# Back of the envelope estimates

The TED talk by Sanjoy Mahajan summarizes the philosophy… striving for too much rigor can stop progress in science and technology. “Perfect” is the enemy of “good enough”. These estimates have been given several different names, including Fermi Estimates, Back of the Envelope Estimates and Street Fighting Estimates. Each name has good reason: First, the stamp of approval from a famous physicist who was capable of all levels of rigor. Second, a reminder that all the information you need can be summarized on a small piece of paper. Third, there are no rules about what you can/cannot do when making an estimate.

- Sanjoy Mahajan's book, “The Art of Insight in Science and Engineering”. PDF copy is available for free online.
- Sanjoy Mahajan has another book called “Street-Fighting Mathematics”. Ethan has a copy.
- Back-of-the-envelope physics by Clifford Swartz. Ethan has a copy.
- In Search of Simplicity articles by Weisskopf, MIT.

To be good at making quantitative estimates, you need to know some basic quantities:

- Solar energy flux: ~ 1 kW/m2
- Energy released by combustion ~ 0.1 eV per bond
- Time for light to travel from the sun to the earth: ~ 8 minutes
- Time for light to travel from the moon to the earth: ~ 1 second.
- Density of water, 1 kg/liter
- etc.

It is unfortunate that the United State has not adopted SI units. Physical laws are usually expressed in SI units. To apply these laws to everyday quantities, you need familiarity with mass in kilograms, length in meters, volume in liters (10^-3 cubic meters) etc.

#### No physics required

- How quickly do children grow taller in mm/day? (✔ Abby)
- How many atoms are in the human body? (✔ Heather)
- How small can a 1TB memory be? (✔ Tristan)
- How far does a car travel before one layer of rubber (the thickness of the polymer strand) wears off the car tire?
*Hint:*The carbon-carbon bond length is 1.4 Angstrom.

#### Force and pressure

- Bed of nails: What is a comfortable spacing between the nails?
- Bouyancy in air: What would the bathroom scale read if I wasn't surrounded by air? (The density of air is 1.3 kg/m^3)
- Washbowls and Coriolis force: How strong is the Coriolis force on a draining basin of water?
- Estimate the weight of a car from the tire pressure and surface area in contact with the ground.
- The earth is being pushed away from the sun by radiation pressure, and pulled toward the sun by gravity. How big is the radiation force compared to the gravitational force?
- What is the force or acceleration you receive in a car crash?
- Estimate the velocity of the air being pushed downwards by a hovering helicopter. (✔ Ethan)

#### Earth

- Allegedly, Eratosthenes measured the radius of the earth in Egypt about 200 years BCE. He did this by remarking that the sun shone directly down a deep well at a one location 500 miles from another in Alexandria, where the sun cast a shadow about 1/12 the length of a long rod. He knew that the distance between the two locations was about 500 miles. Estimate the radius of the earth from this information.
- The radius of the earth can be estimated from observations that Tal made at Lake Tahoe. Sail boats with 6 m tall masts disappear when they sail to the far side of the lake. The far side of the lake is about 10 km away.

#### Climate/Atmosphere

- What is the mass of the atmosphere?
- How long would it take humans to use up 10% of the O2 in the atmosphere?
- How much CO2 does a square kilometer of forest remove from the atmosphere each year (in kg)?
- How much CO2 does an automobile add to the atmosphere each year (in kg)?
- If the CO2 in the atmosphere increases by 1 ppmv due to the burning of coal or oil, how many kg of coal or oil were burned?
- If the CO2 in the atmosphere increases by 1 ppmv due to the burning of coal or oil, how much will the O2/N2 ratio change?

#### Mechanics

1. | The little prince wants to jump off his moon and travel somewhere else. How big can the moon be? (at some critical moon size, his legs won't be strong enough to jump off the moon). (✔ Landon) |

- How much would the ocean surface rise if the ice caps melted?
- Sherlock Holmes is riding a 747 jumbo jet. He is sitting in his chair about to take of. He holds his pocket watch like a pendulum in front of him. He watches it swing 30 degrees towards his nose during the peak acceleration of the aircraft. What is the peak acceleration?
- Consider a 50 kDa protein. Estimate the diffusion coefficient.
- Estimate the strength of the recoil against your shoulder when you fire a rifle.

#### Quantum Mechanics

- What visible wavelength is absorbed/emitted by the chromophore shown above? Note, electrons in the p_z orbitals of the 8 carbon atoms are delocalized. These delocalized states are the highest energy occupied states in the system.
- Estimate the ground state energy of hydrogen and the size of the hydrogen atom

#### The Atmosphere

#### Electricity and Magnetism

- What is the lifetime of a classical atom (before the electron loses 13 eV of energy to synchrotron radiation)?
- How much electrostatic charge can you put on a balloon?

#### Energy

- How much heat does a human output when sitting in one place?
- What is the energy transfer rate when you refuel your car compared to the energy transfer rate when you recharge your iphone or laptop?(✔ Lee)
- How much power can a modern wind turbine generate? (hint: consider the kinetic energy of the air passing through the area swept by the blades) (✔ Dan)
- How much energy is stored in a AA-battery?

#### Dimensional Analysis & Easy Cases (Chpt 1 & 2 of Mahajan)

- Use dimensional analysis and physical arguments (easy cases) to figure out how Debye screening length scales with thermal energy (kT), the charge of an ion (e), the dielectric constant of the medium, and the concentration of ions.
- Use dimensional analysis to figure out Kepler's third law.
- Use dimensional analysis to show that energy emitted per unit area from a blackbody scales as T^4. Note that blackbody radiation involve electromagnetism and thermal physics. Therefore, we expect it to depend on Plank's constant, kT, and the speed of light.

#### House construction

1. Consider a typical house with fairly good thermal insulation. The average R value for the ceiling, walls and floor is “15” (you'll have to look up the definition and units of R value). It is 0 deg C outside. The central heating for this house is controlled by a thermostat. The heater turns on if the temperature drops below 18 deg C and turns off when the temperature exceeds 19 deg C. How often does the heater turn on?

2. The city supplies each house with water at 50 psi. Assume every very fixture (sink, shower, washing machine etc.) draws water at 2 gal/min through a single supply pipe (0.75 inch inner diameter, 75 feet long). Every fixture requires a minimum pressure to operate (about 8 psi). How many fixtures can be run at the same time?

#### Some of Ethan's estimates

Rough calculations that help develop physical intuition