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Physics, astronomy, and order-of-magnitude problems
Dr. Jay Maron

Index

A "*" indicates a link to a solution.

AMNH solar system problems

Artificial gravity     Launch cost     Bamboo     Light travel time     Sunlight mass     Visual resolution     Earth equatorial centripetal acceleration     Dinosaurs     Pinatubo     Rice     Earth temperature     Age of the universe     Strength of gravity     Death Star

AMNH Space, Time, and Motion problems

Artificial gravity     de Broglie wavelength     Launch cost     Bamboo     Death Star     Sunlight mass     Visual resolution     Cosmic ray muons     Photon momentum     Relativistic velocity addition     Cold atoms     Rice     Earth temperature     Age of the universe     Strength of gravity     Death Star     Dark matter

Units

Metric estimation

Without using any kind of measurement device, try estimating things like mass and length for various objects. If you don't have the object handy, imagine it in your mind. For the following objects, what is

Height of a 2 liter bottle of Mountain Dew.
Diameter and mass of a quarter, a ping pong ball, a tennis ball, and a soccer ball.
Balls. Coins
Distance between two train tracks.
Distance between Manhattan streets, such as from 42nd Street to 43rd street.
Length and width of Central Park. (Central Park extends from 59th Street to 110th Street and from 5th Avenue to 8th Avenue).
Length of Manhattan Island.
Height of Tyrion Lannister.
Height of the letters on a stop sign and the distance between opposite edges of the sign.
Length and width of a standard sheet of paper.
Diameter of a light on a stoplight.

Light from the sun

What is the mean distance from the sun to the Earth, what is the speed of light, and how long does it take light to cover this distance?

New Horizons mission to Pluto
```Speed of the New Horizons spaceship  =  14.5 km/s               Speed when it passed Pluto
Distance to Pluto                    =  39 Astronaumical Units
Distance to Alpha Centauri           =  4.4 light years         The nearest star
```
At New Horizon's present speed, how long does it take to travel 39 astronaumical units and 4.4 light years?
Age of the universe

The Virgo cluster of galaxies is 54 million light years away and is moving away from us at 1254 km/s. At this speed, how many billion years does it take the cluster to cover a distance of 54 million light years? This is an estimate of the age of the universe.

Volume

Most beer bottles are 12 ounces in volume and 5 percent alcohol by volume. Define one "Bond" to be the volume of alcohol in one bottle of beer. How many mL of alcohol is one Bond?

Wine is typically 13% alcohol and Scotch is typically 40%. How many ounces of wine or Scotch correspond to one Bond of alcohol?

How many Bonds of alcohol are in a 750 mL bottle of wine, a 750 mL bottle of Scotch, and a 15.5 gallon keg of beer?

Solution

Cost of a penny

Using Wikipedia, find the diameter, thickness, and mass of a penny and calculate its density.

A penny is 2.5% copper and 97.5% zinc. The density and price/kg of these metals is

```          Density   Price/kg
g/cm3      \$/kg

Zinc        7.14       2.0
Copper      8.96       6.2
Gold       19.3    43000
```
What is the cost of the metal in a penny?

What is the price of a penny made of gold?

Newton's laws

Fight Science

In the "Fight Science" youtube video "Wushu Master vs. Snake" the strike of a wushu master and a snake are measured.

```                   Distance    Time
(inches)    (ms)

Wushu master        15.5       94.7
Snake                4.65      29.5
```
If constant acceleration is assumed, what is the acceleration and final speed for each strike? (Use units of meters and seconds).
Momentum

In this scene from the film "A Team", a tank fires a shell to use the recoil for propulsion.

```TankRecoilSpeed * TankMass  =  ShellSpeed * ShellMass
```
Using data from the web, what are typical values for TankMass, ShellSpeed, and ShellMass? What would you estimate is the recoil speed of the tank?

Starting from the point when the tank is dropped from the plane to the point where it reaches the lake bottom, plot the height, vertical velocity, and vertical acceleration as a function of time.

Cannon

A 3 meter long cannon with a mass of 2000 kg fires a 10 kg cannonball at speed of 600 meters/second. What is the recoil speed of the cannon? What is the average acceleration of the cannonball while it is in the barrel?
Solution

Punt

A punter kicks a ball with a vertical velocity of 30 meters/second and a horizontal velocity of 20 meters/second. For how long is the ball in the air and how far horizontally does it travel before it hits the ground? Assume zero air drag.

Solution

Chuck Norris

Chuck Norris has a mass of 110 kg and he has a jet pack that generates 1800 Newtons of upward force. He is also on the surface of the Earth and subject to Earth gravity, which we shall assume to be 10 meters/second2.

What is his vertical acceleration?
Solution

Chuck Norris jumps from an airplane, free-falls until he reaches terminal velocity, falls at terminal velocity for a few seconds, and then opens a parachute. After opening the parachute he decreases speed until he reaches a new terminal velocity as determined by the parachute. Draw a qualitative graph of the vertical position, velocity, and acceleration as a function of time.
Chuck energy

Chuck has a mass of 110 kg. How much gravitational potential energy does he gain if he climbs a 12 meter staircase in 5 seconds, and what is the power in Watts?
Solution

Baseball
```Speed of a baseball pitch thrown by Randy Johnson   45 meters/second
Initial height of the baseball                     2.2 meters
Distance from the pitcher to the batter             20 meters
```
The pitch is thrown horizontally so that if there were no gravity, it would travel horizontally and stay at a height of 2.2 meters. If gravity is included, how much does the pitch drop vertically when it arrives at home plate?
Solution

Climb a set of stairs as fast as you can. Measure the height of the stairs and the time required to climb them and calculate the power.

Acceleration

Obtain a youtube video of any of the following:
A Usain Bolt 100 meter sprint
A Formula-1 car accelerating from 0 to 75 m/s.
A drag racing car accelerating from 0 to 100 m/s.
A cheetah accelerating from 0 to 20 m/s.

Analyze the video frame-by-frame to generate a plot of position vs. time and use it to estimate the velocity and acceleration vs. time.

Plot the power vs. time, where

```Power  =  Force * Velocity  =  Mass * Acceleration * Velocity
```

Circular motion

Earth's equator

The Earth has a radius of 6371 km. What is the speed of the equator due to the Earth's spin, and what is the corresponding centripetal acceleration?

Space station
```Mass of Earth          =  5.972⋅1024 kg
Radius of Earth        =  6372 km
Gravity constant       =  6.674⋅10-11 Newton meters2/kg2
Surface gravity        =  9.807 meters/second2
```
The space station is at a height of 400 km above the surface of the Earth. What is the gravitational acceleration at this height?

What is the orbital velocity at an altitude of 0 km and 400 km?

What is the gravitational potential energy of a 1 kg object 400 km above the surface of the Earth?

What is the kinetic energy of a 1 kg mass orbiting at an altitude of 400 km?

What is the cost of electricity in Joules/\$?

What is the electricity cost to propel an object at orbital velocity at an altitude of 400 km?

Using Wikipedia, how much \$/kg does it require to launch material into space? Find data for several rockets.

