
What I’m presenting here
is a collection of ideas that may make it easier for you to solve physics
problems, and for me to understand your work. (You benefit on both counts.)
I’m not expecting (or asking for) 5page solutions. Be efficient
with your words and equations. The key points to remember are these:
 Don’t let
mathematics be a substitute for a physical understanding. Mathematics
is a convenient language for expressing the ideas of physics. Pictures
and words can help you to avoid this pitfall.
 It’s okay,
if not beneficial, to make mistakes while drafting 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.
 You will get better
with practice. Just as with music or sports, one has to continually
practice to improve. Patience is the key here.
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.
Suggested
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 xaxis 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 charge 1 on charge 2 or charge 2 on charge
1?
 Indicate the fundamental
physics principle or concept. Write a sentence that describes the basic
concept at play “Here we see two charges interacting by a forces
described by Coulomb’s Law.” 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 charges collide?” 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 first charge is moving at nearly 10% the speed of light (and
therefore has momentum to overcome the impulse applied to it), I think
that the two charges will collide.”)
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 reallife 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 the charges collide, the
answer is “yes” or “no”, not 7 msec.) 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 reallife experience. For example, cars cannot travel 3 x 10^{8}
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.
One final note about the models your
turn in neatness counts. Any paper that contains 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 wellorganized, easy to read paper. The models and solutions which
you submit 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.
