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The concept of order of magnitude estimates, a simple yet powerful tool used in science to approximate values and determine the reasonableness of calculated results. The importance of everyday guesses, the legacy of enrico fermi and his famous calculations, and the usefulness of order of magnitude calculations in various scientific contexts.
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Module 1, Lesson 6
Objective: By the end of this lesson you will be able to perform simple back-of-the-envelope calculations that can be used to approximate values without tedious calculations and to determine if calculated values are reasonable. You will also recognize the value of these calculations in doing science and incrementally building models that describe the world around us.
Ordinarily, when we do estimates, we tend to say things like "about a dozen," or "around four thousand." If we look closely, we can see that not all estimates, even verbal ones, are of the same accuracy. The vaguest sorts of estimates are essentially how many zeros are in any measurement. This is known as an order of magnitude estimate - finding the closest power of ten to the number you're looking for. These calculations are usually simple enough that you can perform them on very little paper, such as the back on an envelope. Suppose you're at a crowded hockey stadium. An order of magnitude estimate means that, instead of saying, "There are 4335 people at this hockey game", you can say, "There are a few thousand people at this hockey game." You don't have enough information to say with any precision how many people there are, or even how many thousands of people there are. But you do know enough to say with certainty that there are too many to count in the hundreds, and not enough to count in the tens of thousands.
Enrico Fermi is one of the great great physicists in history. His work spanned many fields, and included both experimental and theoretical results. Fermi was a master of back-of-the-envelope calculations and famously used them to calculate the tonnage of TNT the first atomic blast was equivalent to. As Fermi recalls in My Observations During the Explosion at Trinity on July 16, 194 , “About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during, and after the passage of the blast wave. Since, at the time, there was no wind I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2 1/2 meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T.”
Emilio Segrè, who witnessed the event together with Fermi, states in his biography Enrico Fermi, Physicist that Fermi had done the necessary calculations in advance, "having prepared himself a table of numbers, so that he could tell immediately the energy liberated from this crude but simple measurement." Fermi also started asking order of magnitude questions of his students, such that they because to be known as Fermi Problems. These are designed to teach the skill of making justified guesses about incomplete sets of data. The classic Fermi problem is “How many piano tuners are in Vancouver?” How do we start to answer this? Google perhaps? Let’s try a more systematic approach. By making a series of assumptions
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