Horizon

Suppose you are on a ship in the middle of the ocean. How far can you see a floating buoy as a function of your height above the ocean surface? Express the answer as

Radius of the Earth = R = 6371 km Height above the ocean surface = H Distance for which you can see a buoy = D = C H½ km

Use the approximation that H << R.

Artificial gravity

In space, artificial gravity can be generated by spinning the spaceship. NASA experiments show that the spin period has to be 30 seconds for the astronauts to not get dizzy. If a spaceship has a spin period of 30 seconds and if the artificial gravity is 10 meters/second2, what is the spin radius?

The Earth orbits the sun with a speed of 30 km/s and the gravitational constant is 6.67*1011 Newton meters2 / kg2. Calculate the mass of the sun.
Friction

Drag race

A 800 kg car has an infinitely powerful engine and the tires have a coeffient of friction of 1.4. What is the maximum acceleration?
Solution

Cornering
For a car going around a corner,
```Radius of curvature           =  8 meters
Tire coefficient of friction  =  .9
Car mass                      =  1300 kg
```
What is the maximum cornering speed? Solution

Spin
(Also Exam #2, practice)
Ant man is standing on the top of a bowling ball with a radius of 217 mm and rolling at 8 m/s. What does he experience for the centripetal acceleration?
Rolling

A soccer ball with a radius of .11 meters rolls with a speed of 20 m/s. What is the spin frequency and the angular velocity?

```$\theta 2$  =  Angular velocity
Ek =  Kinetic energy
=  .5 M V2
Es =  Spin energy
=  .5 I $\theta 2$
Et =  Total energy
=  Ek + Es
```
What is Ek/Et?
Ramp

Suppose a ball rolls without slippling down a ramp.

```H  =  Initial height of the ball
g  =  Gravity constant
V  =  Velocity of the ball when it reaches the bottom of the ramp
```
What is V?
Billiard ball

A billiard ball with a radius of 30 mm rolls with a speed of 3 meters/second. What is the ball's spin rate in revolutions per second and radians per second?

Suppose a football receiver with a vertical leap of .8 meters jumps and catches a ball at the top of the leap.

```V  =  Velocity of the ball         =   20 m/s
Mb =  Mass of the ball             =  .42 kg
M  =  Mass of the football player  =  100 kg
```
What is the recoil velocity of the receiver?

How long does it take the receiver to drop .8 meters after catching the ball?

How far is the receiver knocked horizontally by the recoil?

Suppose the receiver catches the ball far from his body so that the recoil also delivers angular momentum. If the ball is caught 1 meter from his axis of rotation, how much angular momentum does the ball deliver?

If the receiver has a moment of inertia of 2.5 kg m2, what is his spin frequency after catching the ball?

Center of mass
```                   Position   Mass
(meters)   (kg)

Chuck Norris         11       110
Tyrion Lannister     15        60
```
Where is the center of mass?
Earth-moon system
```M  =  Mass of the Earth         =  5.97e24 kg
R  =  Radius of the Earth       =     6371 km
m  =  Mass of the moon
=  .0123 M
r  =  Radius of the moon
=  .273 R
D  =  Distance from the Earth to the moon
=  60.3 R
L  =  Distance between the center of mass of the Earth and moon and the center of the Earth
```
What is L/D?

If we assume that the Earth and moon are spheres of uniform density, what is
Angular momentum of the Earth's spin?
Angular momentum of the moon's spin?
Angular momentum of the moon's orbit around the Earth?

Spin

Using data from the web, what is a typical speed and spin rate for the following balls? Measure spin rate in revolutions per second.

A baseball curveball.
A golf ball hit by a driver and a 9 iron.
A tennis groundstroke hit with extreme topspin.
A ping pong ball hit with extreme spin.
A football thrown with an extreme spiral.
A soccer ball kicked with extreme speed and spin.

```           Diameter  Mass   Density
(mm)   (g)    (g/cm3)

Ping pong      40      2.7    .081
Squash         40     24      .716
Golf           43     46     1.10
Racquetball    57     40      .413
Billiards      59    163     1.52
Tennis         67     58      .368
Baseball       74.5  146      .675
Whiffle        76     45      .196
Football      178    420      .142
Rugby         191    435      .119
Bowling       217   7260     1.36
Soccer        220    432      .078
Cannonball    220  14000     7.9

F  =  Spin in revolutions per second
V  =  Ball velocity
Ve =  Velocity of the edge of the ball due to its spin
=  2 Pi R F
Z  =  Spin ratio
=  Ve/V
```
For a ball rolling on the ground, Z=1.

Calculate Z for the data you collected.

Bowling ball

Suppose a bowling ball slides along the floor with zero spin and then friction causes the ball to start spinning.

```V  =  Initial speed
```
What is the speed of the ball once it starts rolling smoothly?
Formation of the moon

The Earth's moon was formed from the debris of a giant collision. After the collision, any material within the Earth's Roche radius formed a ring and then fell back to the Earth. Material outside the Roche radius coalesced into the moon. If angular mometum is conserved as the moon spirals outward, then what was the spin period of the Earth when the moon was located at the Roche radius?

Pressure

What is the mass of the Earth's atmosphere and what is the Earth's surface area? How can you use this to calculate the atmospheric pressure?
What is the pressure at the bottom of a 4 meter deep pool, relative to the surface?
A typical speed for an oxygen molecule in the atmosphere is 400 m/s. If a molecule is launched upward at this speed and doesn't collide with any other molecules, how high does it go?

This sets the characteristic height of the atmosphere.

Waves

A Cello A-string is 690 mm long and has a frequency of 20 Hertz. What is the speed of a wave on the sring?
The lowest-frequency wave on a bass guitar is 41 Hertz and the speed of sound is 340 meters/second. What is the wavelength of the wave?
Draw a diagram showing the spectrum of human color vision and photosynthesis.
Draw a diagram showing a wave with a frequency of 2 Hertz and a wave with a frequency of 4 Hertz, and their sum. Indicate the locations of time = 1 second, 2 seconds, etc.
Draw a diagram showing the phenomenon of beat frequencies.
Draw a set of diagrams showing the spectrum of blackbody radiation as a function of temperature. Describe the properties of blackbody radiation as a function of temperature.
Gases and thermodynamics

Dinosaur extinction

If an asteroid hits the Earth with a speed of 20 km/s and a density of 2 g/cm3 and all of its kinetic energy goes into heating the atmosphere, what size asteroid is required to heat the atmosphere by 100 Kelvin?

Atoms & nuclei

How many protons and neutrons are in an atom of
Hydrogen
Deuterium
Tritium
Carbon-12
Carbon-14
Nitron-14
Oxygen-16
Uranium-235
Uranium-238
Plutonium-238
Plutonium-239
Draw a diagram showing how the Rutherford experiment works.
Draw a diagram showing the phenomenon of beta decay.
If a proton travels East along Chambers street and there is a magnetic field pointing down toward the Earth's core, what direction is the magnetic fiorce?
An electron travels north along 8th Avenue and generates a magnetic field at 5th Avenue. What is the direction of the magnetic field at 5th Avenue?
Draw a diagram showing how a nuclear fission chain reaction works.
Draw a diagram illustrating the concept of critical mass for fission.
Draw a diagram of a fission reactor. What could go wrong in a fission reactor? Describe the events of the Chernobyl disaster and the Fukushima disaster.

