Problem Solving Algorithm

         In general, solving a physics problem (and to a certain degree, the exercises as well) involves taking a realistic situation and constructing a simplified model that captures its essence. Often the complicated details of a problem are deliberately suppressed so it can be solved with simple physics principles. Creating a model that is sufficient to describe the essence of a problem without being overly complex is not an easy task. Doing this will require practice.
        The good news is that the skills you refine here solving physics problems will be applicable to other courses and other facets of life. In problems the situation described may appear new to you; it may appear that you have never seen a similar problem. This is what happens in life- each day brings something new you’ve never encountered before.
         The algorithm presented here is meant to make your job easier. Think of this the basic scaffolding that will allow you to build various masterpieces. At first this algorithm might feel awkward, but trust me it will be beneficial in the long run. (The same is true of other skills- the proper grip on a tennis backhand may at first feel uncomfortable, but you use an incorrect grip your game will be limited.) Studies have shown that experienced problem solvers (unlike novices) usually solve problems using a framework that is independent of the problem or person.
         The approach is based on constructing a model of the situation described though three different representations of the problem- pictorial, conceptual and mathematical. In this way you can tap into three different parts of the brain. A fourth component, evaluating the answer or model, is also present. The model you build (not the number you get at the end of some algebra) is the “answer.” With this view, algebraic mistakes are not as significant as conceptual mistakes (making incorrect assumptions for example). Essentially you are trying to predict the system’s behavior with your model- this is what it means to do science.
         A fifth component to the process is taking a sufficient amount of time to understand the problem. This often means rereading the problem several times, as well as taking some time to visualize the situation in your mind’s eye. The better one understands the problem, the better chance they have of building the correct model.
         It’s okay, if not beneficial to make mistakes within a solution. Everybody does it. Just like solving a puzzle, nobody puts the pieces down in the final position until after trying them in several other places first. Expert problem solvers rarely solve a problem in the linear, always perfect way textbooks present example solutions. You shouldn’t feel as though you’re doing anything wrong if you take a few wrong turns. Recognizing why they are wrong turns can be quite educational.

Problem Solving Algorithm
 

1. Pictorial Representation

           Draw a picture that shows the essence of the situation. It does not need to be a work of art. People can be stick figures, cars can be squares; you’re just trying to get a feel for the problem here. Be sure to indicate your coordinate system. Is up the positive direction? Which way does the x-axis point? Etc.
          Also, this is a good place to list known values & define variables. Choose a naming convention that is easy to remember and understand. For example, using subscripts can help to make sense of a large number of variables.

2. Conceptual or Verbal Representation

          The main point of this representation is to describe the situation (& model) in words. You should be as complete as possible- describing the situation as well as the relevant physics concepts. Here are a few more details:
   
   * Identify the system. Are you looking for the force of the table on the book or the book on the table?
   * Indicate the fundamental physics principle or concept. Write a sentence that describes the basic concept at play- “Here we see the conservation of momentum in a two-body collision.” By identifying the principle early in the solution you can help yourself stay on track.
   * Identify any assumptions or simplifications. For example, are you going to ignore air resistance? Making simplification can make an apparently complicated problem much easier. Just be careful that you retain the essence of the problem and don’t oversimplify.
   * Hypothesize what will be the solution or outcome. Often you will be asked to predict the outcome- “will the car stay on the road as it takes the corner?” By stating in words what you think will happen (by using your intuition) you might be able to catch a mathematical error later in the model. (“Given that the fact that the car isn’t travelling very fast, I believe that it will stay on the road.”)

3. Mathematical Representation

            Here is where you will set up and solve various equations that model your situation. Notice that this is only one part of a complete solution or model. Without the pictures and words the equations are meaningless. Never forget that the equations are merely one representation of the system.
          When you write down the equations make sure you begin with the fundamental principle you identified before. ("Conservation of momentum implies that P_i= P_f"). We want to let the physics guide our math. Then solve the equations symbolically for any unknown variables. This is the step where you will be using your algebraic skills to rearrange the equations to get something useful. You should also explain what you are doing as you carry out each step- "using equation #2, substitute P_i into equation #3")
           Finally, plug in known values and calculate a numerical answer (if needed). Wait until the final step to plug in the numbers. A solution is much easier to follow if you use variables throughout the solution.

4. Evaluation

          This is not really another representation, rather this section is where you make sure that your model and numerical results match your real-life experience. You should check the answer to any numerical calculation- does the answer have the correct units, sign, direction, etc.?
         Make sure that you answer the questions asked (if the question asks if a cheetah can catch a gazelle, the answer is “yes” or “no”, not 5.6 m/s) And finally, you should ask: Does the answer make sense? Is the answer reasonable? How does it compare to your hypothesis? This is where you try to reconcile your intuition with your mathematics. If they differ, it would be worth reviewing your model. Often even the “best” problem solvers will get to this step and find that their model and results do not match their real-life experience. For example, cars cannot travel 3 x 10⁸ m/s. If your equations produce this result, you should go back and reexamine your model. When you perform this step, you will largely be having a conversation with yourself. Document this “conversation on your final paper; in other words, write down your reasoning for believing that your model is reasonable.

       While expert problem solvers usually go through the algorithm in roughly the order I have given, you may find that as you construct your model, you will jump back and forth between various parts to add things you have forgotten. This is normal and appropriate. Also, this algorithm has been written to be as complete as possible, you will certainly encounter problems where various steps aren’t as applicable.
      I should point our that our text also gives some good advice on solving problems. There is page 47, which emphasizes the idea of breaking down complicated problems and situations to simpler parts. Essentially this is learning how to see the trees in the forest. Problems are often very intimidating at first glance, so a good first step can to be looking for “sub-problems” that are easier to handle and bring you closer to understanding the original problem.
       On final note about the models your turn in- neatness counts. Any paper that is full of scratch work, will loose points. Think of the process of writing a paper- you first make notes for yourself, then work on a rough draft or two. What you hand in is a well-organized, easy to read paper. The models and solutions which you hand in in this class should be thought of as final drafts. You are bound to need to perform some scratch work and make revisions, these are your rough drafts which you should keep. You don’t need to write up your models on a computer (you can, but it is difficult to do), just make sure that it organized, and easy to read.