
The fundamental units are the meter, second, kilogram, and Coulomb. They were originally defined in 1793 as the "Standard International" (SI) units, or "MKS" units.
Quantity Unit Definition Length Meter The Earth's circumference is 40 million meters Time Second There are 86400 seconds in one Earth day Mass Kilogram The mass of a cube of water 10 cm on a side is 1 kilogram Charge Coulomb The force between two charges of one Coulomb each and separated by 1 meter is 9 billion Newtons
Density of water =1000 kg/meter$3$ = 1 g/cm$3$ Density of air = 1.2 kg/meter$3$ = .0012 g/cm$3$
The fundamental units are length, mass, time, and charge, and all other units are derived from these.
Quanity Composition Units Length = X meters Mass = M kg Time = T seconds Charge = C Coulomb Speed = V = X/T Length / Time meters/second Momentum = Q = M V Mass * Speed kg meters/second Acceleration = A = V/T Speed / Time meters/second^{2} Force = F = M A Mass * Acceleration Newtons = kg meters/second^{2} Energy = E = F X Force * Distance Joules = kg meters^{2}/second^{2} Power = P = E/T Energy / Time Watts = kg meters^{2}/second^{3} Area = S = X^{2} Length$2$ meters$2$ Volume = Υ = X^{3} Length$3$ meters$3$ Density = ρ = M/Υ Mass / Volume kg / meters^{2} Pressure = Φ = F/S Force / Area Pascals = Newtons/meter$2$ = Joules/meter$3$ Angular momentum = L = M V X Momentum*Length kg meters$2$/second Torque = Γ = F X Force * Length kg meters$2$/second$2$ Frequency = f = 1/T 1 / Time Hertz = 1/second
Meter = 39.37 inches = 1.0936 yards = 3.281 feet = 1/1609 miles Mile = 1609 meters = 1760 yards (exact) Yard = 3 feet (exact) = .9144 meters Foot = 12 inches (exact) = .3048 meters Inch = 25.4 mm (exact) Minute = 60 seconds Hour = 60 minutes Day = 24 hours Year = 365.25 days Ton = 1000 kg (exact) Kilogram = 1000 grams (exact) = 2.205 pounds (pounds interpreted as mass) Newton = .2248 pounds (For Earth gravity at the surface) (pounds interpreted as force) Pound = 16 ounces (exact) (interpreted as mass) = .4535 kg 4.448 Newtons (interpreted as force) Ounce = 28.35 grams (interpreted as mass) Meter/second = 2.24 miles/hour Km/hour = .6214 miles/hour Miles/hour = 1.609 km/hour Pascal = .0001450 pounds/inch^{2} (pounds interpreted as force) Pound/inch^{2} = 6895 Pascals Bar = 101325 Pascals (Atmosphere pressure at sea level) = 14.50 pounds/inch^{2} (pounds interpreted as force) Earth gravity= 9.807 meters/second^{2} = 32.2 feet/second^{2} Standard sheet of paper = 11 x 8.5 inches = 27.94 x 21.59 cm
Meters Earth Earth Light travel radii orbits time (AU) Nucleus 2⋅10$15$ Atom 2⋅10$10$ Green light 5.5⋅10$7$ Neuron .00002 Dime thickness .00135 Dime diameter .0178 Quarter diameter .024 Tennis ball diameter .067 Soccer ball diameter .22 Average person 1.78 Central Park width 800 Mount Everest 8848 Moon radius 1737000 .273 Mars radius 3390000 .532 Earth radius 6371000 1.0 Jupiter radius 6.991⋅10^{7} 10.9 Moon distance 3.844⋅10^{8} 60.3 .00257 1.5 seconds Sun radius 6.957⋅10^{8} 109 .00474 2.3 seconds Earth orbit 1.496⋅10^{11} 23481 1.0 8 minutes Jupiter orbit 5.2 40 minutes Neptune orbit 30.1 3 days Light year 9.461⋅10^{15} 63241 1 year Alpha Centauri 4.4 years Nearest star Galaxy thickness 1000 years Galaxy center 27200 years Galaxy diameter 100000 years Andromeda distance 2.54 million years Virgo cluster distance 54 million years Size of universe 14 billion years
