1 The MAPS Approach to Newtonian Mechanics

Understanding vs. Memorizing

Imagine that you had to solve a novel physics problem today, either in real life or on a test, and that you had a physics text that contained the relevant physics as well as a selection of solved problems. Even if you had a search engine to aid your retrieval of this material, would you feel confident that you could solve your problem?

Probably you wouldn't feel confident if your problem differed in any significant respect from all of the ones in your collection, or if you had to recognize this feature in the real world. For example, you might have the equation for the range of a projectile fired at a certain speed and angle above horizontal. But your problem might be to shoot at enemies attacking from 100 meters beyond your troops who were dug in under some tall trees. In this case you'd need to be able to calculate whether the explosive projectiles would clear the trees on their way to the target, and if not how much slower than usual they must be fired so that they would. In order to solve this problem, or even to recognize that it was a problem, you’d have to visualize the situation and understand how the range formula is obtained. If you understand that the horizontal and vertical components of the motion are independent, you'd find the time when the projectile would be over your troops from its horizontal equation of motion, then find its height from the vertical equation of motion, then adjust the muzzle velocity to avoid the trees. (The failure to recognize and solve this problem cost the US several fatalities in Vietnam – see Friendly Fire by C. D. B. Bryan.)

Please take a moment to reflect on what you have been trained to do when you study physics or review your physics homework. Very likely it is to memorize the formulae in the textbook and the solutions to the problems you’ve done. But the above example should show you that this is not sufficient – somehow you need to understand the material as well. Understanding involves being able to visualize the problem and to see the deep regularities (e.g. conservation of energy) that underlie the solution to many of these example problems. Paradoxically, if your understanding is at this level, you only need to remember the typical schema (plan of attack) involving each concept in order to be able to solve many problems – you don’t need to know the detailed solution of many different example problems involving that concept.

Constructing Understanding

Education experts agree that understanding must be constructed in your mind by your own thought processes. Passive memorization is not enough – rather you must place new knowledge within the context of what you already know and understand. For example, you probably understand intuitively that if your opponent in a snowball (or water balloon) fight is crouching behind a low fence, then you must lob your projectile slowly so that it descends at a steep angle, allowing it to pass over the fence and land on the target. Obviously this intuitive understanding should be transferred to the artillery problem discussed above.

The iimportant lesson here is that Newtonian Mechanics is about the everyday motion of things around you, the energy crisis, and the additional distance it takes to stop your car when going downhill vs. uphill – all subjects about which you have some intuitive/experiential knowledge already. You must insist to yourself that learning Newtonian mechanics requires you to connect it with your intuition. If you don’t, you ignore a rich source of existing understanding and greatly reduce your ability to usefully apply your new learning to new problems.

An example of this struggle for intuitive connection of Newtonian mechanics and real world experience is evident from this poetic summary of Newton’s First Law from the perspective of a student studying physics for the first time:

Objects in motion remain in motion in the classroom and come to rest on the playground.

If she thinks about this until she realizes that the difference is that the air track in class has been designed to reduce friction to insignificance but that there is considerably more friction even on a rolling ball on the playground, then her physics class can enrich her view of the everyday world, and vice versa. On the other hand, if her mental model is that what happens in the classroom is according to one set of rules while the real world operates according to another, then Newtonian mechanics will be a foreign set of concepts and equations that describe only the teacher’s reality and which will soon be forgotten.

Constructing Your Understanding

In order to construct your own knowledge, you must think about what you are reading in the text or hearing in lecture, and you must learn to step back from a problem both while beginning to solve it and also after you have finished.

Text and Lecture: How does the material at hand fit into your overall mental outline of the course? What formulae are simply special cases of others? For example, a = F/m is the same relationship as F=ma. Similarly, both are special cases of F=dp/dt, the fundamental law of change for momentum (as this text will emphasize in teaching you the Momentum Model.)

Problem Solving: When approaching a problem the most important thing is to change your objective from “Find the Answer” to Plan the Solution. That is, concentrate on the process of solving it – what principles did you apply and why? Very valuable is a retrospective look: ask yourself what one or two sentences of advice would have enabled you to solve the problem more quickly – this is what you really learned by doing the problem.

The S.I.M. Approach to Problem Solving

The first step in solving any problem is to develop an understanding of the situation and to conceptually plan your attack. For those new to physics, this first step is often the hardest! As with learning any new skill, when learning to solve real problems it is important to be systematic. Luckily, there is a simple systematic approach that is usually a useful strategy for starting a mechanics problem. This approach is the SIM Strategy at the heart of problem solving in MAPS pedagogy:

• S. Choose a system to consider.
  
  In every mechanics problem, you will choose to focus on the motion of one or more objects. These objects will make up the system under consideration. Sometimes this choice will be easy. For problems involving the motion of a single car or baseball or box, the car or baseball or box will be the system. Other times, the choice can be hard. When two boxes collide, you may want to focus on only one of the boxes, or both independently, or you may want to consider them together as one system. Or you may decompose the system into intervals where first one and then another treatment is best.

• I. Describe the interactions this system experiences.
  
  Interactions are influences that change the motion of the system. The most common way to describe interactions is to think of them as forces and represent them using a free body diagram, but later you will become familiar with energy and torque as alternate descriptions. There are only a few types of interactions that we will study in this course, including gravity, contact forces (pushes, pulls, friction, etc.) and spring forces. It is important to recognize that interactions always occur between two objects. For instance, if your system is subject to a gravitational interaction, it must be near another object (often the earth). If your system is subject to a contact force, there must be another object nearby to do the pushing or pulling.

• M. Choose a model from the hierarchy that will help you to solve the problem.
  
  Models summarize the important physical content of the course, and each has certain allowed interactions and contains an associated equation of change. For example, if you know that your system experiences a constant acceleration, the equations of change describe the system’s position and velocity as functions of time. To help you get a big picture overview of the course, we have organized the most important models that you will learn into a hierarchy of models.

Strategy and the S.I.M.

When starting problems using the S.I.M. approach, it is important to understand that you cannot think of this as a “1, 2, 3” prescription for success. Choosing the most advantageous system when solving a problem will usually require you to understand the interactions in the problem. Choosing an appropriate system and model will always require a good understanding of the interactions. The way that you describe the interactions will require you to think about the model you want to employ. These relationships are fundamental to a strategic approach to problem solving.

Using the S.I.M.

We want you to begin every problem with the S.I.M., so it is important that it be brief.

• Describing your system can be a sentence or two, or perhaps just clear labels on a picture showing what is inside and outside the system.
• Describing the interactions is the most important part, but it should be possible to be clear in one paragraph. It will often help to draw free body diagrams. Briefly discuss the interactions that are relevant to the system and the model, and what objects are participating in the interactions (remember, there are always two objects participating in any interaction).
• Choosing the model is as simple as picking one of the models from the hierarchy.

To get a sense of how these specifications work in practice, please have a look at the worked examples in this WIKI, which all use the S.I.M. approach.

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.