Draw a diagram showing how to make Plutonium-239 from Uranium-238.

Suppose we have 128 kg of radioactive material with a halflife of 2 years. How much of the original material is left after 6 years?
What are the technical obstacles for constructing a uranium fission bomb? What are the obstacles for a plutonium fission bomb? Sketch a diagram showing how to detonate a uranium and plutonium bomb.

Particles

What is the approximate lifetime of a virtual electron?

The formula for relativistic velocity addition is:

```Speed of light                                  =  C
Velocity of a spaceship                         =  VS
Velocity of a cannonball fired by the spaceship =  VC    (In the frame of the ship)
Velocity of the cannonball in the rest frame    =  VR  =  (VA + VB) / (1 + VA VB C-2)
```
If a spaceship moving at .5 C fires a cannonball at a speed of .5 C (with respect to the ship), what is the speed of the cannonball in the rest frame?
Cold atoms
For an ideal gas,
```M  =  Mass of a gas molecule
=  1.4e-25 kg for a rubidium atom
V  =  Characteristic velocity of gas molecules
E  =  Mean thermal energy of gas molecules
=  .5 M V2
k  =  Boltzmann constant
=  1.38e-23 Joules/Kelvin
T  =  Temperature
W  =  Quantum-mechanical wavelength
Q  =  Characteristic momentum of a gas molecule
=  M V
h  =  Planck constant
=  6.62e-34 Joules seconds
=  Q W
```
For a gas in thermal equilibrium, each degree of freedom has mean energy .5 k T. For motion in 3D there are three degrees of freedom and so
```E  =  .5 M V2  =  1.5 k T
```
Atomic cooling experiments can reach temperatures of ~ 10⋅10-7 Kelvin. For this temperature, what is the quantum-mechanical wavelength of a rubidium atom?

De Broglie and Compton wavelengths

```Particle wavelength  =  W
Speed of light       =  C  =  2.998⋅108 meters/second
Planck constant      =  h  =  6.626⋅10-34 Joule seconds
Particle momentum    =  Q
Particle velocity    =  V
Particle frequency   =  F
Particle mass        =  M

De Broglie formula:  h = Q W
Wave formula:        V = F W
Compton wavelength   W = h/(MC)
```

The Compton wavelength is the wavelength of a particle that is "barely relativistic", where we assume Q = M C. This sets the range of forces delivered by virtual particles. For example, the strong force is delivered by pions and its range is equal to the Compton wavelength of a pion.

What is the wavelength of a baseball moving at 40 m/s?

What is the wavelength of an X-ray with an energy of 10000 electron Volts?

What is the wavelength of an electron with an energy of 10000 electron Volts?

What is the frequency of a green photon with a wavelength of 555 nm?

What is the speed of a proton and an electron with a kinetic energy of 5 MeV?

What is the Compton wavelength of an electron, a proton, a pion, and a Higgs boson?

Reference: www.jaymaron.com/particles/particles.html

Physics challenge questions

If a planet is a sphere with a uniform density of 5.5 g/cm3, calculate the gravitational acceleration at the surface as a function of planet radius.
Renewable energy

Bamboo is a fast-growing tree. Using Wikipedia, how much wood can 1 km2 of bamboo forest produce per year and much energy is released from burning 1 kg of bamboo? How much energy is used by civilization in one year and how many kg2 of bamboo forest does it take to equal it?

Astronomy

Units

The Earth's orbit radius is 1.5e11 meters and the orbit time is one year. Calculate the Earth's orbit velocity in meters/second.

What is the distance from Earth to Mars when they are as close as possible?
How long does it take light to travel this distance?
How long does it take a spaceship traveling at 10 km/s to cover this distance?

If a sprinter covers 40 yards in 4.4 seconds then what is the average speed in meters/second?

Convert 65 miles/hour to meters/second, showing the steps in the conversion.

```Distance to Alpha Centauri           =  4.4 light years     The nearest star
Speed of the New Horizons spaceship  =  14.5 km/s           Speed when it passed Pluto
```
At New Horizon's present speed, how long will it take to travel 4.4 light years?
Astronomy challenge questions

Pinatubo

How much CO2 was released by the Pinatubo volcano and what was the volume of magma ejected?

What is the mass of CO2 in the Earth's atmosphere and how many Pinatubos would it take to double the atmosphere's CO2?

Rice and water footprint

Using data from Wikipedia,
How many kg of water does it take to grow one kg of rice?
How many Calories are in 1 kg of rice?
Approximately how many Calories you eat in one day and one year?
How many Joules are there in one food Calorie?
How many Joules do you require in one year?
How many kg of rice do you require in one year?
How much water does it take to grow a year's worth of rice?
How many kg of water flow through the Mississippi per year?
If all of the water in the Mississippi is used to produce rice, how many people can the Mississippi support?

```Energy in one day                        =   2500  Calories/day
Energy in one Calorie                    =   4200  Joules/Calorie
Energy in one day                        =   10.5  MJoules/day
Energy/Mass of rice        =  e          =     15  MJoules/kg
Mass of rice in one day    =  m          =     .7  kg/day
Mass of rice in one year   =  M          =    256  kg/year
Water footprint of rice    =  w          =   1000  kg water per kg rice
Water for 1 year           =  W  =  M w  = 2.56e5  kg/year
Mississippi flow rate      =  F          = 1.67e7  kg/second
= 5.0e14  kg/year
People supported           =  N  =  F/W  =1.95e11  people  =  195 billion people
```

Launch cost

Using Wikipedia, what is the \$/kg required to launch material from the surface of the Earth into low Earth orbit? Find data for several rockets.

```B1  =  Brightness of Pluto in Watts/meter2 when it is 30 AU from the sun
B2  =  Brightness of Pluto in Watts/meter2 when it is 40 AU from the sun
```
What is B1 / B2?
Atmospheric pressure

```Mass of the Earth's atmospere    =  M  =  5.15e18 kg
Surface area of the Earth        =  A  =  5.10e14 m2
Gravitational constant           =  g  =  9.8 m/s2
Pressure of Earth's atmosphere   =  P  =  101000 Pascals  =  M g / A
=  15 pounds/inch2
=  1 Bar
Mass above one meter2 of surface =  m  =  M / A
```
One bar is defined as the Earth's mean atmospheric pressure at sea level

Properties of atmospheres:

```       Density   Pressure    m       Gravity
(kg/m2)    (Bar)   (tons/m2)   (m/s2)

Venus    67       92.1      1050      8.87
Titan     5.3      1.46      109      1.35
Earth     1.22     1          10.3    9.78
Mars       .020     .0063       .54   3.71

Mass of atmosphere above one meter2 of surface  =  m  =  10.3 tons for the Earth

P  =  m g
```
You don't need a pressure suit on Titan. You can use the kind of gear arctic scuba divers use. Also, the gravity is so weak and the atmosphere is so thick that human-powered flight is easy. Titan will be a good place for the X games.

