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    Potential Scenario
    215

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    46

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    1980-Peastrel-Lynch-Armenti-Terminal velocity of a shuttlecock in vertical fall
    We have performed a straightforward vertical fall experiment for a case where the effects of air resistance are important and directly measurable.
    fallingterminal velocityshuttlecockvertical fall
    Modeling Scenario
    1154

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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
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    169

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    3-009-BallDropInWater-ModelingScenario
    We conduct an analysis of a falling ball in liquid to determine its terminal velocity and to ascertain just what radius ball for a given mass density is necessary to attain a designated terminal velocity.
    resistancegravityfalling bodyterminal velocitybuoyancy
    Modeling Scenario
    358

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    351

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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
    Modeling Scenario
    245

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    141

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    3-027-BobbingDropping-ModelingScenario
    We present two exercises in which we ask students to model (1) falling object experiencing terminal velocity and (2) bobbing block of wood in liquid. We model the motion using Newton's Second Law of Motion and Archimedes' Principle.
    drug resistancedirectedbuoyancyfree fallstatic equilibriumdisplacement
    Article or Presentation
    164

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    50

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    1980-Peastrel-Lynch-Armenti-Terminal_velocity_of_a_shuttlecock_in_vertical_fall
    This paper offers a set of data on a falling shuttlecock (the moving object in badminton) in air. How the authors collected the data is described fully and can be replicated easily by modern equipment.
    parameter estimationdata collectionresistancefallingterminal velocityNewton's Second Law of Motionshuttlecock
    Modeling Scenario
    335

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    149

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    1-105-AnimalFall-ModelingScenario
    This project uses Newton's Second Law of Motion to model a falling animal with a resistance term proportional to cross sectional area of the animal, presumed to be spherical in shape.
    animalNetwon's Second Law of Motionfalling bodyterminal velocityair resistanceNewtonair frictionfall
    Modeling Scenario
    280

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    148

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    3-063-FallingBuildingIce-ModelingScenario
    We model the fall of a piece of ice which is falling from a high building in New York City.
    resistancefallinggravityiceterminal velocityskyscrapers
    Modeling Scenario
    537

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    294

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    1-050-BargingAhead-ModelingScenario
    As captain of a barge, you need to determine how fast to transport your barge up river against the current in order to minimize the expended energy.
    optimizationkineticsenergytrophic efficiencyworkriverterminal velocityfrictioncurrentforcebargeworlkspeed
    Modeling Scenario
    203

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    211

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    3-095-ShotInWater-ModelingScenario
    This project uses Newton's Second Law of Motion in conjunction with a quadratic model for the resistance experienced by a bullet moving through water to analyze a classic action movie scene.
    projectile motionterminal velocityprojectilewater resistancereduction of orderfrictionair resistance
    Modeling Scenario
    468

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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
    Potential Scenario
    190

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    44

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    1994-Roland_Minton-A Progression of Projectiles-Examples from Sports
    There is a progression of complexity and issues in projectile motion modeling from no resistance, to resistance proportional to velocity, then to velocity squared,
    sportsresistanceprojectilebaseballdraglinear resistancequadratic resistancetennis
    Modeling Scenario
    214

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    114

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    3-020-ChordPathTime-ModelingScenario
    Using Newton's Second Law of motion we can determine the time it takes for a mass to slide along a chord of a vertical circle from high point to any point along the circle. Initially, the result is nonintuitive and challenging to understand.
    Unit CirclepathfallinggravitychordNetwon's Second Law of Motion
    Article or Presentation
    131

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    47

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    1999-F_Brauer-What_Goes_Up_Must_Come_Down
    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 that are nonlinear functions of velocity.
    resistancefallingprojectilevertical projectile
    Modeling Scenario
    342

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    270

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    3-017-StackedCoffeeFiltersFalling-ModelingScenario
    Data on free falling 2, 4, 6, and 8 stacked coffee filters is offered. Students form a model using a resistance term proportional to velocity, velocity squared, or velocity to some general power. Parameters need to be estimated and models compared.
    dataresistancefalling bodycoffee filterstacked
    Modeling Scenario
    356

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    215

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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
    156

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    61

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    1989-R_Blickhan-Spring Mass Model For Running-Hopping
    A simple spring—mass model consisting of a massless spring attached to a point mass describes the interdependency of mechanical parameters characterizing running and hopping of humans as a function of speed.
    spring massjumpjumping
    Potential Scenario
    128

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    34

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    1999-Long-Weiss-The Velocity Dependence of Aerodynamic Drag A Primer for Mathematicians
    This paper is exactly what it says it is, namely, a Primer for Mathematicians. There are ample models, data, attention to such issues as Reynolds’ numbers, and attention to aerodynamics issues.
    drag forceaerodynamics
    Potential Scenario
    176

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    56

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    1972-Suresh_Sethi-Optimal Control of the Vidale-Wolfe Advertising Model
    This paper considers an optimal-control problem for the dynamics of the Vidale-Wolfe advertising model, the optimal control being the rate of advertising expenditure to achieve a terminal market share within specified limits.
    Optimal Control Theoryadvertisingsales
    Modeling Scenario
    244

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    182

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    3-042-CatapultLaunch-ModelingScenario
    We maximize the range of a projectile by backing up an incline in the opposite direction of the range to give some initial lift. Find the position on the hill from which to launch the projectile to give the best lift.
    projectile motioncatapultuphilllaunch angle