meters/second Mach Walk 1.5 Running sprint 10 Cycling sprint 20 Cheetah 30 Fastest land animal 70 miles/hour 31 Baseball pitch 45 100 miles/hour Human neuron 100 Sound at altitude 295 Speed of sound at altitude 10 km to 20 km 747 airplane 266 .9 Sound at sea level 340 1.0 At sea level and 15 degrees Celsius F35 Lightning 475 1.6 Stealth fighter F16 Falcon 590 2.0 Concorde 606 2.05 F22 Raptor 670 2.3 Stealth fighter F15 Eagle 740 2.5 SR71 Blackbird 980 3.3 Orbit speed 7800 26.4 Minimum speed to orbit the Earth Escape speed 11200 38.0 Minimum speed to escape the Earth's gravity Ion rocket 100000 Fastest spacecraft we can build Fission rocket 10^{7} Fusion rocket 10^{7} Light 3⋅10^{8} 1020000Aircraft typical fly at altitude 10 km to 20 km, where the speed of sound is 295 m/s. Mach 1 for aircraft is defined using this speed.
kg Earth Solar masses masses Electron 9.109⋅10^{31} Proton 1.673⋅10^{27} Neutron 1.675⋅10^{27} 1 ounce .0283 Tennis ball .058 Soccer ball .44 1 pound .454 Typical human 67 Sumo wrestler 230 Ton 1000 Honda Civic 1200 Elephant 5000 Bradley tank 27000 Argentinosaurus 70000 Largest dinosaur Blue whale 200000 Moon 7.35⋅10^{22} .0123 Mars 6.42⋅10^{23} .107 Earth 5.92⋅10^{24} 1 Jupiter 1.90⋅10^{27} 318 .00096 Sun 1.99⋅10^{30} 330000 1.0 White dwarf max 2.9⋅10^{30} 1.44 Milky Way black hole7.4⋅10^{36} 4.2 million Milky Way 2.5⋅10^{42} 1.2 trillion Andromeda 2.5⋅10^{42} 1.2 trillion M87 galaxy 10 trillion Virgo galaxy cluster 1200 trillion
Mass Diameter Height Density Price/kg Copper Nickel Zinc Manganese g mm mm g/cm^{3} $/kg fraction fraction fraction fraction Penny 2.5 19.05 1.52 5.77 4.0 .025 .975 Nickel 5.000 21.21 1.95 7.26 10.0 .75 .25 Dime 2.268 17.91 1.35 4.62 44.1 .9167 .0833 Quarter 5.670 24.26 1.75 6.29 44.1 .9167 .0833 Half dollar 11.340 30.61 2.15 7.90 44.1 .9167 .0833 Dollar 8.100 26.5 2.00 7.53 123.5 .885 .02 .06 .035 Dollar bill 1.0 .11 .88 1000The above objects are all to scale. The dimensions of a dollar bill are 155.956 mm * 66.294 mm * .11 mm.
Mass = M Diameter = D Height = H Volume = Vol = π H D^{2} / 4 Density = M / VolThe density of metals is:
Density Price Year of g/cm^{3} $/kg discovery Air .0012 Water 1.0 Magnesium 1.74 2.8 1808 Aluminum 2.70 1.7 1827 Titanium 4.51 10 1910 Zinc 7.14 2.0 1300 Manganese 7.21 2.3 1774 Iron 7.9 .3 1200 Nickel 8.91 15 1751 Copper 8.96 6 5000 Silver 10.49 640 Ancient Lead 11.3 2 6500 Tungsten 19.25 50 1783 Gold 19.30 43000 Ancient Platinum 21.45 37000 1735 Osmium 22.59 12000 1803 Densest elementGold was the densest element known until the discovery of tungsten in 1783 and was hence valuable as an uncounterfeitable currency. Silver can be counterfeited with lead because lead is more dense and cheaper than silver.
