(Fermi Problems)
Sometimes we need to make a quickcalculation. We may want to do a rough check of a complicatednumerical calculation to see if the answer is reasonable. Or we maynot have access to all the data needed, and an estimate, or a veryrough approximation, is in order. In these cases we want to do anorder of magnitude estimate, where variables are rounded offto the nearest power of 10 or to some other easily handled number.For example, the radius of the earth is 6.4 x 107 mor about 108 m to an order of magnitude. The height of aperson is roughly 2 m or about 100 m to an order ofmagnitude. This is not to imply that a typical height is really 1 m,but rather that it is closer to 1 m than to 10 m or 0.1 m.
Judicious approximations and order ofmagnitude estimates are things that play an important role in aphysicist’s approach to problems. The final result of acalculation with variables so dramatically rounded off should not betrusted to better than a factor of ten or so, but the estimate canstill be extremely useful. Such estimates are often used inenvironmental problem solving where problems often call for aquantitative answer but the statement of these problems has thatfuzzy quality characteristic of real-world situations, and we cannothope for a solution to better than an order of magnitude.
The Nobel Prize winning physicist,Enrico Fermi, was particularly good at such "back-of-the-envelope"calculations. Legend has it that when the first atomic bomb wastested towards the end of the Second World War, Fermi estimated theenergy released in the explosion by dropping bits of torn paper andnoting their horizontal displacement when the shock wave from theblast reached his position, several miles from the explosion. He wasright, to better than an order of magnitude.
To give a specific example of such acalculation, suppose a discussion about recycling leads to thequestion of how much aluminum is used in the United States each yearin the manufacture of soft drink cans. It would be difficult to getaccurate data, but we can still get a useful approximation to withinan order of magnitude. Consider that the population of the UnitedStates is about 250 million people. I would estimate that each dayabout 20% of these people have a soft drink in a can, so that about50 million cans are thrown away each day. In one year this amounts toabout 20 billion cans. (Here I estimated the number of days in a yearto be 400 &endash; remember, this calculation need be correct only toan order of magnitude.) Estimating the weight of a typical can to beabout 1 ounce, this is about 20 billion ounces of aluminum. With 1 lbequal to 16 oz (or about 20 oz for this rough calculation) thisamounts to about 1 billion pounds of aluminum thrown away each year.We shouldn’t give this calculation more respect than itdeserves; it is correct only to an order of magnitude. All we aresaying is that the amount of aluminum thrown away is closer to109 pounds than to 108 pounds or1010 pounds. We can even put a monetary value on this. Saya can of soda costs about 50 cents. I would estimate the cost of thecan itself at no more than 10% of this, or 5 cents. With the canweighing 1 oz, that amounts to about 1 dollar per pound for aluminum.Write your Congressperson and tell him or her that we could saveabout a billion dollars a year recycling aluminum soft drinkcans!
Many more interesting problems can befound at the Universityof Maryland Fermi Problem Site.