15 <strong>Topic XIV. Fermi Problems</strong>
Topic XIV. Fermi Problems
- Estimating quantities based on what we know.
- Physicists train their students in doing “Fermi problems,” back-of-the-envelope estimates of quantities that arise in physics problems and in life. This is useful as an approach to performing “sanity checks” of claims in the world and of your own ideas, beliefs, and inventions. Checking numbers with quick Fermi estimates may be even more important in a world in which it is difficult to evaluate the credibility of numbers available by Googling.
- Addressing the Question: How do we find out how the world works?
- Estimating quantities
- Exemplary Quotes
- “He’s suggesting that the Federal budget deficit is due to the money we spend on job training programs. But that’s ridiculous! Even if every single person out of work -- let’s imagine that it is 10% of the working-age population (say 10 million people out of work) -- went to a job training program that cost as much as a year of college at a good university (say, $40,000), that would cost 400 billion dollars. Hmm... well that’s not quite as small as I expected, but it’s still not trillions of dollars, and furthermore I am sure we aren’t spending that much on each person for job training. Let see, can I estimate that cost per person in some more realistic way than using college costs...”
- Cautionary Quotes: Mistakes, Misconceptions, & Misunderstandings
- "I don't know how many pianos there are in Chicago so I can' t estimate how many piano tuners are currently working there."
- "Keyla asked me how many firecrackers were shot last night. I tried googling it, but couldn't find out. I guess we'll never know now, will we?"
- A. ATTITUDES
- Be confident in one’s ability to make a reasonable magnitude estimate of quantities for which one has no intuitive guess or direct knowledge.
- B. CONCEPT ACQUISITION
- Fermi Estimates: A systematic estimate of a quantity based on what you know. The typical goal is to get within an order of magnitude of the right answer. (This often proves possible even for topics about which you know very little.)
- a. Decompose the problem into multiple components that you can estimate. (Break down unfamiliar components into familiar components).
- b. Estimate components using approximations.
- c. Combine estimated components to calculate Fermi estimate.
- d. Compute upper and lower bounds (maximum and minimum quantities above/below between which you are fairly confident the correct estimate should be).
- Order of Magnitude: factor of ten.
- C. CONCEPT APPLICATION
- Identify quantities that would and would not be appropriate to estimate with a Fermi calculation.
- Provide rough estimates for real-world quantities using “back-of-the-envelope” (Fermi) approximations.
- Evaluate the credibility of quantitative statements using “back-of-the-envelope” approximations.
- Use Fermi estimates to identify first, second, third order causes for example problems, and estimate their effect sizes.
- Suggested Readings & Reading Questions
- No readings
- The purpose of today’s homework assignment is to give you some practice with the process of Fermi estimation which was outlined in the Santos reading. Make Fermi estimates on Question 1 and Question 2, below, following the steps: (i) identify what quantities to multiply together, (ii) make rough estimates of each of those quantities, (iii) do the math to get your answer, and (iv) state your answer clearly. Except for step iv, you do not need to write in complete sentences. Your answer can be formatted just like the sample answer to the sample question given below.
- Question 1: How much money do you spend on coffee (or other warm beverage, if you are not a coffee drinker) in a year?
- Question 2: If you took all of the household garbage in the US generated in a year and spread it out in the San Francisco Bay, what percentage of the surface of the bay would be covered in garbage?
- Sample question: What is the total length of fingernail growth that one could achieve in a lifetime? Answer to sample question:
- i) Quantities to multiply together: (length grown per clipping ) x ( of clippings per week) x ( of weeks per lifetime). ii) Estimates: length grown per clipping: .002 meters. of clippings per week: 1. of weeks per lifetime: 4160. iii) Do the math:(.002) x (1) x (4160) = 8.3 meters. iv) Answer: I estimate that a person’s fingernails grow about 8.3 meters (27 feet) in a lifetime. hw
- Clicker Questions
- How many pounds of food was thrown out (sent to landfills or incinerators) in the United States last year?
- A.Less than 100 million pounds
- B.Between 100 million pounds and 1 billion pounds
- C.Between 1 billion pounds and 10 billion pounds
- D.Between 10 billion pounds and 100 billion pounds
- E.More than 100 billion pounds
- Which of these three does the government spend the most on (including federal, state, and local government spending)? The second most?
- Most on Education, then Incarceration, then Social Security
- Most on Education, then Social Security, then Incarceration
- Most on Incarceration, then Education, then Social Security
- Most on Incarceration, then Social Security, then Education
- Most on Social Security, then Education, then Incarceration
- Most on Social Security, then Incarceration, then Education
- Discussion Questions
- Class Exercises
- Together with the whole class, the professor shows how to develop Fermi-problem estimates of a given quantity, e.g. the amount per year that Americans spend on gas for personal transportation.
- In small groups, students work on several Fermi problems to develop facility with the approach.
- In small groups, students use Fermi estimates to re-think the first-order, second-order, etc. parsing of US government spending on education, incarceration, and social security (following up the final activity from Topic XIIV, Orders of Understanding). The students frequently reach completely different orderings than they did in the previous class—and come within ~20% of the actual amounts spent.