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    3-030-SecondOrderIntro-ModelingScenario
    We outline the solution strategies involved in solving second-order, linear, constant coefficient ordinary differential equations, both homogeneous and nonhomogeneous and offer many application and...

    Keywords: parameter estimationnon-homogeneousSpanishfrequencybuoyancyinverse problemthrustunderdampedfirst passagespring mash dashpotrealcomplexhomogeneoussuperpositionparachuteoverdampedcritically dampedbuoyquadratic equationrootsrocket flight

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    9-002-GroundWaterFlow-ModelingScenario
    The goals of this project are to compare a conceptual one-dimensional groundwater flow model to observations made in a laboratory setting, and to discuss the differences.

    Keywords: sgroundwater flowDarcy's Lawheat equationaquiferhydraulic headFourier's Lawsand tank

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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.

    Keywords: whole picturetankmixingflowcolumn of watersalt mixinggraduated cylinderapertureTorricelli's Law

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    3-072-EarthQuakePartI-ModelingScenario
    This modeling scenario considers a one-story building as a simple structure; the roof is modeled as a single point mass. Movement of the roof can be modeled similar to a mass-spring system.

    Keywords: earthquakevibrationamplituderesonanceundampedspring-mass system

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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...

    Keywords: Torricelli's LawConservation of EnergytConservation of MassBernoulli’s Theorem

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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.

    Keywords: fallingcolumn of waterTorricelli's Lawmultiple spigotsspigots

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    3-043-BallisticModeling-SpongeDart-ModelingScenario
    The goal of this project is for students to develop, analyze, and compare three different models for the flight of a sponge dart moving under the influences of gravity and air resistance.

    Keywords: projectile motionair resistanceballisticsdimensional analysisNewtonian mechanicssponge dartNerf gunNerf dartBuckingham's Theorem

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    9-012-PDEGuitarTuning-ModelingScenario
    We lead students through a derivation of a partial differential equation which models the motion of a string held at both ends, a case of the one-dimensional wave equation, and then play it on...

    Keywords: extensionsine wavesfrequencyguitardampingFourierstringtuning

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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...

    Keywords: VolumesphereTorricelli's Lawtimingspherical tankfluidempty

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    1-011-Kinetics-ModelingScenario
    We make the connection between chemistry course and differential equations coursework. We do this through modeling kinetics, or rates of chemical reaction. We study zeroth, first, and second order...

    Keywords: kineticschemistryLaw of Mass Actionchemical kineticsorder of reactionchemical reaction

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

    Keywords: projectile motionprojectilelaunch anglemaximum range

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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...

    Keywords: gravityprojectile motionfalling bodymaximum heightftiming

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    5-040-TunedMassDamper-Part-I-Modeling Scenario
    We offer an opportunity to build mathematical models to mitigate dangerous displacements in structures using structural improvements called Tuned Mass Dampers. We model the motion of the original...

    Keywords: structurespring-mass-dashpotspring massearthquakevibrationtuned mass tunerrestoring forcecontroldampe

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    3-076-CircuitBuilding-ModelingScenario
    In this project students will establish a mathematical model for an electric circuit as a second-order ordinary differential equation with constant coefficients.

    Keywords: simulationexperimentMultisimcircuitinductancecapacitorKirchhoff's Voltage Lawresistor

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    3-092-WirelessPower-ModelingScenario
    We present an engineering application (wireless power transmission) modelled by a coupled system of two linear second-order differential equations with constant coefficients. One equation is...

    Keywords: DESMOSwireless power transmissionelectric circuitsKirchhoff's Voltage Law

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    3-035-StadiumDesign-ModelingScenario
    For a given baseball playing field outline how high must the outfield fence be at each point in order to make a homerun equally likely in all fair directions?

    Keywords: projectile motionbaseballparametric equationstadiumhome runno reistantfairnessno resistance

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    2005-P_Howard-Modeling with ODE
    In these notes we consider three critical aspects in the theory of ordinary differential equations: developing models of physical phenomena, mathematically well-posed, solving ODE numerically .

    Keywords: Fluid Mechanicscompartmentpopulationsvariational methodsHamiltonian mechanics

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    3-044-DeepWell-ModelingScenario
    We drop a pebble in a deep well. Given the time elapsed from release of the pebble until we hear the splash determine the depth of the well.

    Keywords: gravityprojectile motionfalling bodyspeed of sound

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    2017-Agmour-EtAl-Optimization of the Two Fishermen's Profits Exploiting Three Competing Species Where Prices Depend on Harvest
    The main purpose of this work is to define the fishing effort that maximizes the profit of each fisherman, but all of them have to respect two constraints: the first one is the sustainable...

    Keywords: FishingharvestNash equilibriumoptimizeprofits

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    5-040-TunedMassDampers-PartII-ModelingScenario
    Studentsbuild mathematical models to mitigate dangerous swaying in structures using structural improvements called Tuned Mass Dampers (TMD). We model the motion of the original structure as a...

    Keywords: structureresistancespring-mass-dashpotearthquakevibrationcontroldampingtuned mass damperstiffnessreduction of vibration

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