The price of the metal in a penny is
Metal price = Penny mass * (Copper fraction * Copper price/kg + Zinc fraction * Zinc price/kg) = .0025 kg * ( .025 6 $/kg .975 2 $/kg ) = .0052 $For a penny made of pure copper the price of the metal is 1.5 cents. A penny made of gold, silver, or zinc has a value of:
Price/Mass Price $/kg $ Zinc 2 .005 Copper 6 .015 Silver 640 1.6 Gold 43000 108
Ball Ball Court Court Ball diameter Mass length width density (mm) (g) (m) (m) (g/cm^{3}) Ping pong 40 2.7 2.74 1.525 .081 Squash 40 24 9.75 6.4 .716 Golf 43 46 1.10 Badminton 54 5.1 13.4 5.18 .062 Racquetball 57 40 12.22 6.10 .413 Billiards 59 163 2.84 1.42 1.52 Tennis 67 58 23.77 8.23 .368 Baseball 74.5 146 .675 Pitcherbatter dist. = 19.4 m Hockey puck 76 163 61 26 1.44 25 mm thick Whiffle 76 45 .196 Football 178 420 91.44 48.76 .142 Rugby 191 435 100 70 .119 Bowling 217 7260 18.29 1.05 1.36 Soccer 220 432 105 68 .078 Basketball 239 624 28 15 .087 Cannonball 220 14000 7.9 For an iron cannonball
grams/cm^{2} Air on Mars .00002 Air at Everest .0004 10 km altitude Air at Denver .001 1 Mile altitude Air at sea level .00127 Ice .92 Water 1.0 Rock 2.5 Magnesium 1.7 Aluminum 2.7 Titanium 4.5 Iron 7.9 Silver 10.5 Lead 11.3 Gold 19.3 Tungsten 19.3 Osmium 22.6 Densest element Earth 5.52 Moon 3.35 Mars 3.95 Europa 3.103 Ganymede 1.94 Callisto 1.83 Titan 1.88 Balsa .12 Corkwood .21 Cedar .32 Pine .37 Spruce, red .41 Oak, red .66 Hickory .81 Bamboo .85 Oak, live .98 Ironwood 1.1 Lignum Vitae 1.26
For a 1 kg mass, Newton's law is:
Mass = M = 1 kg Gravitational acceleration = g = 9.8 m/s^{2} Gravitational force = F = M g = 9.8 Newtons = 2.203 poundsThe pound is a unit of force. 1 pound = 4.448 Newtons. In Earth gravity a mass of .454 kg produces a force of 1 pound.
