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    Modeling Scenario
    297

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    406

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    3-060-DataToDifferentialEquation-ModelingScenario
    Students use knowledge of second-order linear differential equations in conjunction with physical intuition of spring-mass systems to estimate the damping coefficient and spring constant from data.
    Association & Data Fittingmassspring-mass systemspring\total distancenumerical differentiation
    Modeling Scenario
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    459

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    3-099-PullBack-ModelingScenario
    We guide students through the development of an empirical model for the velocity and distance traveled of a simple pull-back toy. Students can record videos and extract data using their own pull-back toy or use data included.
    dynamicspull-back toy
    Modeling Scenario
    365

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    3-103-PullBackCars-ModelingScenario
    This activity offers analysis of a toy pull-back car: solution of a differential equation from model; data collection and parameter estimation; and adapting the model to predict maximum speed and distance traveled for a new pull-back distance.
    modelingdynamicshands-ontoypull-back carpullback
    Modeling Scenario
    1093

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    179

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    3-051-ProjectileMotions-ModelingScenario
    We consider several instances of projectile flight without resistance, one on level ground and one from edge of cliff to determine maximum distance and placement.
    trajectorylaunching radiantprojectile motionprojectileno resistanceangle of launch
    Potential Scenario
    338

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    58

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    2017-Jun_Liu-Hammer Throwing parameters optimization model research based on flight dynamical differential equation
    With progress of times, sports techniques are also rapidly developing, in order to let Chinese hammer throwers more quickly improve themselves levels.
    sportsprojectilehammer throwrelease angle
    Potential Scenario
    139

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    44

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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
    249

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    117

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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
    280

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    167

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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
    Potential Scenario
    146

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    84

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    2017-Bruce_Emerson-Wonderful World of Differential Equations
    This is a terrific set of noted with models and applications woven into the material at every opportunity and data as well. This is introductory differential equations with attention to detail and motivation.
    Newton's Second Law of MotionNewton's Law of CoolingreactionsPoiseuille’s Lawstopping distancerate of changeproblemsmodeling opportunities
    Potential Scenario
    169

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    67

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    2017-Andras_Domokos-Differential Equations - Theory and Applications – Notes
    This is a terrific set of notes with models and applications woven into the material at every opportunity and data as well. This is introductory differential equations with attention to detail and motivation.
    Newton's Second Law of MotionNewton's Law of Coolingbiochemical reactionsPoiseuille’s Lawstopping distance
    Modeling Scenario
    356

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    216

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    3-054-Relay-ModelingScenario
    We use a differential equations of one dimensional projectile motion and an integration of velocity for total distance to model the relay between an outfielder and an infielder in throwing the ball to home plate.
    distanceprojectile motionrelaytimebaseballoutfieldhome plateminimization
    Potential Scenario
    168

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    59

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    2017-CW_Groetsch-Hammer and Feather - Some Calculus of Mass and Fall Time
    Does a heavy object fall faster than a lighter one? In 1971 NASA astronaut Dave Scott, Commander of the Apollo 15 lunar mission, used a hammer and a falcon feather on the surface of the moon to give a dramatic illustration that this is not so.
    gravityfree falllinear resistancequadratic resistancehammerfeatherMoon resistanceGalileo
    Potential Scenario
    103

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    47

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    1994-T_Gruszka-A Balloon Experiment in the Classroom
    The following experiment involves a balloon, a stopwatch, and a measurement device such as a meter stick,
    resistancefalling bodydragballoon
    Modeling Scenario
    1159

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    3-065-UpDown-ModelingScenario
    We model the height of a launched object which is subject to resistance proportional to velocity during its flight. We ask questions about the motion as well, e.g., highest point or apex and terminal velocity.
    terminal velocityNewton's Second Law of Motionvelocitymotionapex
    Modeling Scenario
    310

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    199

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    1-013-SleuthingWithDifferentialEquations-ModelingScenario
    We present several situations in which differential equation models serve to aid in sleuthing and general investigations.
    projectile motionaccelerationNewton's Law of Coolingspeeding
    Article or Presentation
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    70

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    2004-Mark_McCartney-Using_second-order_ordinary_differential_equations_to_model_traffic_flow
    A simple mathematical model for how vehicles follow each other along a stretch of road is presented. The resulting linear second-order differential equation with constant coefficients is solved and interpreted.
    differential equationstraffic flowvehicle followsecond order
    Modeling Scenario
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    280

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    1-145-FastPitch-ModelingScenario
    We consider the problem of comparing pitch velocities using measurement methods in different eras of baseball.
    Measurementresistanceair resistancebaseballpitcherfast
    Modeling Scenario
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    820

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    3-002-ModelsMotivatingSecondOrder-ModelingScenario
    Ordinary differential equations involve second derivatives and second derivatives appear in many contexts, chief among them are the study of forces and resulting motion. This is principally because of Newton's Second Law of Motion.
    resistancespring massoscillationHooke's Lawdampening
    Modeling Scenario
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    287

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    3-011-EulerBallThrowing-ModelingScenario
    If a tennis ball is thrown through the air it will hit the ground due to gravity. Using Euler's method, write a short script (Python, Matlab, R, etc.) to find the trajectory of the ball which will maximize the distance the ball lands from the...
    codingdragonVectorsEuler's methodball
    Modeling Scenario
    296

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    158

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    3-015-StyrofoamBallFall-ModelingScenario
    We are given data on a falling Styrofoam ball and we seek to model this motion.
    footballresistancegravityfalling bodysum of square errorsNewton's Second Law of Motionsum of forcesstyrofoam