ASTR 120
# Problem Solving Hints

This page is meant to give you advice to help you improve your problem
solving skills and your homework writeups. I expect you to follow
these points for ASTR120 homeworks, and encourage you to employ them
in your other science classes as well.

** Write up Neat Homeworks. ** Take pride in
your homework writeups and do the best job that you can on them. Take
the time to solve the homework problems roughly on scratch paper,
and then copy them over neatly, filling in additional details on your
final copy.

** Show Your Work. ** Often it is useful to give
written descriptions of what you are doing, and why you are doing
it. This is especially useful at the beginning of a problem where it
will force you to think about the problem physically and formulate
your approach mathematically. Descriptions will also maximize the
chances that the TA can follow what you have done in a problem
(especially if you go off on a wild tangent!) and will help her to
give you constructive comments on your work. Give enough detail, and
show enough mathematical steps, that students less advanced than you
could understand your solution!

**Use Common Sense. ** Usually you will have some
physical insight into how the solution to a problem should look. Is
the number too big? Too small? Compare your solution to a problem to
what you expect from physical insight. Trust your instincts! If an
answer surprises you, go back through your work and look for an
error. Test your calculations. If you cannot find an error, make a note
of your concerns near your final solution and the TA will comment on
them.

**Check Units. ** Any mathematical expression that
you write must be dimensionally correct. Check your equations
occasionally as you go through a derivation. It takes just a second to
do so, and you can quickly catch many common errors. All distances
must have units of length (meters, miles, etc.) all times must have
units of time (seconds, years, etc.) and so on. Remember this general
rule: the argument of all functions (e.g. trigonometric functions,
exponentials, logs, hyperbolic functions, etc.) must be
dimensionless. Taking the cosine of something with units of mass or
length makes no physical sense.

**Check Limits. ** Check all of your final answers and
important intermediate results to see if they behave correctly in as
many different limits as you can think of. Sometimes you will know how
a general expression should behave if a particular variable is set to
zero, infinity, or some other value. Make sure that your general
expression actually displays the expected behavior!

**Take Advantage of Symmetries. ** Symmetries are
fundamental in physics (and astronomy!). Problems can have symmetry
about a point (spherical symmetry), a line (cylindrical or axial
symmetry), or a plane (mirror symmetry). You can use symmetries in two
ways: 1) to check your final answer to a problem or, with a little
more effort, 2) to simplify the derivation of that final answer. As
an example, time-independent central forces (like gravity) have
spherical symmetry because the force depends only on the distance from
the origin. In this case, spherical symmetry means that once we find
one solution (e.g. a particular ellipse for gravity), all other possible
orientations of this solution in space are also solutions.

**Get Help from Others. ** Work on the homework
problems on your own first and get as far as you can on them. This is
the best way to improve your problem solving skill and prepare for in
class tests. But by all means get help from other people (other
students, the TA, or me) when you are stuck! By trying the problems
first, you will be able to ask more intelligent questions and better
understand the ideas of other students and/or the hints that the TA
and I might give.

** Go over Homework Solution Sets.** When you get
homeworks back, go over the solution sets and your corrected
homework together. Use the solution set to see how to get past points
where you were stuck, and make sure that you could easily do a similar
problem if given the chance, say on a midterm. Even if you get a
particular problem correct, there is always much to learn by following
through someone else's solution. The TA spends a lot of time writing up
solution sets so that you can all improve you problem solving
abilities. Take advantage of the opportunity!

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