Mass = M = .454 kg Gravitational acceleration = g = 9.8 m/s^{2} Gravitational force = F = M g = 4.448 Newtons = 1 pound
Meters/second^{2} Ceres gravity .27 Europa gravity 1.31 Titan gravity 1.35 Moon gravity 1.62 Mars gravity 3.8 Venus gravity 8.87 Earth gravity 9.8 Bugatti Veyron 15.2 0 to 100 km/h in 2.4 seconds Red out 30 Max longterm acceleration in the direction of blood rushing to your head Blackout 50 Max longterm acceleration while sitting Formula1 car 50 Highspeed breaking and cornering with a downforce wing Blackout with g suit 90 Max longterm acceleration while sitting with a gsuit Max longterm (front) 120 Max longterm acceleration while lying on one's front Max longterm (back) 170 Max longterm acceleration while lying on one's back Max shortterm 500 Max shortterm acceleration Bullet 310000 9x19 Parabellum handgun, average acceleration along the barrel
Energies in MJoules = 10^{6} Joules
Raise 1 kg of water by 1 Kelvin = .00419 = 1 Food Calorie = 1000 calories Sprinting person = .004 (80 kg moving at 10 m/s) Phone battery = .018 (5 Watt hours) Laptop battery = .180 (50 Watt hours) 1 kg of supercapacitors = .04 1 kg of Lithium battery = 1.3 1 kg of TNT = 4.2 1 kg of sugar = 20 = 5000 Food Calories 1 kg of protein = 20 = 5000 Food Calories 1 kg of alcohol = 25 = 6000 Food Calories 1 kg of fat = 38 = 10000 Food Calories 1 kg of gasoline = 48 = 13000 Food Calories Uranium fission bomb (Little boy) = 7⋅10^{7} = 16 kilotons of TNT Plutonium fission bomb (Trinity) = 8⋅10^{7} = 20 kilotons of TNT Uranium fission bomb (Fat man) = 9⋅10^{7} = 22 kilotons of TNT Fusion bomb = 4⋅10^{10} = 10 megatons of TNT 1 kg of antimatter = 9⋅10^{10} = 20 megatons of TNT Krakatoa volcano, 1883 = 8⋅10^{11} World energy production in 1 year = 6⋅10^{14} Dinosaurextinction asteroid = 5⋅10^{17} Typical energy of a supernova = 1⋅10^{38} Intense gamma ray burst = 1⋅10^{41}
Watts Human cell 10^{12} Laptop computer 10 Human brain 20 Incandescent Light bulb 60 Human at rest 100 1 horsepower 746 Strenuous exercise 1000 Maximum human power 1600 World power per person 2500 Tesla S Ludicrous 397000 532 horsepower Wind turbine 1⋅10^{6} Blue whale 2.5⋅10^{6} Boeing 747 1.4⋅10^{8} Hoover Dam 2.1⋅10^{9} U.S. power consumption 3.4⋅10^{12} World power consumption 1.5⋅10^{13} Earth geologic heat 4.4⋅10^{13} World photosynthesis 7.5⋅10^{13} Hurricane 1.0⋅10^{14} Earth solar power 1.7⋅10^{17} Total solar power falling on the Earth
Kelvin Celsius Fahrenheit Absolute zero 0 273.2 459.7 Water melting point 273.2 0 32 Room temperature 294 21 70 Human body temperature 310 37 98.6 Water boiling point 373.2 100 212 Kelvin Absolute zero 0 Helium boiling point 4.2 Hydrogen boiling point 20.3 Triton 38 Pluto 44 Titania 70 Nitrogen boiling point 77.4 Oxygen boiling point 90.2 Titan 94 Europa 102 Hottest superconductor 135 HgBaCaCuO Ceres 168 Mars 210 Water melting point 273.15 Earth average 288 Room temperature 293 Water boiling point 373.15 Venus 740 Wood fire 1170 Copper melting point 1358 Iron melting point 1811 Bunsen burner 1830 Tungsten melting point 3683 Highest melting point among metals Earth's core 5650 Innercore boundary Sun's surface 5780 Solar core 13.6 million Helium4 fusion 200 million Carbon12 fusion 230 million
Pressure in bars Air on Mars .0063 Air at Everest .30 10 km altitude Air at Denver .8 1.6 km altitude Air at sea level 1.0 15 pounds per square inch Air on Titan 1.46 Inside a football 1.9 15 + 13 pounds per square inch 10 meters underwater 2.0 Air on Venus 92.1 Seawolf nuclear sub 50 Maximum depth of 490 meters 11 km underwater 1100 Mariana trench, deepest part of the ocean1 bar = 101300 Pascals
Energy density (MegaJoules/kg) Antimatter 90 billion Hydrogen bomb 25,000,000 theoretical maximum yield Hydrogen bomb 21,700,000 highest achieved yield Uranium 20,000,000 nuclear fuel Hydrogen 143 Natural gas 53.6 Gasoline 47 Jet fuel 43 Fat 37 Coal 24 Carbohydrates & sugar 17 Protein 16.8 Wood 16 Lithiumair battery 9 TNT 4.6 Gunpowder 3 Lithium battery 1.3 Lithiumion battery .72 Alkaline battery .59 Compressed air .5 300 atmospheres Supercapacitor .1 Capacitor .00036The energy cost to convert water to hydrogen and oxygen is 13.16 MJ/kg. If hydrogen and oxygen are reacted to produce one kg of water, the energy produced is equivalent to a 1 kg mass moving at 5.13 km/s.
