## The Physics of Decision making

### Quick

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.

#### 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: $$\frac{50 \rm{mm}}{500 \rm{sheets}} = 0.1 \frac{\rm{mm}}{\rm{sheet}}.$$

### 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.