-ModelingScenario
    We consider a column of water with three holes or spigots through which water can exit and ask students to model the height of the column of water over time.
    fallingcolumn of waterTorricelli's Lawmultiple spigotsspigots
    Article or Presentation
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    581

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    2020-TeachingModule-ModelingFallingColumnOfWater
    We discuss how to model a falling column of water empirically and analytically from first principles in physics laws.
    modulecolumn of waterTorricelli's LawTorricelliTeaching Module
    Modeling Scenario
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    192

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    1-059-ContainerShapeFallingWater-ModelingScenario
    We examine many different physical situations to determine the time it takes a fixed volume of water to flow out of different shape containers through the same size exit hole at the bottom of the container.
    containersTorricelli's Lawshapedischarge coefficient