Unit conversions and context
- Necessary to solve numerical problems. But if that's *all* you want to do, Google is your friend.
- Perhaps more important: To support decisions we need to compare quantities using the same units.
2.5 Cents Per Kilowatt Hour: America's Cheapest, Cleanest Fuel 'Holds Steady' - Good communication practice: add context (another kind of comparison) to numbers.
For example: The average German household pays 30.5 eurocents/ kWh (34 cents / kWh) for electricity. In Indiana, the average rate is 11.8 cents / kWh.
About the 2017 total solar eclipse...
Wind turbines kill birds!
Anecdote/example - the power of the particular
...a flock of 15 American White Pelicans ...caught [the biologist's] eye, flying toward the nearby Peñascal wind farm. As he watched, a pelican at the flock's tail end was swiped by a massive turbine blade and "literally 'erased'" from the air.
Numbers - "objective" information
Wind turbines kill an estimated 140,000 to 328,000 birds each year in North America, making it the most threatening form of green energy.
...but there's no story about a particular bird.
Adding context to a number - the bigger picture
Is 300,000 a lot of birds? What other risks do birds face, besides wind turbines?
America's cats, including housecats that adventure outdoors and feral cats, kill between 1.3 billion and 4.0 billion birds in a year.
According to Peter Marra, Conservation biologist at the Smithsonian, 2013
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100 million to a billion birds die each year when they run into glass-covered or highly illuminated buildings.Patrick Standish
Recognize the complexity
So, does that mean we can ignore the effect of wind turbines? Or should we advocate against all wind turbine construction?
What can you bring into this discussion to make it more complex? What other considerations?
- Anecdote - story - subjective information
- Numbers - what can we measure? Averages and trends - objective information
- Context - the bigger picture.
- Complexity - Trade-offs, solutions are rarely simple, unintended consequences are common.
Unit conversions
Many times you can use Google, but you should also know the math of how to do this. Here are some common conversions of different units:
1 inch = 2.54 cm
100 cm = 1 meter
1 m = 39 inch ~ 36 inches = 1 yard = 3 ft
2.2 lb = 1 kg
1 mile = 1.6 km
1000 m = 1 kilometerI'd like to convert my weight in pounds (170 lbs) to kilograms. There are two ways to think about this:
Fractions / ratios
Some people say "$x$ kilograms is to 170 lbs as 1 kg is to 2.2 lbs". You write down two fractions, one for each "is to" phrase, like this: $$\frac{ x \text{ kg}}{170\text{ lbs}}=\frac{1\text{ kg}}{2.2\text{ lbs}}.$$ Now, we use algebra to solve for $x$: $$x=\frac{170\times 1}{2.2} =77.3\text{ kg}.$$
Conversion factor reasoning
Follow these steps:
- Write down the number with the units that you know, and the units that you'd like to have on the rate. We'll stick an empty fraction in there that is going be our conversion factor:
$$170\text{ lbs}\times \frac{\text{ ? }}{\text{ ? }}= \text{____ kg}.$$
Thinking "algebraically" just about the units: I need a fraction that has 'lbs' on the bottom (that will cancel with the lbs on top) and 'kg' on the top, so as to equal the same units as we already have on the right:
$$170\text{ lbs}\times \frac{\text{ ? kg }}{\text{ ? lbs }}= \text{____ kg}.$$
Now, we look to the conversion that might look like "1 kg = 2.2 lbs", and fill in the numbers that go with each unit, like this:
$$170\text{ lbs}\times \frac{\text{ 1 kg }}{\text{ 2.2 lbs }}= \text{____ kg}.$$
You can think of the fraction as saying "There's 1 kg per 2.2 lbs" or "We're going to multiply by 1 kg for every 2.2 lbs". The next step is to calculate out the numbers, and get the answer:
$$170\times\frac{1}{2.2}=\frac{170}{2.2}=77.3\text{ kg}.$$
- Google to find the number of kilocalories per megajoule (MJ).
-
Now, use one of the techniques above to convert 127 MJ to kilocalories.
Write this below as ___ kilocalories / 1 US gal.
127 MJ = 30,400 kcal so
30,400 kcal / 1 US gallon of gas. - How many gallons of gasoline would release 2000 kilocalories when burned?
We want gallons for 2000 kcal. We could do this: $$\frac{30,400\text{ kcal}}{1\text{ gallon}} = \frac{2000\text{ kcal}}{x\text{ gallons}}. $$ Solving for "$x$": $$x=\frac{2000*1}{30,400}=0.066\text{ gallons}$$
- What is the approximate cost of this much gasoline? (You'll have to estimate the cost of gasoline.... ___\$ / 1 gallon of gas
$$0.066\text{ gal} * \frac{\$2.50}{1\text{ gallon}}=\$0.17.$$
- Approximately how far could a typical car (~30 mpg means 30 miles / 1 gallon) travel on this much gasoline?
$$0.066\text{ gal}*\frac{30\text{miles}}{1\text{ gal}}=2.0\text{ miles}.$$
Your turn:Find out how long a "10 k" (10 kilometer) race is in miles.
How many meters are there in 3.0 kilometers?
About how many hours are there in one year? (This will require using two conversion factors. Or perhaps making 2 conversions.)
The metric system
You should know / memorize these metric prefixes
These prefixes correspond to powers of ten:
milli- = $10^{-3}=\frac{1}{1,000}$
micro- = $10^{-6}=\frac{1}{1,000,000}$
nano- = $10^{-9}=\frac{1}{1,000,000,000}$
pico- = $10^{-12}$
centi- = $10^{-2}=\frac{1}{100}$kilo- = $10^3=1,000$
mega- = $10^6=1,000,000$ (1 million)
giga- = $10^{9}=1,000,000,000$ (1 billion)
tera- = $10^{12}$How much gasoline...?
Talking about Chemical energy, it is well known about how much chemical energy from food that people need to survive. An average woman needs about 2000 kilocalories each day (=2000 "C"alories) to keep her weight constant.
I wonder how this compares to the energy in gasoline?
The main constituent of gasoline is 'octane'. ('oct'=8) $$2\,C_8H_{18}+ 25\,O_2 \rightarrow 16\,CO_2+ 18\,H_2O+\Delta E.$$ When gasoline is burned, it releases approximately $\Delta E =$ 127 MJ (megajoules) / 1 US gallon of gasoline.