Frequency Note (Hertz) Whale songs 10 Human ear lower limit 20 Bass lowest note 41 E Bass guitar lowest note 41 E Cello lowest note 65 C Bass singer lowest note 82 E Viola & tenor lowest note 131 C Violin & alto lowest note 196 G Soprano lowest note 262 C Violin D string 293 D Violin A string 440 A Violin E string 660 E Human ear upper limit 20000
terameter = Tm = 10$12$ meters gigameter = Bm = 10$9$ meters megameter = Mm = 10$6$ meters kilometer = km = 10$3$ meters meter = m = 10$0$ meters centimeter = cm = 10$2$ meters millimeter = mm = 10$3$ meters micrometer = μm = 10$6$ meters nanometer = nm = 10$9$ meters picometer = pm = 10$12$ meters femtometer = fm = 10$15$ meters 1 million kg = 1 Mkg 1 million dollars = 1 M$
Examples of scientific notation.
1 = 10^{0} = e0 10 = 10^{1} = e1 100 = 10^{2} = e2 123 = 1.23⋅10^{2} = 1.23e2 0.123 = 1.23⋅10^{1} = 1.23e1 11000 * .012 = 1.1⋅10^{4} * 1.2⋅10^{2} = 1.32⋅10^{2} = 132The abbreviation "e" for "10^" comes from Fortran and is standard in all programming languages.
A measurement consists of a quantity and an estimated error. For example, you might measure the length of a room to be
Length = 6.35 ± .02 meters"6.35" is the measurement and ".02" is the estimated error.
Care should be taken to use an appropriate number of digits. For example,
Length = 6.3 ± .02 meters Not enough digits in the measured quantity Length = 6.34 ± .02 meters Minimum number of digits to state the measured quantity Length = 6.342 ± .02 meters It is wise to to include an extra digit Length = 6.3421 ± .02 meters Too many digits. The last digit is unnecessary.The fractional error is defined as
Fractional error = Error / Measured quanitity = .02 / 6.34 = .0032Rounding:
6.3424 → 6.342 6.3425 → 6.342 6.3426 → 6.343If the last digit is even then round down, and if odd then round up. This prevents bias in rounding. For example:
6.3405 → 6.340 6.3415 → 6.342 6.3425 → 6.342 6.3435 → 6.344 6.3445 → 6.344
1 mile = 1609 meters 1 hour = 3600 seconds 1609 meters 1 hour 1 mile/hour = 1 mile/hour *  *  = .447 meters/second 1 mile 3600 seconds
If you have data that is not in SI units, then the safest procedure is to convert everything to SI units do the calculation. You can't go wrong with this. For example, if a car moving at 70 mph travels for 2 hours, how far does it go?
Speed of a car = V = 70 mph = 31.3 meters/second Time traveled = T = 2 hours = 7200 seconds Distance traveled = X = V T = 140 miles = 225360 metersOne first converts 70 mph and 2 hours to SI units, then apply X=VT to arrive at X=225360 meters, and then convert this to mph.
Alternatively, you can do the calculation in nonSI units but care must be taken to make sure the units are consistent.
