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06 April 2010

The difference between a tautology and a justification

Here's an optics homework question I asked the other day:

A concave mirror has a focal length of 10 cm.  It is desired that an image of an object be created that is upright and magnified three times.  (a) Should the object be placed greater or less than 10 cm from the mirror?  Justify your answer.

Now, in and out of class, we have reviewed this type of situation ad nauseum.  We have showed with equations, experiment, and ray diagrams that an object closer to the concave mirror than a focal length produces an enlarged virtual image; any object placed beyond a focal length from the mirror will produce a real image.  My students hopefully KNOW these mirror facts as well as they know that a field goal is worth three points.

Consider this justifcation, then:

The object should be placed less than 10 cm from the mirror, because that produces an enlarged, virtual image.

Such a justification earns no credit!  This response is a tautology, circular reasoning that essentially says "it is because it is."  I mean, *I* know that this student is reasoning from personal experience or memorized facts, but the student could just as easily be guessing and restating the conditions from the question stem.

A correct justification here must go beyond a restatement of the facts, no matter how simple the reasoning might be.  Possible correct justifications:

*... because an object placed inside the focal point of a concave mirror will *always* produce a virtual, enlarged image.  That's a property of concave mirrors.

*... look at my ray diagram here.  You see that the image is virtual and enlarged, and the object is inside the focal point.

* ... look at this calculation here.  I get an object distance of nearly 7 cm, which is clearly less than 10 cm.

* ... in my class we looked at images in concave mirrors, and we saw that only objects close to the mirror produced upright images.

Don't fall into the tautology trap.

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