Fun with Units
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Random derivations (some false) based on unit multipliers.

# Calories and Kilograms

Einstein taught us E = mc2. In that equation E stands for energy, and Calories is a measure of energy; m stands for mass, and kilograms is a measure of mass; and c is just some constant. To consume 1 kilogram requires about 2.15 × 1013 21.5 trillion, or 21,500,000,000,000 Calories. That’s a number too big to comprehend, even if we do hear numbers nearly that large in economic news all the time, so let’s rework it a bit.

• If you use 2000 Calories per day it will take about 29,400 days to lose a single gram.

• A small (3mg) grain of dry rice, consumed at a rate or 2000 Calories a day, would last for almost 90 years.

• It would take seven billion people each working hard (3000 Calories hard) all day long to jointly burn a kilogram in a single day.

Moral of this story: it is more appropriate to talk about losing weight then to talk about burning it. Also, if you had an antimatter stomach a single meal would be enough to keep you and several generations of your descendants full of energy well into old age.

# Fuel and Diamonds

Say I put ten gallons of gasoline into my car, drive 300 miles, then find all ten gallons are gone. Where did the gasoline go? Out my tailpipe, after a bit of chemical reaction. In fact,
 10 gallons 300 miles
= 0.0784 mm2, or about ten times the cross sectional area of straight human hair. Conceptually, there’s a trail of gasoline about a third of a millimeter wide being trailed from my car’s tailpipe everywhere it goes.

When my gar is driving at 60mph on the freeway it uses about 35 horsepower (or 26 kilowatts) to do that.
 35 horsepower 60 miles per hour
= 973 newtons of force. 973 newtons of force applied by 0.0784 mm2 means the gasoline is pressing at 12.4 gigapascals, just a little bit more pressure than is needed to turn carbon into diamonds. Gasoline is about 90% carbon.

Moral of this story: if we didn’t let the gasoline expand into the air before it pushed against the pistons in the engine we could be dribbling a trail of hair-width diamonds from our tailpipes instead of fumes. Also, chemistry is more complicated than unit multipliers suggest.

# Confusing Pounds for a very old Earth

One pound = 4.448 Newtons = 4.448 kg m / s. One pound = 0.4536 kg. Thus 0.4536 kg = 4.448 kg m / s. Canceling terms by dividing both sides by 0.4536 s / kg, This derivation contains multiple errors; see the comments below. we find 1 second = 2.0176 kilograms. This looks like a useful constant; let’s use it to find the age of the earth. The earth is 5.972198×1024 kilograms, so it is 2.96×1024 seconds or 93.86 quadrillion years, about seven million times older than current estimates of the age of the universe.

Moral of this story: If the earth is older than the universe but the universe was created first then the earth must just age really really fast. As dog-years are shorter than person years, so earth-years are shorter than universe-years. Or is it universe years that are shorter? Also, there are several kinds of pound.