Pandora is the moon in the film "Avatar". Using data from the web, add Pandora to the above table.

Asteroid defense

Suppose an asteroid is a sphere of uniform density

```Mass of a sphere                 =  M
Radius of a sphere               =  R
Density of a sphere              =  D  =  2000 kg/meter3    (rock)
Gravity constant                 =  G
Gravitational energy of a sphere =  Egrav  =  .6 G M2/R
Energy of a hydrogen bomb        =  Ebomb  =  4e16 Joules   (equivalent to 10 Megatons of TNT)
```
For what value of R is Egrav = Ebomb?
Death Star

In Star Wars, a death star destroys planet Aldebaran. Suppose we assume that the death star is powered by antimatter.

```Mass of antimatter         =  Manti
Speed of light             =  C  =  3.00e8 meters/second
Energy in antimatter       =  Eanti  =  M C2
Mass of the Earth          =  M  =  6.0e24 kg
Radius of the Earth        =  R  =  6371 km                (From Star Wars)
Density of the Earth       =  D  =  5500 kg/meter3
Gravitational energy of the Earth = EEarth  =  .6 G M2 / R          (Assume a uniform density)
```
For what value of Manti is E = Eanti?
```              MicroWatts  Half life   Mantle     PicoWatts/kg
/kg      (billion    abundance  of mantle
years)    (ppb)

Uranium-238      94.6       4.47      30.8        2.91
Uranium-235     569.         .70        .22        .125
Thorium-232      26.4      14.0      124          3.27
Potassium-40     29.2       1.25      36.9        1.08
Total                                             7.38
```
The Earth loses heat at a rate of .087 Watts/m2, for a global heat los of 4.42e13 Watts.

80% of the Earth's heat is from radioactivity and 20% is from accretion.

```Time elapsed                        =  T
Half life of the material           =  Thalf
Original mass of material           =  M0
Mass of material left after time T  =  M  =  M0  exp(-T/Thalf)
```
Based on the above data, what will be the radioactive intensity of the Earth in PicoWatts/kg one billion years from now?
Atmospheres

Composition of the Earth's atmosphere

```Matmos  =  Mass of the Earth's atmosphere
Fatmos  =  Mass fraction of nitrogen in the Earth's atmosphere
Mnitro  =  Mass of nitrogen in the Earth's atmosphere

Area    =  Surface area of the Earth
Density =  Density of the Earth's crust
=  2000 kg/m3          (This is approximate.  It varies from 1500 to 3000)
Fcrust  =  Mass fraction of nitrogen in the Earth's crust
Depth   =  Depth below the surface of the Earth
Mcrust  =  Mass of the Earth's crust down to a depth "Depth"
=  Area * Depth * Density
Mnitro  =  Mass of nitrogen in the Earth's crust down to a depth "Depth"   =  Mcrust * Fcrust
```
Using wikipedia, what are the values of the above numbers?

For what value of "Depth" is Matmos = Mnitro?

Composition of the Earth's crust.

```          Parts per thousand

Oxygen      460
Silicon     270
Aluminum     82
Iron         63
Calcium      50
Magnesium    29
Sodium       23
Potassium    15
Titanium      6.6
Hydrogen      1.5
Manganese     1.1
Phosphorus    1.0
Fluorine       .54
Barium         .5
Carbon        1.8
Strontium      .36
Sulfur         .42
Zirconium      .13
Tungsten       .0011
Chlorine       .17
Chromium       .14
Rubidium       .06
Nickel         .09
Zinc           .079
Copper         .068
Cerium         .060
Neodymium      .033
Lanthanum      .034
Yttrium        .029
Nitrogen       .020
```

Humans generate energy by reacting hydrocarbons with oxygen. Consider the reaction of octane, which has 8 carbons and 18 hydrogens.

```C8H18  +  25 O2  ->  9 H2O  +  8 CO2

Moctane  =  Mass of octane in the reaction
Moxygen  =  Mass of oxygen in the reaction
```
What is Moctane / Moxygen?
What is the distance from Earth to Mars when they are as close as possible?

How long does it take light to travel this distance?

How long does it take a spaceship traveling at 10 km/s to cover this distance?

Asteroid risk
```R  =  Radius of an asteroid
P  =  Probability per millenium of a hit from an asteroid at least as large as R
```
Using data from the web, what is P for R = .01, .1, 1, 10 km?
Using data from the web, estimate the probability that a random star has a planet in the habitable zone that has a mass of 1.5 Earth masses or less. What data did you use and how did you analyze it?

How many stars are there in the Milky Way?

Using data from Wikipdedia, how much energy does 1 kg of diesel fuel have and how much thermal energy does 1 kg of uranium produce in a nuclear reactor? How many kg of diesel does it take to equal the energy of 1 kg of uranium?
A violin string is .32 meters long and has a frequency of 440 Hertz. The speed of sound is 340 meters/second. What is the speed of a wave on the string? What is the wavelength of the sound wave?
Using Wikipedia, how many light years is the sun from the center of the Milky Way, and how fast is the sun moving on its orbit around the galactic center?

How fast is the Milky Way moving with respect to the cosmic microwave background radiation?

Dinosaur extinction

What is the heat capacity of air and what is the mass of the Earth's atmosphere?

If an asteroid hits the Earth with a speed of 20 km/s, a diameter of 10 km, and a density of 2 g/cm3, and if all of its kinetic energy goes into heating the atmosphere, by how many degrees Kelvin does the temperature of the atmosphere rise?

Earth temperature

How many Watts of sunlight strike the Earth?

If the Earth were an ideal blackbody with uniform temperature, what is the blackbody radiation power?

Setting the above two quantities equal, what is the resulting temperature of the Earth?

Solar fusion

What are the fusion reactions that convert hydrogen to helium in the sun? Using Wikipedia, what is:

```Mass of 4 protons                     =  m
Mass of a helium nucleus              =  M
Fraction of mass converted to energy  =  F  =  (M-m)/M
```
Using E=MC2, how much energy is released in the fusion of 1 kg of hydrogen into helium?
Astronomy

What are the different types of chemical reactions used by rockets?
What is a typical exhaust speed for each chemical reaction?
If you were the President of the United States, what measures would you take to defend against asteroids?
Describe a "Hohmann maneuver", including a diagram.
What are the advantages of launching a rocket from a high-altitude airplane vs. launching it from the ground?
What elements exist on Mars that are essential for life?

What elements would we have to bring from Earth if we want to grow crops on Mars?

What molecules are used by all Earth life forms?

What biochemical processes are the same for all Earth life forms?

