Guesstimation: Solving the world’s problems on the back of a cocktail napkin … ·...

Guesstimation: Back of the envelope solutions to the world’s problems

Princeton University Press, 2008 Princeton University Press, 2012

Lawrence Weinstein Old Dominion University

Weinstein - NSTA - 2017

Why estimate (for adults)? We are continually bombarded by numbers, good, bad and bogus.

• Our deficit is $1 trillion • Every American produces 100 cubic-feet of garbage a year

Do they make sense? How should we respond? Answers/estimates lead to actions. Estimates fall in one of three ‘Goldilocks’ categories: 1) Too big 2) Too small 3) Just right (in the middle) ‘Too big’ or ‘Too small’ lead to obvious actions. ‘In the middle’ requires more thought (and perhaps even actual calculation).


Why estimate (for students)?

Understand the problem without worrying about three significant figure calculations.

Focus on the science part of the problem, not

the algebra and arithmetic. Learn how to calculate without calculators. Learn to solve “impossible” problems. Develop common sense and a feel for

numbers.


How to estimate (almost) anything

1) Write down the answer (to within a factor of 10). 1) Establish lower and upper bounds, 2) Then take the geometric mean for your answer.

How many clowns can fit in a Cooper Mini?

This ensures that your estimate will be within a factor of 10.

2) If you can’t write down the answer, break the question

into smaller pieces and go to step 1

Dare to be imprecise!

1 < nclowns <100

® nclowns » 1´100 = 10

Making Arithmetic Easy Million = 1,000,000 = 106 Billion = 1,000,000,000 = 109

Trillion = 1,000,000,000,000 = 1012

Counting zeroes gets tiresome, use scientific notation

(3´104)´ (2 ´105) = (2 ´ 3)´104+5 = 6 ´109


Arithmetic is now easy! •Worry mostly about the exponents •Use only ONE significant figure 3×3 = 10 9 / 4 = 2 √10 = π

Calculate without a calculator!

Millions, billions, trillions, gazillions What do these big numbers really mean?

Consider seconds: One day = 24 hours × 60 min/hr × 60 s/min ~ 105 s Thus 1 million sec = 106 s × (1 day / 105 s) = 10 days 1 billion sec = 109 s × (1 day / 105 s) = 104 days ~ 30 years 1 trillion sec ~ 3×104 years


• The MegaMillions maximum jackpot was about $200 Million • We spend about $600 Billion on education yearly • Our gross domestic product is about $18 Trillion

Millions, billions, trillions, gazillions What do these big numbers really mean?

How high would a stack of a million $1 bills be?

How high is one bill?

How high are 500 bills?

Million dollar height = 106 bills × (5 cm / 500 bills)

= 104 cm = 100 meters

Billion dollar height = 109 bills × (5 cm / 500 bills)

= 107 cm = 100 kilometers

Trillion dollar height = 1012 bills × (5 cm / 500 bills)

= 1010 cm = 105 kilometers

= 1/4 of the way to the Moon!



www.xkcd.com

Dear news organizations: Stop giving large

numbers without context or proper comparison.

The difference between a million and a billion is the

difference between a sip of wine and 30 seconds

with your daughter and a bottle of gin and all night

with her.

Golf balls around the world Think of the Pacific Ocean as just a large water hazard

How many golf balls would it take to circle the Earth at the

equator?


Golf balls around the world How many golf balls would it take to circle the Earth at the equator?

Golf ball diameter is about 1.5” or about 4 cm Circumference of Earth 1. Remember that it is 3000 miles to California and

three time zones × 24 time zones is 24,000 miles 2. Remember that the radius of the Earth is 4000 miles

and the circumference is 2πR 3. Remember that it is 25,000 miles 1 mile ≈ 1.5 km so 25,000 miles ≈ 40,000 km 40,000 km = 4×104 km = 4×107 m = 4×109 cm

109 (1 billion) golf balls


Pickles to the Moon!

If all the pickles consumed in the US in one year were placed end-to-end, how far would they stretch?


Pickles to the Moon!

If all the pickles consumed in the US in one year were placed end-to-end, how far would they stretch?

Pickles per person: •More than one per year •Less than one per day (400/yr) Take the geometric mean of 20/year 20 pickles/year 3×108 Americans 6” (15 cm) per pickle 20 × (3×108) × 0.15 m = 109 m = 106 km That will take us to the Moon and back!


How much spaghetti do Americans consume yearly? How fast would we have to slurp a single strand?


Meals per year: 1/yr < M < 1/day (400/yr) M = 20/yr Strands per meal: 10 < S < 103 S = 102 Length per strand: l = 1 ft = 30 cm Length per year per person: #Americans = 3×108 Total length: Slurping speed:

L = MSl = (20 / yr)(102 )(0.3 m)

= 6 ´102 m/yr

Ltot = (3´108 )(600 m/yr)

= 2 ´1011m = 2 ´108 km/yr

v = (2 ´108 m/yr)1 yr

p ´107 s

= 6 km/s


How many people in the world are listening to a speaker now?

picking their nose now?


