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    Modeling Scenario
    185

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    3-052-OptimalProjectileFiring-ModelingScenario
    We offer the opportunity to model a projectile's trajectory in several cases, all without resistance.
    projectile motionprojectilelaunch anglemaximum range
    Potential Scenario
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    46

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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
    Potential Scenario
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    1999-F_Brauer-What Goes Up Must Come Down
    This paper is a wonderfully general analysis of the following, “It is natural to ask whether a particle propelled upwards takes longer to fall to earth from its maximum height than it takes to rise to this maximum height for frictional forces.
    projectilefrictionfalling timerising timemaximum height
    Modeling Scenario
    201

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    1-140-LeakyBucket-ModelingScenario
    We seek to model the height of water in a cylindrical tank (bucket) in which water flows out the bottom of the tank through a small bore hole while we are pouring water into the tank at the top of the tank at a constant (or varying) rate.
    flowTorricelli's Lawleaky bucket
    Modeling Scenario
    251

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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
    256

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    1-055-WaterFallingInCone-ModelingScenario
    We offer an opportunity to model the height of a falling body of water in a right circular cone (funnel) and to estimate an appropriate parameter based on data collected from a video of the experiment found on YouTube.
    Torricelli's Lawdischarge coefficientconecontraction coefficient
    Potential Scenario
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    2002-Chai-Optimal initial angle to fire a projectile
    Assume a projectile is fired without air resistance and lands at a height y above its initial vertical position. What is the optimal initial angle of firing to maximize the horizontal distance traveled by the projectile?”
    optimizationprojectile motionlaunch angle
    Modeling Scenario
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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
    Potential Scenario
    140

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    2003-Bohren-Dimensional analysis-falling bodies--fine art of not solving differential equations
    Dimensional analysis is a simple, physically transparent and intuitive method for obtaining approximate solutions to physics problems, especially in mechanics.
    falling bodydimensional analysisdrag
    Modeling Scenario
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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
    Modeling Scenario
    262

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    162

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    1-073-WaterExitBottle-ModelingScenario
    We offer an experiment in which data is collected to ascertain a parameter in the differential equation formulation of Torricelli's Law for water flow out of a cylindrical container.
    datawaterflowTorricelli's Lawbottle
    Modeling Scenario
    280

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    3-045-RampBounce-ModelingScenario
    Students build two projectile motion models (1) a one-dimensional model for a vertically falling ball from a fixed distance until it hits an inclined ramp and (2) a two-dimensional projectile motion model of the ball bouncing off the ramp.
    projectile motioninclined rampcoefficient of restitutionoptimal anglebounce
    Modeling Scenario
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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
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    2012-Lensyl-Urbano-The Draining of a Plastic Bottle-Integrating a Physics Experiment into Calculus
    I punched a small hole (about 1mm radius) in a one gallon plastic bottle and had my students measure the rate at which water drained.
    in-class activityexperimentTorricelli's Lawcylinder of waterexpderiment
    Modeling Scenario
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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
    Modeling Scenario
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    3-041-UpDown-ModelingScenario
    Shoot a projectile straight up in the air. Determine maximum height the projectile will go. Consider time T(a) (0 < a < 1) it takes between when the projectile passes distance a.H going up and then coming down. Develop T(a) as a function of a.
    gravityprojectile motionfalling bodymaximum heightftiming
    Modeling Scenario
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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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    3-140-TwoSpringsOneMassFixedEnds-ModelingScenario
    Students build a model of a two spring, single mass with fixed end configuration and then plot solutions to experience the motion.
    massundampedFree Body Diagramspringstwo springsoscillationspring constantfixed ends
    Modeling Scenario
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    4-039-FallingDarts-ModelingScenario
    we develop, solve, and analyze a second order differential equation model for free fall incorporating air resistance. Students solve the model using two methods -- reduction of order and separation of variables, and method of undetermined...
    optimizationresistanceterminal velocitymuzzle velocitydrag coefficientfree fallair drag