Describe what we know about planets around other stars. Describe the methods used to detect them, including diagrams.
We have 5 rocks from Mars. How did they get to the Earth? How do we know they're from Mars?
Draw an approximate timeline showing the evolutionary tree of mammals, amphibians, reptiles, and birds.
Describe the characteristics of a tetrapod, including diagrams. Show as much detail as possible.
Plot the following all on one plot:
Spectrum of the three color vision cells
Spectrum of photosynthesis
Spectrum of blackbody radiation at 300 Kelvin (the temperature of the Earth)
Spectrum of blackbody radiation at 6000 Kelvin (the temperature of the sun)
Estimate the number of rotations the Milky Way has completed in its lifetime.
Copernicus and Galileo

In Copernicus' time, what astronomical evidence was consistent with the heliocentric model and what evidence was consistent with the geocentric model? (It's possible for evidence to be consistent with both)

What discoveries did Galileo make that relate to heliocentrism? How is each discovery inconsistent with geocentrism?

Describe how to use the Hubble constant to estimate the age of the universe. Estimate the age in billions of years.
Andromeda is moving toward us at a speed of 120 km/s and is at a distance of 2.54 million light years. HOw long will it take to get here?
At approximately what altitude is the Earth's atmospheric pressure half the value at sea level? What altitude has the same pressure as the surface of Mars?
List the hazards to life if you are standing on the surface of Mars without a spacesuit.
Draw a timeline showing the important discoveries in astronomy up to the year 1700, and identify the discoverer.

Describe the life cycle of a star that ends its life as a neutron star.

Cosmology

Draw a timeline of the universe and the Earth, starting from the big bang and ending at the Big Chill. Show as many important events as you can.

Parallax

To measure the parallax of a star, its angle on the sky is measured twice, 6 months apart. The parallax angle is the difference between these two angles.

The resolution of a 10 cm telescope is 1 arcsecond. Increasing the size of the telescope doesn't increase the resolution because air turbulence distorts the image.

If a star has a parallax angle of 1 arcsecond, how far away is it in light years?

Latitude and longitude

Define "noon" to be the instant when the sun is highest in the sky for a given day.

The summer solstice is the day of the year when the sun is highest in the sky.

Suppose a 10 meter vertical pole is planted on Wall Street, New York City. At noon on the summer solstice, what is the length of the shadow?

Suppose there are two synchronized clocks, one in New York City and another directly to the east in Brooklyn, and suppose you use these clocks to measure the time of sunset.

Sunset for the clock in New York City = 6:00 PM Sunset for the clock in Brooklyn = 5:57 PM

What is the distance between the clocks?

What properties of a planet affect its ability to host intelligent life?

For each property there is a "Goldilocks zone" that is ideal for habitability. What are the consequences of if there is too much or too little of each property?

Why did Venus and Mars become uninhabitable?

Order of magnitude physics problems

A "*" indicates a link to a solution.

Atmospheres
Earth
Mars
Titan
Veuns

What is the minimum mass for a planet to have an atmosphere? Assume a temperature of 300 Kelvin. What is the minimum mass for a gas giant?     *

Eyes

The characteristic diffraction angle for a circular aperture is:

```Wavelength         =  λ
Aperture diameter  =  D
Diffraction angle  =  θ  =  1.22 λ / D
```
What is the wavelength of green light with a wavelength of 555 nm?

What is the diameter of a pupil at maximum dilation?

What is the diffraction angle?

Conduct an experiment and determine the minimum angle for which you can resolve two dots.

For 20/20 vision, what is the angle of the smallest letter you can resolve?

Neptune is at the edge of detectability by the the human eye. How bright is Neptune in Watts/meter2 and how many photons/second does this correspond to?

Bullets

How far can a 9 mm diameter lead bullet travel underwater before losing half its speed? Estimate the mass of the bullet and the recoil momentum of gun. How far does a soccer ball travel in air before losing half its speed?     *

Maximum speed of a car

What is the maximum speed of a car with a 400 horsepower engine?

Freeway

How far can a car go on 50 liters of gasoline at 75 mph?

Asteroid destruction

If you detonate the bomb at the center and if the asteroid is too large, gravity will bring the asteroid back together. For a uniform-density sphere,

```Gravitational energy  =  .6 G Mass2 / R
```
Suppose a hydrogen bomb has the energy of 10 megatons of TNT, which is 4*1016 Joules. What would you estimate is the largest value for the radius of an asteroid that this bomb can shatter?

In "Star Wars", a Death Star shatters a planet. If the planet is identical to the Earth, how much energy does this take? If the energy were provided by a sphere of antimatter with the density of iron, what is the radius of this sphere?

Elasticity of metals

For a typical metal, estimate the value of the tensile modulus divided by the density.

Analyze the deflategate incident. If a ball is inflated at room temperature and the game is at 4 Celsius, how many psi do the balls lose when they are brought to the game?
Cosmic ray shielding

Estimate the stopping distance of a cosmic ray proton in iron in units of tons/meter2.     *

Estimate the critical mass of Uranium-235.

Violin strings

Estimate the minimum and maximum frequency of a string on a violin?
How does the minimum and maximum frequency scale with string length?     *

Electric cars

How far can a car go at 30 mph if it is powered by 100 kg of supercapacitors?     *

Flight

What is the minimum power/weight ratio required for human-powered flight?     *

Loudspeakers

If a speaker with diameter D plays a note of frequency F, what is the maximum sound power that it can produce?     *

SR-71 Blackbird

For a passenger aircraft such as a 747 the maximum useful speed is just less than the speed of sound. If you go faster than this then the plane is less fuel efficient. What is the maximum thrust/meter2 of a jet engine moving at the speed of sound?

The SR-71 uses ramjet engines. Estimate its maximum speed.

Ion drives

Estimate the maximum acceleration of an ion drive powered by Strontium-90. Assume that the mass of the ion drive is less than the Strontium-90 required to power it. (For the Chang-Diaz ion drive they are approximately equal).

Trees

Estimate the maximum height of a tree.

Moon stability

The moon is spiraling outward and in the distant future it will be stolen by the sun. Estimate the maximum stable orbit radius for the moon.     *     *

Tidal locking

If a planet is close enough to a star it becomes tidally locked to the star. Mercury is just barely close enough for this and it orbits such that

3 * Orbit period = 2 * Spin period

If it were any closer it would be forced into a lock such that

Orbit period = Spin period

All of the solar system's moons are locked to their planets in this way. None of the planets are locked to their moons except Pluto.

For a low-mass star, the habitable zone is closer to the star than it is for the sun. If the star is sufficiently small, a planet in the habitable zone will be close enough to be locked to the star and will experience extreme weather.

The luminosity of a star scales as mass to the 3.5 power.

Let L = Luminosity of a star / Luminosity of the sun (Watts).

If a star is such that the habitable zone is at Mercury's orbit, what is L? What stellar mass corresponds to this luminosity?

Atomic cooling

Atomic cooling experiments can reach temperatures of ~ 10-7 Kelvin. For this temperature, what is the quantum-mechanical wavelength of a rubidium atom?

What is the maximum height of a mountain as a function of a planet's radius?     *
Sunlight

Using Wikipedia, what is the mass, luminosity (Energy/time), and lifetime of the sun? What is the distance from the Sun to the Earth and what is the radius of the Earth?

