The Physics of Decision making

Quick

The Ledge If you still have to go to chapel at 10:00 today, can you make it to Chicago and back before Chapel?

[image: view from "the Ledge" in Chicago's Willis tower.]

The process of decision making

...frequently involves:

  • Making a 'rough' estimate of something, such as driving time to Chicago and back,
  • in order to make a comparison, e.g. time available vs driving time,
  • which falls into one of the Goldilocks categories (see 'guesstimation'): too hot (too large), too cold (too small), or just about right*.

*Only in the case of 'just about right' is further thought required!

Did you bother to use a formula like: $$ \text{time}=\frac{\text{distance}}{\text {speed}}$$ ??

So, this process of estimation from experience can be used as a check when you are using an equation, to see if the equation or your calculation actually makes any sense.

Estimation--

Use your experience of something that pertains to the problem, and *conversion factors* such as 1 thumb ~ 1", 1"=2.54 cm, 10 mm in each 1 cm.

How thick is a sheet of paper?

Using my thumbs...

  • Ream is about two "thumbs" thick, ~2"
  • 2 inches * (2.54 cm / 1 inch) ~ 5 cm and 5 cm * (10 mm/ 1 cm)= $50$ millimeters
  • Ream of paper has (guess if necessary) 500 sheets
  • Thickness of one sheet: $$\frac{50 \rm{mm}}{500 \rm{sheets}} = 0.1 \frac{\rm{mm}}{\rm{sheet}}.$$

[Writing]

The Ledge

The adventure-loving building inspector.

Units

Often, you don't need formulas to figure out an answer. You just need to know the units that are required.

How long does it take for light, traveling at $3 \times 10^8$ m/s to reach Earth from the Sun--a distance of $1.5 \times 10^{11}$ m?

The answer to "How long" must have units of time. ("Seconds" rather than minutes or hours in the numbers we have so far.):

time (s) = $\frac{1.5 \times 10^{11}\rm{ m}}{ 3 \times 10^8 \rm{ m/s}} = 0.5 \times 10^3 \rm{s} = 5 \times 10^2 \rm{s} = 500 \rm{s}$

600 s$*\frac{1 \rm{ minute}}{60\ \rm{s}}$is 10 minutes, so, 500 s sounds like about 8 minutes.

Unit conversions

What does per mean? in each

Seconds per year?

1 year * 365 days / yr * 24 hours / day * 60 minutes / hour * 60 seconds / minute $\approx 400 * 20 * 4000 sec = 32 \times 10^6 sec$.

60 sec = 1 minute

1 minute / 60 sec

Exponents / Powers of 10

$10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000$

$10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001$

Think of the 'power of ten' as the number of times you move the decimal point from its initial position in $1.0$ to the right or left.

$10^{-3}?$: Take 3 steps towards smaller numbers... $1.0\to 0.1 \to 0.01 \to 0.001=10^{-3}.$

$10,000?$ How many times do you have to move the decimal point to reach 1.0?
$10,000\to 1,000\to 100 \to 10 \to 1.0$: 4 steps so 10,000$=10^4$.

So, move the decimal point in 1.0 zero steps? That means... $10^0 = 1.0$!

Multiplying powers of ten

$10^ 3 \times 10^1 = 10^{3+1} = 10^4$

This is the same problem as... $1000 \times 10 = 10000$

$10^3 \times 10 ^{-1} = 10^{3+(-1)}=10^2$

Multiplication problems turn into addition problems!

Division:

$10^3 / 10 ^5 = 10^ {3-5} = 10^{-2} = 0.01$

This is the same problem as 1000 / 100,000 = 0.01.

Division problems turn into subtraction problems!

Scientific notation

$32,000=3,200 \times 10^1=320 \times 10^2 = 32 \times 10^3 = 3.2 \times 10^4=0.32 \times 10^5$

Multiplying two numbers

$32,000 * 68 = 3.2 \times 10^4 * 6.8 \times 10^1 = (3.2 * 6.8) \times (10^4 * 10^1) \approx 21 \times 10^{4+1} = 21 \times 10^5 = 2.1 \times 10^6 = 2,100,000$

Division

$(1.5 \times 10 ^7) / (3 \times 10^8) = (1.5/3) \times (10^7/10^8) = .5 \times 10^{7-8} = .5 \times 10^{-1} = .5\times 0.1=0.05$

Metric system

Even if you have a problem purely in English units, it is often easier to convert to metric units first, because then you can start using powers of 10 to convert to bigger / smaller units.

1 inch = 2.54 cm
1 m = 39 inch ~ 36 inches = 1 yard = 3 ft
2.2 lb = 1 kg
1 mile = 1.6 km

Welcome to the 21st century

Nowadays, the much easier way to convert units is to use Google or Wolfram Alpha to do it for you.

For example

160 lbs in kg

oil consumption in USA

Metric prefixes

These prefixes correspond to powers of ten:

milli- = $10^{-3}$; micro- = $10^{-6}$; nano- = $10^{-9}$; pico- = $10^{-12}$, also centi- = $10^{-2}$

kilo- = $10^3$; mega- = $10^6$; giga- = $10^{9}$; tera- = $10^{12}$

A kilometer is $10^3$ m = 1000 m.