How many people in the world are listening to a speaker

picking their nose now?

The fraction of time you spend doing something typical equals

The fraction of people doing that something right now

Time spent nose picking: (7×109) ×(10-4) = 7×105 = almost a million people!

1< t <100 s/day

® t » 1´100 =10 s/day = 10 s

105s=10-4

Energy Measured in Joules, calories, BTU, etc

•Energy of height (potential energy)

•Energy of motion (kinetic energy)

•Chemical energy

•Food energy

– Food calorie = 103 physics calories = 4×103 J

•Gravitational potential energy

U = GMm/R


PE =mgh

KE =1

2mv2

How many calories does it take to climb a mountain?

• The potential energy is PE = mgh

• Let’s climb a 3000 foot mountain = 103 m

• People weigh 100 – 250 pounds ~ 102 kg

• PE = (102 kg)(10 m/s2)(103 m) = 106 J

• But that’s only 250 food calories

– Or one 24-ounce bottle of soda!



What is the kinetic energy of a 1 km asteroid hitting the Earth (the dinosaur-killer)?

We need to estimate the mass and the velocity.

M = rV = (104 kg/m3)´ (103m)3 =1013kg

Assume an iron asteroid (divide by 3 for a rocky one):

The asteroid’s velocity comes from the Sun’s gravity. So does the Earth’s velocity:

Vasteroid »VEarth =2pREarth-Sun

1 year=

2p ´1.5 ´1011m

p ´107 s= 3´104 m/s

KE =1

2Mv2 = 0.5 ´ (1013kg)(3´104 m/s)2 = 5 ´1021J

This is 106 megatons of TNT!

What is the kinetic energy of a drifting continent?


We need to estimate mass (volume and density) and speed. Volume:

Depth: Width: 5000 km, Height: 5000 km

Density: more than water (103 kg/m3) and less than iron (104) Speed: the Atlantic Ocean opened 3000 km in 108 years

V = (5 ´106 m)(5 ´106 m)(3´104 m) =1018 m3

m = rV = (3´103kg/m3)(1018 m3) = 3´1021kg

v = (3´106m) / (108 yr ´p ´107s/yr) =10-9 m/s

KE = (1/ 2)mv2 = (1/ 2)(3´1021kg)(10-9 m/s)2 =103J

1 km < d <103km® d » 30 km


Why do we drive gasoline-powered cars?

Gasoline energy density:

At 14 g/mole, 1 kg contains 102 moles Energy per kg: e = nNAereac

= (102 mole/kg)´ (6 ´1023 # /mole)´ (3eV)´ (1.6 ´10-19 J/eV)

= 3´107 J/kg

Reality: e = 4.5 ´107 J/kg = 3´107 J/L

C (coal) < gasoline < CH4 (methane) CH2

1.5 eV per chemical reaction: CH2 + O CO2 + H2O


Why do we drive gasoline-powered cars (2) ?

Battery energy density:

100 times less than gasoline

D-Battery powered flashlight (hand torch) 1 Watt < power < 10 W 1 hour < time < 1 day Stored energy: 5-10 D-batteries per kg

E = Pt = (3W) ´ (5hr)´ (60s/min)´ (60min/hr)

= (10W-hr)´ (4 ´103s/hr)

= 4 ´104 J

e = (4 ´104 J/batt)´ (7 batt/kg) = 3´105 J/kg

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Gasoline energy transfer rate:

Why do we drive gasoline-powered cars (3) ?

At the gas pump: Fuel transferred: ~12 gal = 50 L Time: 10 s < t < 103 s

P = E / t = em / t

=(3 ´107 J/L) ´ (50 L)

102 s= 107 W

That is 10 MegaWatts!

Home power cords only carry 103 W We’ll need a very thick power cord!

Can we grow our gasoline?


How much cropland would we need to grow corn for ethanol?

How much energy do we each consume? How much energy do our cars consume?

We each consume 2500 Calories per day.

2.5 ´103Cal = 2.5 ´106 cal =107 J

Our car consumes 1 to 2 gallons (4 to 8 L) of gas per day

(6 L)´ (3´107 J/L) = 2 ´108 J

We would need 20 times more cropland to grow all our ethanol.

Why are we burning food in our cars?

Excuse me.

I’m going to

need this to

run my car.

How many foul balls land in beer cups at ball games?

• Foul balls per game: 1 < n < 103 n = 30

• Not caught by fan: 0.1 < c < 1 c = 0.3

• Seats w/ beer: 0.01 < b < 1 b = 0.1

• Cup:seat area: a = (0.1 m)2/(1 m)2 = 10-2

30 teams, 80 games/team-season

2000 games/season

20 beer-landings/season


N = ncba = (30 /game)(0.3)(0.1)(10-2)

=10-2 /game


How much trash does the US generate yearly?

Landfill space needed for a century’s trash?