The logarithm is the inverse of the exponential function.
log_{10} 0 = ∞ 10^{2} = .01 log_{10} .01 = 2 10^{1} = .1 log_{10} .1 = 1 10^{0} = 1 log_{10} 1 = 0 10^{1} = 10 log_{10} 10 = 1 10^{2} = 100 log_{10}100 = 2 10^{log10x} = log_{10} 10^{x} = x e^{ln x} = ln e^{x} = x e^{1} = .368 e^{0} = 1 e^{1} = 2.718 e^{2} = 7.389 e^{3} = 20.086
Many phenomena are besty understood by constructing a 2D table of numbers. For example, suppose you're wondering how to compare the alcohol content of a 6 pack of beer, a bottle of wine, and a bottle of Scotch. The following section constructs a table to show the alcohol content.
A typical bottle of beer has a volume of 12 ounces, is 5% alcohol, and contains
.6 ounces of alcohol. We use this amount as a reference unit and define
.6 ounces of alcohol to be one "Bond".
Volume of the drink = V Fraction of alcohol = F Volume of alcohol = V_{alc} = F V Volume of one beer = V_{beer} = 12 ounces Fraction of alcohol in beer = F_{beer} = .05 Volume of alcohol in one beer = V_{Bond} = .6 ounces One ounce = 29.6 mL One "Bond" of alcohol = .6 ounces One wine or Scotch bottle = 25.4 ounces = 750 ml Alcohol Volume Alcohol Alcohol $ $/Bond fraction (oz) (oz) (Bonds) Beer (12 oz) .05 12 .6 1 .67 .67 Budweiser Wine glass .13 4.6 .6 1 8 8.0 Napa Valley Scotch shot .40 1.5 .6 1 8 8.0 Laphroaig Beer pitcher .05 64 3.2 5.3 16 3.0 Budweiser Beer keg .05 1984 99.2 165.3 100 .60 Budweiser Wine bottle .13 25.4 3.3 5.5 3 .55 Charles Shaw Vodka bottle .40 25.4 10.1 16.9 15 .89 Smirnoff Scotch bottle .40 25.4 10.1 16.9 50 3.0 Laphroaig Distilled ethanol .95 25.4 24.1 40.2 15 .37 Everclear
Speed of light 2.9979e8 m/s Gravitational constant 6.6738e11 m^{3}/kg/s^{2} Planck constant 6.6261e34 J s Earth surface gravity 9.8067 m/s Electric force constant 8.9876e9 N m^{2} / C^{2} Magnetic constant 4 Pi e7 N/A^{2} Proton mass 1.6726e27 kg = 938.272 GeV Neutron mass 1.6749e27 kg = 939.565 GeV Electron mass 9.1094e31 kg Electron charge 1.6022e19 C Atomic mass unit 1.6605e27 kg Bohr radius 5.2918e11 m = hbar^{2} / (ElectronMass*ElectronCharge^{2}*Ke) Boltzmann constant 1.3806e23 J/K Avogadro number 6.0221e23 particles/mole Gas constant 8.3145 J/K/mole StefanBoltzmann constant 5.6704e8 Watts/m^{2}/K^{4} Wein constant 2.8978e3 m K Mole of Carbon12 .012 kg Exact Planck length 1.6162e35 m Planck mass 2.1765e8 kg Planck time 5.3911e44 s Planck charge 1.8755e18 C Planck temperature 1.4168e32 K Water heat capacity 4200 J/kg/K Steam heat capacity 2080 J/kg/K At 100 Celsius Ice heat capacity 2110 J/kg/K At 10 Celsius Air heat capacity 1004 J/kg/K StefanBoltzmann 5.67e8 Watts/meter^{2}/Kelvin^{4} = (2π^{5}/15) Boltzmann^{4} / SpeedOfLight^{2} / PlanckConstant^{3} Wein 2.898e3 Kelvin meters Electron spin 5.2729e35 Joule seconds = PlanckConstant / (4 Pi) Pi 3.14159 Euler number 2.71828