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    Modeling Scenario
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    3-064-GearTrain-ModelingScenario
    Students model an input-output mechanical system of gears with a second order, non-homogeneous, ordinary differential equation with constant coefficients. The model incorporates friction and moments of inertia of the gear train components.
    frictiongearspur geargear traintorquegear ratioangular accelerationangular velocitymoment of inertiadouble reduction gear
    Modeling Scenario
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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
    Potential Scenario
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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
    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
    358

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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
    Modeling Scenario
    289

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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
    Potential Scenario
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    2018-Kovacs-Insperger-Retarded neutral and advanced differential equation models for balancing using an accelerometer
    It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models.
    feedbackAccelerometersfunctional differential equationssemi-discretizationD-subdivision method
    Modeling Scenario
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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
    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
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    155

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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
    Modeling Scenario
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    451

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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
    Article or Presentation
    140

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    50

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    2014-C_W_Groetsch-S_A_ Yost-Vertical Projection_in_a_Resisting_Medium_Revelations_on_Observations_of_Mersenne
    This article, inspired by a 17th-century woodcut, validates empirical observations of Marin Mersenne (1588–1648) on timing of vertically-launched projectiles for a general mathematical model of resistance.
    resistanceprojectile motionprojectilefluidMersenne
    Modeling Scenario
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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
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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
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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
    Modeling Scenario
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    3-033-S-TimeUpTimeDown-ModelingScenario
    We seek to compare for the time a projectile takes to go vertically up with the time it takes to return to its starting position.
    projectile motionapexvertical motion