1 The MAPS Approach to Newtonian Mechanics

Getting Yourself Ready to Understand Modeling Applied to Problem Solving    [!copyright and waiver^SectionEdit.png!]

Learning Objectives

By the end of this Lesson, you should be able to:

• Describe the strategy behind the letters S.I.M.
• Know roughly what elements constitute a model used for representing the core concepts of mechanics.
• Explain why you have to be actively involved in the process of learning to apply modeling to solve problems.

Learning Newtonian Mechanics well enough to find answers to questions that are real problems for you requires you to be an active participant in constructing your own knowledge and skill. If your previous study of physics may led you to believe that solving physics problems involves mainly plugging in the right formula, you may need to review the difference between [problems and exercises].

If you think just memorizing things (e.g. definitions, formulae...) is enough, to enable you to solve real problems, read the section immediately below; if you already realize you have to do more than memorize, then skip it:

Models and Physical Models

A model is a simplified description of a complex entity or process.  Models often highlight some particular feature of the modeled entity or process while blatantly ignoring others.  For example some scale models of a particular airplane (e.g. an SR-71) are faithful in appearance, but can't fly.  Other models of that same plane can fly but don't look very much like an SR-71.  Still other models are not tangible, e.g. computer models of the supersonic air flow over the wings and into the air inlets for the engines of the SR-71.  As George Box (an industrial statistician) once said, "All models are wrong.  Some are useful."

This WIKI aims to enable you to apply these models to the physical world. It therefore adds another category: what are the key restrictions and requirements about the system and its interactions, and the typical physical cues, that trigger the mind to recognize that a particular model applies.

Models in Newtonian Mechanics

Newtonian mechanics is a restricted domain that is concerned only with describing certain effects of interactions between objects. The power of Newtonian mechanics is that the small number of idealized frameworks (mechanical energy conservation, constant acceleration, momentum conservation, etc.) presented in the course are sufficient to describe many seemingly disparate situations.

In our modeling approach to mechanics, the various idealized frameworks are envisioned as basic models that can be used as approximations to a large number of real-world situations. The job of the problem solver is to take a real world situation and idealize it in such a way that one of the models adequately describes the system evolution. For example, both a baseball falling through the air and a jet moving down a runway might reasonably be idealized to fit the model of motion with constant acceleration.

Examples of physical models relevant to Newtonian mechanics are:

• One-Dimensional Motion with Constant Acceleration
• Simple Harmonic Motion
• Mechanical Energy, External Work, and Internal Non-Conservative Work
• Point Particle Dynamics (applying Σ F = m a to a point particle)
  For a formal organization of these models, see Hierarchy of Models. 

A model consists of the following pieces:

1. the situations, conditions, and idealizations under which the model applies
2. specification of the independent and dependent measurable state variables that characterize the system and which the model interrelates
3. what physical theories underlie the model and the resulting equations and representations
4. descriptions of the model and interpretation of its predictions as expressed in all various useful representations
5. the behavior/change in state, geometric, temporal, and interaction structure
   

A model will generally model only some of the structure in a physical system. For example, the engine of a car can be regarded as a "heat engine" to turn heat into mechanical energy (work), or as the "powerplant" - a source of a certain amount of power that can accelerate the car.  The particularization of the model therefore relies critically on the selection of which state variables to include or exclude.

Specification of Basic Models for Mechanics

Name:  Each model must have a name 

Description:  What separates this model from the others.

Compatible Systems: The restrictions needed to ensure a given system can be adequately described by the model.

Relevant Interactions: The types of interactions that must be considered when evolving the system using this model.

Laws of Change: The mathematical rules that govern the evolution of a system that is described by the model. Often this is differential equation or an integral equation ( for example, F = dp/dt ).

Hierarchical Organization of the Models

A Models Hierarchy for Mechanics is used to help the learner understand that there are only four basic conceptual domains in Newtonian mechanics:

• Motion
• Momentum
• Mechanical Energy
• Angular Momentum

and that each domain has an associated class of relevant interactions (the class of interactions that cause evolution of the principle quantity)

• Motion is altered by acceleration, or equivalently, net force.
• Momentum is altered by external forces.
• Mechanical Energy is altered by non-conservative work.
• Angular Momentum is altered by external torque.

Pedagogical Usefulness

The key pedagogical reason for using models is to provide a framework within which students can organize the many facts and procedures they learn in introductory physics into a small number of useful models. They can then relate these models to the few overall theories that underlie the material, and think about the real world by recognizing situations or simplifications where these models apply.   This leads to an understanding of the world through the ability to simplify and model physical situations that are new.  From an expert/novice perspective models organize the many formulae and graphs on the novice's formulae sheet into a much smaller number of "chunks" of related things that are of reflective of nature's organization. 

Models are idealizations of physical reality that involve a particular structure or pattern.  Models can be mathematical, logical, pictorial, or they can be actual physical objects.  Models only approximate reality; they represent an idealization of reality (where we can, for example, exclude friction, or ignore the bending of rigid bodies, and so on), but generally they are applicable to many situations. If they weren't, the model would not be useful.  Models generally involve a cluster of several concepts and theories (As an example, harmonic motion involves kinematics, F = ma , and a linear restoring force).  Models are almost always expressed through several representations, and the cross-connections among these representations provide a rich envisioning of the situation.  For example, motion with constant acceleration may be represented with standard equations, strobe pictures of the object, graphs or tables of kinematic variables vs. time, or a concise verbal description.  The quantum mechanical two-level system may be represented using the equations following from time-dependent perturbation theory, or by the Bloch vector, or by using the density matrix.   

Physicists and educational psychologists agree that understanding a model implies fluency with, and ability to transfer among, all of its commonly-used representations.  A physicist familiar with any model can recognize/describe/understand/quantitatively predict situations that fit within the model's assumptions with little effort, but is typically confronted with a "problem" if even a small discrepancy exists between the situation at hand and the relevant model. 
Being able to understand and use a model involves:

1. understanding the various representations and their interrelationships
2. developing an ability to recognize physical situations where the model applies (even novel ones)
3. being able to map the reality onto the model (i.e. to ignore unimportant things)
4. being able to carry through the solution in any of the model's representations
5. at a high level, being able to generalize the model

Understanding/Learning a Model

Understanding a model requires each student to reconstruct and interrelate its components in their own minds, and to understand the relationships among its representations.  This is usually achieved by a laboratory course in which a group of several students follows a guided discovery procedure (but never a completely cookbook prescription) followed by a discussion skillfully conducted by a trained person.  DEP feels that interactive lecture demonstrations together with problems involving transfer of representations should be able to perform much of this function.   

However it is done, successful learning involves the student understanding the pieces of the model and being able to use it in context-rich problems (i.e. extract the relevant variables from a real world story as well as the minimalist presentation typically found in textbook problems).  Hestenes would contend that a key to modeling is that students become skillful at constructing models for new situations, and so he would disapprove of our approach.  Making a model and learning to apply it typically takes two weeks. This limits the number of models we can study to about 6 for the course.  Our hope is that we can teach roughly one model/week by starting with only 4 general models and indicating how several other models are subcases of the general models.