SIMIODE: Resources

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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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

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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

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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

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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

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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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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

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

Potential Scenario

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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

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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

Modeling Scenario

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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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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

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1-120-CircularRollerCoaster-ModelingScenario

Students study the dynamics of a circular roller coaster and work out the equations of motion in the ideal case as well as considering the interesting complication of including kinetic friction. This problem is an excellent introduction for students

population dynamicsenergyFluid Mechanicsmfrictionroller coasterintegration by8 parts

Modeling Scenario

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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?

projectile motionbaseballparametric equationstadiumhome runno reistantfairnessno resistance

Article or Presentation

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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