How many kilograms of sunlight fall on the Earth each second?

If we assume that the sun's luminosity throughout its lifetime is equal to its present luminosity, how much energy does it produce in its lifetime? How many Earth masses is this? What fraction of the sun's mass is this?

Proton ring collider

For a proton ring collider, what is the proton energy as a function of ring radius, assuming a ring-averaged magnet strength of 10 Tesla.     *

Electron ring collider

For an electron ring collider, the maximum energy is limited by synchrotron radiation. What is the maximum energy as a function of radius?     *

Earth-scale electron collider

The maximum energy of an electron linear collider is limited by synchrotron radiation from the curvature of the Earth. What is the maximum energy?     *

Muon collider

For a muon in a ring collider, how many times does the muon go around the ring during one halflife?     *

Cosmic ray muons

Cosmic rays crashing into the atmosphere produce muons, many of which reach the surface. Using data from Wikipedia, what is the typical altitude of muon production? What is the halflife of a muon and how far does it travel during this time if it is moving at close to the speed of light?

Cosmic rays

What is the lowest-energy cosmic ray that can reach Earth's orbit?

Planet X in the Kuiper Belt

What is the furthest distance we could detect an Earth-like planet in the Kuiper belt?     *

Asteroid deflection

Construct a method for deflecting an asteroid and estimate the maximum-sized asteroid it can deflect, if we have 1 year of advance warning.     *

Is there room for an Earth-mass planet between the Earth and Mars (without disrupting the existing planets)? Between Jupiter and Saturn? Use the "My Solar System" simulator at phet.colorado.edu to construct a test system that behaves the same as the real solar system.     *
Crowded rooms
```A   =  Area of a room
H   =  Height of a room
N   =  People in a room
D   =  Surface density of people in the room
=  D/A
C   =  Carbon dioxide fraction in the air
dC  =  Change in carbon dioxide fraction per second
Cmax=  Maximum save carbon dioxide fraction in the air
Ct  =  Time for the carbon dioxide fraction to go from 0 to Cmax
T   =  Temperature in the room
dT  =  Change in room temperature per second
Tmax=  Maximum tolerable change in room temperature
=  5 Kelvin
Tt  =  Time for the temperature to increase by Tmax
W   =  Water vapor fraction in the air
dW  =  Change in water vapor fraction per second
Wmax=  Saturation vapor pressure of water at 293 Kelvin
Wt  =  Time for the water vapor fraction to go from 0 to Wmax
P   =  Power generated per person
=  100 Watts
```
What is Ct, Wt, and Tt?
Quiz bowl

Quiz bowl (astronomy)

For each planet, what is the largest moon?

What is the name of the largest asteroid?
Name as many other asteroids as you can.

What is the name of the telescope we use to find asteroids with?
What is the name of the new telescope that will succeed it?

What are the names of the 2 small galaxies near the Milky Way?

When Uranus was first discovered by Herschel, what name did he give it?
What is the theme for the names of the moons of Uranus?

What is the name of the largest radio telescope?

What is the name of the telescope that will succeed the Hubble? Where will it be located?

The data that Kepler used to prove that planets orbit as ellipses was taken by whom?

What kind of light is the Webb telescope optimized for?

The measurements that Aristarchus did were improved upon by what Indian mathematician?

What is the name of the telescope that finds planets by eclipse?

What are the names of the 5 moons of Pluto?

Name as many Kuiper belt objects as you can
Among all known Kuiper belt objects, which is furthest from the sun?

Where is the Oort cloud?

Who invented the pendulum clock?

On the flags of Australia and New Zealand, what is the name of the constellation depicted?

What is the Messier Catalog number for the Andromeda galaxy?

Name as many galaxy superclusters as you can.

What is the speed of the Earth's orbit around the sun?
How far away is the center of the galaxy?
How fast is the sun moving in its orbit around the center of the Milky Way?
How fast is the Milky Way moving with respect to the cosmic microwave background?

What is the orbital velocity for the Earth?
What is the escape velocity from the surface?

Of the five Lagrange points, which are stable?

Other than the Milky Way and Andromeda, name as many galaxies as you can.

Other than the sun and the stars in the Alpha Centauri system, name as many stars as you can.

In units of solar masses, what is the minimum mass to be a star?

For rocket engines, what is the most powerful possible chemical reaction?

Quiz bowl (periodic table)

Name as many noble gases as you can.

What elements are liquids at room temperature?

Gold was an uncounterfeitable currency until the discovery of what element?

Which three elements are the active ingredient in fertilizer?

Name the elements that are more dense than gold.
What is the densest element?

What is the most abundant element in the Earth's crust?

What is the most abundant element in a metallic meteorite?

What is the element to the right of iron?
What is the element below chromium?
What is the element below oxygen?
What is the element to the right of carbon?
What is the element with the lowest proton number that has no stable isotopes?

Name as many metals as you can that have a proton number that is less than that for iron.

Name as many metals as you can that have a proton number that is greater than that for iron.

Quiz bowl (history of science)

In the 16th century, who brought decimal numbers to Europe?

In what century were cannons first used?

Before the discovery of iron, what metal was used for swords?

Who discovered that electric currents attract?

Other than Neil Armstrong, name as many astronauts as you can that have been on the moon.

Who discovered that an electric current attracts a compass?

In what year did humans first walk on the moon?
What is the most recent year that humans walked on the moon?

Quiz bowl (physics)

At the Large Hadron Collider, what particle was recently discovered?

In units of Celsius, what is the temperature of absolute zero?

Quiz bowl (miscellaneous)

How many different DNA codons are possible?

What is the order of atoms in a protein backbone?

Pizza

What would you estimate is the mass of a 16 inch diameter pizza with sausage, Canadian bacon, Parmesan cheese, olives, mushrooms, onions, and peppers? What is an estimate of the mass of each ingredient?

Dark matter

What is the speed of the sun's orbit around the center of the galaxy and what is its distance from the center of the galaxy? Using this, what is the mass of material interior to the sun?

How many stars are there interior to the sun, what is their average mass, and what is the total mass of stars interior to the sun?

