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 ASTR430 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
Show Your Work.
Give written descriptions of what you are doing, and why you are doing
it. This is often 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 I can follow what you have done in a
derivation (especially if you go off on a wild tangent!) and will help
me 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 derivation!
Check Units. Any equation 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. Remember this general rule: in all
physically valid solutions, 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.
Use Common Sense. Usually you will have some
physical insight into how the solution to a problem should
look. Compare your derived solution to a problem to what you expect
from physical insight. Trust your instincts! If a derived equation or
numerical value looks funny, go back through the derivation and look
for an error. If you can't find an error, make a note of your concerns
near your final solution and I will comment on them.
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, 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 I might give.
Go over Homework Solution Sets. When you get
homeworks back from me, 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. I spend 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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