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Problem Solving

         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. And the reverse is true as well, the problem solving skills you have from other experiences will facilitate your physics problem solving. The reason for this transfer is that problem solving, not matter the context, has the same core elements.
       Think of this the problem solving as 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.
      What follows are some suggestions for how to implement the general problem method to physics. In each step, there is a list of questions that you may want to ask yourself as you solve a physics problem, particularly, if you get “stuck” along the way.

Problem Solving Algorithm

  1. Understanding the problem (Recognizing what is asked for.)
  • Construct a mental image of the objects and sequence of events
  • Sketch a picture
  • Construct diagrams or graphs to show important spatial and temporal relationships
  • Define all of the mathematical symbols
    • What objects are involved?
    • Are all the important objects known?
    • The important times, spatial relations, interactions and motions known?
    • Does the question ask about a specific physical quantity? (If not, can the question be reformulated so it does?)
    • What is the system of interest?
    • What are the known, and unknown, quantities?
  1. Creating a plan (Responding to what is asked for.)
  • State principles and relationships that will likely be used.
  • Select one of the relationships to relate the desired unknown quantity to known ones.
  • If there are multiple unknown quantities, additional relationships may be necessary.
  • If using mathematical relationships (equations), combine so they allow the target unknown quantity to be found.
    • Which physics principles are relevant?
    • Are any approximations necessary?
    • What reference frame or coordinate system would be most convenient?
    • What relationships and equations will follow from the selected principles?
    • Which relationships relate the unknown quantity to the known ones?
    • Are there any restrictions on the relationships that may conflict with the given information or principles?
    • Are there any quantities that cancel out when combining relationships?
  1. Executing out the plan (Developing the result of the response.)
  • Substitute the numerical values and units for each quantity in the equations and relationships.
  • Calculate the target quantity, both numerical value and units.
  • If possible, simplify the target quantity’s units so it is more readily understandable.
    • Which values should be used for the variables?
    • Do the units need to be converted?
    • Do any units cancel?
  1. Reviewing (Checking. What does the result tell me? )
  • Check that the answer is reasonable. (If not, review the solution and revise where necessary.
  • Determine if the answer is complete.
    • Do the units make sense?
    • Do vector quantities have both magnitude and direction?
    • Does the answer fit the mental picture of the situation?
    • Do the signs of the quantities agree with the chosen coordinate system?
    • Is there a calculation mistake in the execution?
    • Has the question been answered?
    • Could somebody else read and follow the solution?

 

         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 or lop back to previous steps before moving to the next.

         “Okay,… so that’s how I solve a problem, but what do I turn in?” Excellent question! ? After all, you’ll notice that most of the problem solving occurs inside your head. Dr. Jeff can’t very well scan your head with an MRI to see what you’re thinking, but wouldn’t that be cool?
         This is where your documentation comes in to play. In order for Dr. Jeff, or your supervisor later in life, to know that you solved the problem you have to document your ideas. This is what is graded; or, gets you promoted at work. The approach outlined here 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 modes of communication. 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.
         It should be pointed out that this sequence- solve the problem first, then create a final documentation explaining your ideas, is the way to go. Just like an English paper, what you turn in is not a rough draft or a sheet full of scratch marks, rather it is a final draft. In other words, don’t try to work out your solution on the same piece of a paper that you’re planning on turning in. And, to motivate you to practice this… Any paper that is full of scratch work, will loose points.

Problem Solving Documentation
 

  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 Pi= Pf"). 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 Pi 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.

 

 


       I should point our that our text also gives some good advice on solving problems. There is page 47 (volume 1), 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.