Solutions

If the cannonball moves to the right and the cannon recoils to the left,
```Cannon mass * Cannon recoil velocity  =  - Cannonball mass * Cannonball velocity
2000     * Cannon recoil velocity  =  -       10        *      600

Cannon recoil velocity = -3 meters/second
```

```Initial vertical velocity  =  Acceleration * Time to reach the top of the arc
30              =       10      * Time to reach the top of the arc

Time to reach the top of the arc  =  3 seconds
```
The football travels upward for 3 seconds and downward for 3 seconds before hitting the ground. The total hang time is 6 seconds.
```Horizontal distance traveled  =  Horizontal velocity  *  Hang time  =  20 * 6  =  120 meters
```

Upward forces are positive and downward forces are negative.
```Total force  =  Jet force  +  Gravity force
=    1800     +  Mass * Gravityconstant
=    1800     -  110  * 10
=     600

Acceleration =  Total force / Mass
=  600 / 110
=  5 m/s2     (upward)
```

```Acceleration  =  Velocity / Time
=  20 / 5
=  4 m/s2

Distance  =  .5 * Acceleration * Time2
=  .5 * 4 * 52
=  50 meters
```

```Energy  =  Mass *  g * Height
=  110  * 10 * 12
=  13200 Joules
Power   =  13200 / 5
=  2640 Watts
```

```Centripetal force  =  Maximum friction force
Mass V2 / R    =  FrictionCoefficient * Mass * g

V  =  Squareroot(Frictioncoefficient * g *R)
=  Squareroot(.9 * 10 * 8)
=  8.5
```

```Mass  =  Density * Volume
=  19.3 * 33
=  521.1 grams
Price =  Price/gram  *  Grams
=  40 * 521.1
=  20800 \$
```
Speed of a baseball pitch thrown by Randy Johnson 45 meters/second Initial height of the baseball 2.2 meters Distance from the pitcher to the batter 20 meters
```Time to travel from pitcher to catcher  =  Distance / Velocity  =  20 / 45  =  .44 seconds

Vertical drop  =  .5 * Acceleration * Time2  =  .5 * 10 * .442  =  .99 meters
```

```Maximum acceleration  =  Maximum friction force / Mass
=  Friction coefficient * Downforce / Mass
=  Friction coefficient * Mass * g / Mass
=  1.4 * 10
=  14
```

```G  =  Gravity constant
M  =  Mass of Earth
A  =  Gravitational acceleration at the surface of the Earth
=  G M / R2
=  6.67e-11 * 5.97e24 / 6372000**2
=  9.8 m/s2
A  =  Gravitational acceleration at an altitude of 400 km
=  G M / R2
=  6.67e-11 * 5.97e24 / (6372000+400000)2
=  8.7 m/s2

Gravitational energy  =  Mass * g * Height  =  1 * 10 * 400000 = 4e6
Kinetic energy        =  .5 * Mass * Velocity2  =  .5 * 1 * 78000002  =  3e13
```

Estimation
```Distance between two train tracks = 1.435 meters
Central Park length               = 4.0 km
Central Park width                =  .8 km
Traffic light diameter            =20.3 cm
Stop sign size                    =  75 cm      Distance between opposite edges
Stop sign letter height           =  25 cm
Height of Tyrion Lannister        = 135 cm      4 feet 5 inches
Standard sheet of paper, height   =27.9 cm      11 inches
Standard sheet of paper, width    =21.6 cm     8.5 inches
Manhattan island length           =21.6 km
Distance between Manhattan streets=  78 meters  =  Central park length / Central park streets  =  4.0 km / (110 - 59)
```

The density of iron is 8.49e28 atoms/meter3. Suppose two cubes of iron are stacked one on top of the other and each cube is 1 meter on a side, what is the force per iron atom at the surface of contact?

For an electron that is 10-15 meters from another electron, what is the force?

```Fele  =  Electric force between two protons separated by 1 meter
Fgrav =  Gravity frce between two protons separated by 1 meter
Q                =  Fele / grav
```
What is Q?
```Volume of the drink        =  V
Fraction of alcohol        =  F
Volume of alcohol          =  A  =  F V
Volume of alcohol in Bonds =  B  =  A / .5 ounces

Beer:  F V  =  .05 * 12  =  ounces   (One "Bond")

One ounce  =  29.57 mL
One gallon =  128 ounces

In ounces:
F     V      A     B

Beer          .05   12      .6    1
Wine glass    .13    4.6    .6    1
Scotch shot   .40    1.5    .6    1
Wine bottle   .13   25.4   3.3    5.5
Scotch bottle .40   25.4  10.1   16.9
Keg           .05 1984    99.2  165.3
```

Density

Gold has a density of 19.3 g/cm3. What is the mass of a gold cube that is 3 cm on a side?

Gold costs 40 \$/gram. What is the price of the gold cube?

Using Wikipedia, How much CO2 was released by the Pinatubo volcano and what was the volume of magma ejected?

What is the mass of CO2 in the Earth's atmosphere and how many Pinatubos would it take to double the atmosphere's CO2?

What is the mass of the Earth's atmosphere, what is the mass of the carbon dioxide it contains, and what is the mass of elemental carbon contained in the carbon dioxide?

The wave equation is:
```Wavelength * Frequency  =  Wavespeed
```
The wavelength of visible light is 5.55e-7 meters. What is the frequency?

A violin A-string has a frequency of 440 Hertz and the speed of sound is 340 m/s. What is the wavelength of this wave?

A person with 20/20 vision can see 1.5 mm letters at a distance of 1 meter.

For a computer screen 20 cm from your eyes, what is the minimum size of letters you can see?

If each letter is 7 pixels then how many pixels does a 10 cm screen have? (This is a "retinal" display)

A ten centimeter telescope is 20 times more powerful that a human with 20/20 vision. If giant letters are painted on the moon, what is the minimum size of letters that the telescope can see?

Using data from Wikipedia, how many kg of water does it take to grow one kg of rice? How many calories are in 1 kg of rice? Approximately how many calories you eat in one year and how many Joules is this? If you get the energy from rice then how many kg is a year's supply of rice, and how many kg of water does it take to produce it?

How many kg of water flow through the Mississippi per year, and how many people can it feed if it is used to grow rice?

The intensity of sunlight at Earth's orbit is 1360 Watts/meter2.
How much energy in Joules falls on the Earth each second?
```Energy = Mass * Speedoflight2
```
How much mass does this energy correspond to?
Humans generate energy by reacting hydrocarbons with oxygen. Consider the reaction of octane, which has 8 carbons and 18 hydrogens.
```C8H18  +  25 O2  ->  9 H2O  +  8 CO2

Moctane  =  Mass of octane in the reaction
Moxygen  =  Mass of oxygen in the reaction
```
What is Moxygen/Moctane?
```R  =  Radius of an asteroid
P  =  Probability per millenium of a hit from an asteroid at least as large as R
```
Using data from the web, what is P for R = .01, .1, 1, 10 km?
Homework due March 31

What would you estimate is the mass of a 16 inch diameter pizza with sausage, Canadian bacon, Parmesan cheese, olives, mushrooms, onions, and peppers? What is an estimate of the mass of each ingredient?

Using data from Wikipedia, find the energy released when hydrogen and oxygen combine to form 1 kg of water.
```Kinetic energy = ½ Mass Velocity2
```
If a rocket nozzle converts this energy into kinetic energy of exhaust, how fast is the exhaust moving?

If a solar cell produces 1000 Watts, estimate how long it takes to produce enough energy to convert 1 ton of water to hydrogen and oxygen gas?

The Pan-STARRS telescope

How are the Pan-STARRS and LSST telescopes different from conventional telescopes like Keck? What can they do that Keck can't?