1 < trash < 100 pounds per person per week 500 lbs / person - year Double it to include businesses 1000 lbs / person - year (1/2 ton / person-year) × (3×108 people) = 2×108 tons/year

That is Virginia Beach or Los Angeles or 10-4 of the total area of the US

Landfill space is not a problem.

V =(2 ´108 ton/yr)´ (102 yr)

0.2 ton/m3= 1011m3

10 m < h <103m ® A =V / h =109 m2 =103km2


Should you recycle that plastic drink bottle?

What is its value? Energy density: Plastic ~ gasoline Mass: 1 g (paperclip) < m < 100 g (4 oz) m = 10 g m ≈ 1% of a liter of gasoline

That is worth < $0.01

… after collecting, cleaning, sorting and processing. Landfill space is not a problem.

It’s up to you!

Lightening the Load


How much fuel will the airline save if you urinate immediately before flying across country?

Airplane glide ratio r: the number of furlongs it glides for every furlong it descends. 1 < r < 102 r 10

Distance across country: d = 5 ´103km

h = d / r = 5 ´102 km = 5 ´105 mTotal distance descended:

Volume of pee:

Energy needed: E = mgh = (0.3 kg)(10 m/s2 )(5 ´105 m) = 2 ´106 J

Engine efficiency:

Fuel needed: Vfuel =E

ee=

2 ´106 J

0.3(3´107 J/L)= 0.2 L

That’s about 20¢ of fuel.

0.1<V <1 L®V » 0.3 L®M pee » 0.3 kg

0.1< e <1®e » 0.3

How much energy does it take to destroy Alderaan?


How much energy does it take to destroy Alderaan?


Energy to

disassemble a

planet (ours):

Energy to make it glow yellow hot: T ~ 2000 K

heat capacity ~ 0.1 water

= 0.1 cal/gK = 400 J/kgK

Unsurprising:

thermal velocities

<< escape velocities

Where can that energy come from? We need 1032 J

•The US uses 1 TW electricity 1020 s = 3x1012 yr

•At a solar power density S = 103 W/m2, the Earth receives

– P = AS = 𝜋R2S = 3(6x106 m)2(103W/m2) = 1017 W

– t = 1032 J / 1017 W = 1015 s = 3x107 yr

•The sun’s total heat output is

– P = AS = 4𝜋R2S = 4𝜋(1.5x1011 m)2(103W/m2) = 3x1026 W

– t = 1032 J / 3x1026 W = 3x105 s = 4 days

And if the Death star power efficiency is only 99% …


1% of the energy that destroyed Alderaan would vaporize the much smaller Death Star


If the Sun were made of gerbils, the Earth would be incinerated. Compare the power density of mammals and the sun.

Humans:

Sun:

P =(2.5 ´103kcal/day)(4 ´103 J/kcal)

105s/day= 100W

M = 100 kg

P

M=

100 W

100 kg= 1 W/kg

P = (103 W/m2 )4p (1.5 ´1011m)2 = 3´1026 W

M = 2 ´1030 kg (from the orbital period of the Earth)

P

M=

3´1026 W

2 ´1030 kg= 10-4 W/kg

Humans have 104 times the power density of the Sun!

(and gerbils are even worse!)

How much fuel do we need to travel to the stars?


Distance to Alpha Centauri: Time to get there: Velocity needed: Mass needed: a lot more than the Santa Maria (102 tons) and less than an aircraft carrier (105 tons) Energy needed: Fuel needed:

d = 4 light-years

t < (40 yr)

v = d / t = (4 l-y) / (40 y) = 0.1c » 3´107 m/s

m =104 tons =107 kg

KE = 0.5mv2 = 0.5(107 kg)(3´107 m/s)2

= 5 ´1021J

m fuel =5 ´1021J

4 ´109 J/ton= 1012 tons

We need 108 times more (chemical) fuel than payload!

Flying to the center of the galaxy


You are here

Energy Released by a 30M

Black Hole Merger


Energy Released by a 30M

Black Hole Merger

• Inspiral from ∞ to d = 2RS (Schwarzchild radius):

ΔE = GM2/2RS

• RS = 2GM/c2

• ΔE = Mc2/4 = 8M

c2

• ΔE = (8×2×1030 kg)(3×108 m/s)2

= 2×1048 J

• Δt < 1 s PBH > 2×1048 W

• Puniverse = (1026 W/)(1011 /galaxy)(1011 galaxy) = 1048 W

• PBH > Puniverse!



Summary To estimate: •Dare to be imprecise!

• Estimate to within a factor of 10 • Take the geometric mean of lower and upper bounds

•Break up the question into smaller pieces if necessary. •Remember a few key numbers Using estimation and without relying on experts we found: •Landfill space is not a problem •Batteries suck •Burning ethanol in cars is a really really bad idea •You won’t save the world by recycling plastic water bottles •Don’t get between two black holes!

Guesstimation: Solving the world’s problems on the back of a cocktail napkin … ·...

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