Handout
Prof. Rassoul
Fermi Questions -- “Ball-park figure”, “estimation”,
“guesstimation”, “back-of the-envelope (napkin)
calculation” are all “order of magnitude” calculations
One of the fascinating questions about extraterrestrial (ET) civilizations
has been “Where are they?” From the history of the Earth, one can infer that
it takes about 500 million years for a civilization to become a mature
technical civilization so it expands across the galaxy. So why isn’t our 10billion-year-old galaxy filled by such civilizations?! Shouldn’t we have
many ET visitors? Enrico Fermi, the great Italian-American nuclear
physicist (1901-1954) first posed the above simple question (see
question#11). This simple question remains unanswered, except in the
entertainment worlds of “X-files” and “Men-in-Black”.
Fermi’s question is one of the best examples of how even a crude
estimate can expose the essence of a problem. An estimate, even a very
crude one, of a certain quantity often gives us useful information that might
be good enough to isolate the important parts of a problem. Sometimes we
know how to calculate a certain quantity but have to assume or guess the
data we need for calculation. Or the calculation might be too complicated to
carry out exactly, so we make some rough approximation. In either case our
result is a guess, but such a guess can be useful even if it is uncertain by a
factor of ten (an order of magnitude) or more. The results can then be used
to determine whether or not a more precise calculation is necessary. Taylor
and Wheeler (1966) gave the spirit of “order-of-magnitude” calculations as
“make an estimate before every calculation, try a simple physical argument
… before every derivation, guess the answer to every puzzle. Courage: no
one else needs to know what the guess is”.
Here is a collection of several “order-of-magnitude” or “estimating”
exercises from my collection and various physics textbooks. Some are silly,
and some are serious, but almost all require guesswork for needed input
data. State each step in your reasoning and what data you think is necessary
to look up in tables. But don’t try to look up a lot of data; make the best
guesses you can, and remember that you can always modify your initial
assumptions if more precision is needed. The most important lesson in
“back-of-the-envelope” calculation is to turn your imagination loose, and
then to see what is needed to improve the estimate. You can estimate the size
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of the most bizarre quantities, if you wish. Have fun! Here are some examples
(see The Physics Teacher magazine for more; samples are attached with this handout):
 How many liters of water are drunk in the US per year?
Answer --- 10^11 liters (250 x 10^6 people times 1.5 liters of water per day times 365
days).
 How long would it take to walk the distance between Earth and Moon?
Answer --- 10^4 days (2.4 x 10^5 miles divided by 24 miles/day).
 What is the number of postage stamps needed to cover a football field?
Answer --- 10^7 (stamp area of 4 cm^2 over a field of 100 x 50 m^2).
Here are some questions for you
to try (not a homework)!
1. How many hairs do you have on your head? The result might be more
important ($wise) to someone who decide to have a hair transplant operation. [1]
2. How many times does a human heart beat during a lifetime? How many
gallons of blood does it pump? (Estimate that the heart pumps 50 cm3 of
blood with each beat.) The result is essential if you are inventing reliable valves
for a mechanical heart pump. [2]
3. Estimate the volume of gasoline burned annually by private automobiles
in the United States. Use this number to estimate a yearly consumer
expenditure in dollars (You need the United State population, the average number
of people per family, average number of cars per family, average distance traveled by
each car per year, average miles per gallon for a car, average $ per gallon of gasoline.
If you wish, you can modify your result by including the commercial consumption.)
4. Estimate the total length of the interstate highway system. (Estimate the
length and width of the United States and the number of major E-W and S-N
interstates)
5. Estimate the number of university professors (or students, or doctors, or
disk jockeys, and so on) in the United States. (You may need to estimate the
number of college students, faculty/student ratio, and a yellow page book, number of
radio stations per city)
6. The 2nd most famous Fermi’s problem: Estimate the number of piano tuners living
in New York City.
7. How many drops of water are in all the oceans of the earth? How many
words are in your favorite book (physics, chemistry, biology, or Bible,
etc.) How many kernels of corns does it take to fill a 2-L soft drink
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bottle? How many typewriter papers are in a typical shipping box of the
dimension 11”x17”x9”? [2]
8. How many pizzas do students in Florida Tech consume each academic
year? (Or by all in your city, or in the United States?) Use this number to
estimate yearly consumer expenditure in dollars.
9. A billionaire offers to give you $1 billion if you can count it out using
only one-dollar bills. Should you accept her offer? (Assume you can count
one bill every second, and be sure to allow for the fact that you need about 8 hours a
day for sleeping and eating, and that right now you are probably at least 18 years old.)
[3]
10.You are writing an adventure novel in which the hero escapes the border
with a billion dollars worth of gold in his suitcase. Is this possible?
Would that amount of gold fit in a suitcase? Would it be too heavy to
carry? (You may need the following information: gold sells for around $400 an
ounce, where 1 ounce of gold is 30 grams. The density of water is about 1 g/cm3 and
gold is about 20 times denser than water) [2]
11. The most famous Fermi’s problem. Assume that the maximum speed of travel
achievable by an interstellar colonization mission is 0.001 times the speed of
light, and that the average distance between stars is 10 light-years (ly). Model an
expanding galactic civilization as occupying a spherical volume whose size
increase because freshly colonized worlds develop mature local economies and
then launch new colonization missions. From the history of the earth, estimate
the time required for a civilization to become mature. Does spacecraft speed or
development time place a stronger limit on the expansion speed of the
civilization? Estimate the time required for the sphere to achieve a radius of 3 x
104 ly – the size of our galaxy. Where are they?! (A light-year is the distance
traveled by light in 1 year.) [3]
References:
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[1] University Physics, Sears/Zemansky/Young, 7 ed., Addison Wesley, 1987.
[2] Physics for Science and Engineers, Serway, 4 ed., Saunders College
Publishing, 1996.
[3] Physics: The Nature of Things, Lea and Burke, Brooks/Cole Publishing
Company, 1997.
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PHY1050 – Homework
Name:_____________________
Ten points will be awarded for each correct answer (correct order of magnitude). There will be 1 point off for
each order of magnitude difference from the accepted order of magnitude. No answer will score less than zero.
Show your work, but write your order of magnitude (10? ) answers in boxes.
1. How much land area (in square meters) is found on earth?
2. How many revolutions will a 14-inch tire (diameter ~26 inches) have to make during a crossing of the
continental United States?
3. How many square meters of turf (real or artificial) are there in a National Football League Stadium?
4. A water balloon is holding 2,000 cm3 of water. More water is added until the diameter doubles. How
many cm3 of water does it now hold?
5. An automobile travels 100,000 km before the tire tread wears out. What thickness of rubber wears off
the tire each revolution of the wheel? Express your answer in centimeters.
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