A human with 20/20 vision can resolve letters 1.5 mm tall at a distance of 1 meter. If a letter is painted on the moon, what is the smallest letter that can be resolved?
Find a subject from the history of science and use Wikipedia to construct a timeline for it.
For a steel wire,
```Force required to break the wire  =  F     (Newtons)
Cross-sectional area of the wire  =  A     (meters2)
Tensile strength of the wire      =  S  =  2⋅108 Newtons/meter2
```
If a person with a weight of 1000 Newtons (100 kg) hangs from this wire, what is the minimum cross-sectional area for the wire to not break? What is the diameter of this wire?
Kegs

Without using the web, estimate: (show how you arrived at each estimation)
The number of students at the University of Wisconsin.
The number of bars on State Street (half of the bars are located here).
The amount of beer consumed per week in pints and kegs.
The number of 18-wheel trucks required to carry the kegs.

Briefcase

Without using the web, estimate the number of 20 dollar bills that a typical briefcase can carry. If the bills are replaced with gold, how many \$ worth of gold is this? (Assume gold costs 35000 \$/kg).

Brain memory

Your brain contains 1013 bytes of information. If this information is stored on micro SD cards, how many suitcases would it take to carry them?

Solar cells
```Time                                 =  T            seconds
Solar cell efficiency                =  C   =  .25   dimensionless
Area of a solar cell                 =  A            meters2
Average solar intensity in Arizona   =  Ssol =   250 Watts/meter2    (Averaged over day and night)
Power generated by a solar cell      =  P   =  C A Ssol
Energy generated by a solar cell     =  E   =  P T   Joules
United States population             =  N   =    318 million
United States electricity per person =  Pcap =  1700 Watts
United states electrical power       =  PUSA = N Pcap = 4.5⋅1011 Watts
```
What is "A", the area of solar cells required to provide America's electricity?
Without using the web, estimate the time it would take to hike from Manhattan to San Francisco, taking account for rest and sleep. Describe your methodology behind your estimate.
Using data from the web, estimate how many times per century that Manhattan can expect an earthquake of magnitude larger than 8. Describe how you arrived at the estimate.
Using Wikipedia, what is a typical value for the amount of MegaWatts produced by a 1 km2 region of wind turbines? What is the power consumed by the United States? How many km2 of wind turbines are required to supply this power?
Using Wikipedia, how much does it cost to desalinate 1 ton of seawater? If we assume that food is grown in greenhouses with a water footprint of .2 tons/kg of food, and if it takes 400 kg of food to feed 1 person for 1 year, what is the cost to desalinate the water for a year's supply of food?
For a bullet,
```Lead volume              =  Slead     (meters3)
Gunpowder volume         =  Spowder   (meters3)
Gunpowder density        =  Dpowder =   1400 kg/meters3
Gunpowder mass           =  Mpowder =  Dpowder Spowder
Gunpowder energy density =  Qpowder =  5.2⋅106 Joules/meter3
Gunpowder energy         =  Epowder =  Qpowder Spowder
```
If we assume that the lead and gunpowder volumes are equal and that all of the gunpowder energy goes into lead kinetic energy, what is the speed V of the lead?
What is the speed of light, what is the distance from the Sun to the Earth, and how many minutes does it take light to travel from the sun to the Earth?
```Mass               =  M
Gravity constant   =  g  =  10 m/s2
Height             =  H
Gravity energy     =  Eg =  M g H
Human sprint speed =  V  =  10 m/s
Kinetic energy     =  Ek =  ½ M V2
```
In a pole vault, kinetic energy is converted to gravitational energy. We can approximate how high the vaulter goes by setting Ek=Eg. Given this assumption, what is H?
Optional (this one is difficult).
Acceleration lab
Using Wikipedia, what is the radius of a typical neuron cell in the brain, what is its volume, and what is the volume of the brain? What is the value of the brain volume over the neuron volume?

Using Wikipedia, what is the number of neurons in the brain?

The Krakatoa volcanic explosion generated a sound wave that traveled around the Earth 3 times.

What is the speed of sound and how long does it take sound to go all the way around the Earth?

The distance between North American and Europe is increasing due to continental drift. Using Wikipedia, how fast is the drift? What is the distance from Manhattan to Lisbon? How long will it take continental drift to double this distance?
Momentum of photons

What is the momentum of 1 Joule of light?

What is the momentum of a golf ball with 1 Joule of kinetic energy?

Photon energy

If a photon collides head-on with an electron, the electron is set in motion and the photon recoils backwards with a reduced frequency. In this experiment the observables are:

```F  =  Frequency of photon before collision
This photon is traveling to the right at speed C.
Frequency is measured in Hertz or cycles per second
f  =  Frequency of photon after collision
This photon is traveling to the left at speed C.
V  =  Velocity of the electron after the collision
This electron is moving to the right at speed C.
Before the collision, the electron is at rest.
M  =  Mass of an electron
=  9.109*e-31 kg
```
We assume that the photon has a low enough frequency so that the recoiling electron is not relativistic. This applies if the photon is less energetic than an X-ray. In this case,
```V << C

(F-f)/F << 1        The change in frequency is small compared to the initial frequency
```
If you do the experiment and measure the observables you find that (in MKS units)
```F - f  = 1.62e-20 * F2   Hertz
V      = 4.85e-12 * F    meters/second
```
Given this result, how would you construct formulas for the energy and momentum of a photon?

____________ Solution ______________
```E  =  Energy of the initial photon
Q  =  Momentum of the initial photon
```
The photon frequency is almost the same before and after the collision, and so we assume that the change in photon momentum during the collision is 2Q. This momentum is taken up by the electron.
```2 Q = M V

Q  = M V / 2
= 2.21e-42 F

Electron energy  =  hF - hf
=  h(F-f)
```
We can express the photon momentum in terms of the Planck constant.
```Q  =  2.21e-42 F
=  h F / 3e8              (3e8 happens to be the speed of light)
=  h F / C
```
Hence the photon energy and momentum are
```F  =  Photon frequency
W  =  Photon wavelength
C  =  Speed of light
=  F W
E  =  Photon energy
=  h F
Q  =  Photon momentum
=  h F / C
=  h / W
```
The momentum and wavelength are related by
```Q W = h

E  =  Photon energy
=  h F
Q  =  Photon momentum
=  h F / C
=  h / W
```
The momentum and wavelength are related by
```Q W = h
```
This is the "de Broglie" relation, and it is true both for photons and for particles with mass. Every particle with mass has an associated wavelength W which is its quantum-mechanical "size".

Photons and electrons have a different relationship between energy and momentum.

```For a photon,         Energy = Momentum * C

For an electron,      Energy = Momentum2 / (2*Mass)
```

Solutions

Bond:
```Volume of the drink        =  V
Fraction of alcohol        =  F
Volume of alcohol          =  A  =  F V
Volume of alcohol in Bonds =  B  =  A / .5 ounces

Beer:  F V  =  .05 * 12  =  ounces   (One "Bond")

One ounce  =  29.57 mL
One gallon =  128 ounces

In ounces:
F     V      A     B

Beer          .05   12      .6    1
Wine glass    .13    4.6    .6    1
Scotch shot   .40    1.5    .6    1
Wine bottle   .13   25.4   3.3    5.5
Scotch bottle .40   25.4  10.1   16.9
Keg           .05 1984    99.2  165.3
```

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