Units let us know what property is being measured, and they tell us the magnitude scale.

Metric units follow powers of ten. There are 100 cm in a meter, and 1000 mm in a meter. English units have less predictable relationships. There are 3 feet in a yard, and 12 inches in a foot, and 36 inches in a yard.

Metric Units

The metric system was designed to be a unified and rational system of measures. The metric system improves calculation, communication, and conversion. Amazingly, it has been adopted by almost every country on Earth.

Annoyingly, United States have chosen to not fully adopt the metric system, but the American sciences do use metric. We use metric units on this site.

The metric system defines all units from seven base units. The base units come from measuring physical constants. For example, the meter is defined as the distance light travels in (1 / 299 792 458) seconds.

Quantity	Name	Symbol
time	second	s
length	meter	m
mass	kilogram	kg
current	ampere	A
temperature	kelvin	K
amount	mole	mol
light intensity	candela	cd

Derived units are combinations of these base units. For example, velocity is combination of length divided by time (m/s).

Quantity	Name	Symbols
area	meters squared	m²
volume	meters cubed	m³
velocity	meters per seconds	m/s
acceleration	meters per seconds squared	m/s²
momentum	kilogram meters per seconds	kg m/s

Some derived units get a special abbreviation normally written as a capital letter. For example, the unit of force is N, but it stands for kg m/s².

Quantity	Name	Abbreviation	Symbols
force	newton	N	kg m/s²
energy	joule	J	kg m²/s²
power	watt	W	kg m²/s³
frequency	hertz	Hz	1/s
volume	liter	L	10⁻³ m³

Metric Prefixes

We use metric prefixes to indicate multiplication or division by powers of ten. For example we can replace 1000 with the letter "k".

$$ 1000 \, \mathrm{m} = 1 \, \mathrm{km} $$ $$ 22\,500 \, \mathrm{m} = 22.5 \, \mathrm{km} $$

Name	Symbol	Power
giga	G	10⁹
mega	M	10⁶
kilo	k	10³
centi	c	10^-2
milli	m	10^-3
micro	μ	10^-6
nano	n	10^-9

Name	Symbol	Factor		Power
tera	T	1 000 000 000 000		10¹²
giga	G	1 000 000 000		10⁹
mega	M	1 000 000		10⁶
kilo	k	1 000		10³
centi	c		0.01	10^-2
milli	m		0.001	10^-3
micro	μ		0.000 001	10^-6
nano	n		0.000 000 001	10^-9
pico	p		0.000 000 000 001	10^-12

Converting 10 km into meters means multiplying by 1000.

$$10 \, \mathrm{km} = 10\left({\color{DeepPink}1000}\right) \mathrm{m} = 10\,000 \, \mathrm{m}$$ $$10\,000 \, \mathrm{m} = 10\,({\color{DeepPink}1000}) \, \mathrm{m} = 10 \, \mathrm{km}$$

Converting 10 ms into meters means multiplying by 0.001.

$$10 \, \mathrm{ms} = 10\left({\color{DodgerBlue}0.001 }\right) \mathrm{s} = 0.01\, \mathrm{s}$$ $$0.01 \, \mathrm{s} = 10\left({\color{DodgerBlue} 0.001 }\right) \mathrm{s} = 10\, \mathrm{ms}$$

Example: Convert 120 kilometers into meters.

solution

$$ \mathrm{k} = 1000 $$ $$120\, \mathrm{km} = 120(1000)\, \mathrm{m} = 120\,000\, \mathrm{m}$$

Example: How many meters is 450 nanometers?

solution

$$ \mathrm{n} = 10^{-9} $$ $$450\,\mathrm{nm} = 450\left( 10^{-9}\right) \mathrm{m} = 0.000\,000\,45 \, \mathrm{m}$$

In practice, conversions are just moving the decimal to the left for negative exponents and to the right for positive.

$$10 \, \mathrm{km} = 10.\overrightarrow{\undergroup{0}\undergroup{0}\undergroup{0}}{\color{DeepPink}.} \, \mathrm{m} = 10\,000 \, \mathrm{m}$$ $$10\,000 \, \mathrm{m} = 10{\color{DeepPink}.} \overleftarrow{\undergroup{0}\undergroup{0}\undergroup{0}}. \, \mathrm{m} = 10 \, \mathrm{km}$$
$$10 \, \mathrm{ms} = 0{\color{DodgerBlue}.}\overleftarrow{\undergroup{0}\undergroup{1}\undergroup{0}}.0 \, \mathrm{s}= 0.01 \, \mathrm{s}$$ $$0.01 \, \mathrm{s} = 0.\overrightarrow{\undergroup{0}\undergroup{1}\undergroup{0}}{\color{DodgerBlue}.}0 \, \mathrm{s}= 10 \, \mathrm{ms}$$

Example: How many meters is 10 Mm?

solution

$$10 \, \mathrm{Mm} = 10.\overrightarrow{\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}}{\color{DeepPink}.} \, \mathrm{m} = 10\,000\,000 \, \mathrm{m}$$

Example: How many seconds is 10 μs?

solution

$$10 \, \mathrm{\mu s} = 0{\color{DodgerBlue}.}\overleftarrow{\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{1}\undergroup{0}}.0\, \mathrm{s}= 0.000\,01 \, \mathrm{s}$$

Nonmetric Units

Converting outside the metric system is more complex. Other unit systems don't always use powers of ten, so we can't simply move the decimal left or right. To convert we need to multiply by a conversion fraction.

Build a conversion fraction.
Put the old unit on the bottom of the fraction.
Put the new unit on top of the fraction.
Find out how two numbers are equal. Add numbers to the top and bottom of the fraction so that they are equal.
Multiply the number you are converting by the conversion fraction. The old unit should cancel to leave just the new unit.

Let's convert 50 minutes into seconds.

$$ 50 \,\mathrm{min} \to \mathrm{s}$$

In order to cancel minutes we want to build a fraction with minutes on top and the seconds on the bottom.

$$ 50 \,\mathrm{min} \left( \mathrm{\frac{s}{min}} \right)$$

Our fraction can't change the actual value, so it must be equal to one. The top and bottom of the fraction must equal each other.

$$\text{{\color{DeepPink}1 minute} = \color{DodgerBlue}60 seconds}$$ $$ 50 \, \mathrm{min} \left( \frac{{\color{DodgerBlue}60\, \mathrm{s}} }{{\color{DeepPink}1\,\mathrm{ min}}} \right)$$ $$ 50 \, \cancel{\mathrm{min}} \left( \frac{60\, \mathrm{s}}{1 \,\cancel{\mathrm{min}}} \right)$$ $$ 50 \left( \frac{60\, \mathrm{s}}{1} \right)$$ $$3000\, \mathrm{s}$$

Example: Convert 50 minutes into hours.

solution

$$ 50 \,\mathrm{min} \to \mathrm{hour}$$ $$50\,\mathrm{\color{DeepPink}min} \left( \frac{1\,\mathrm{hour}}{60\,\mathrm{\color{DeepPink}min}} \right)$$ $$\frac{50}{60}\,\mathrm{hour}$$ $$0.8\overline{33}\,\mathrm{hour}$$

Example: A marathon is 26.2 miles. How far is a marathon in kilometers?
(1 mile = 1.6 kilometers)

solution

$$26.2 \, \textcolor{DeepPink}{ \mathrm{mile}} \left( \frac{1.6 \, \mathrm{km}}{1 \,\textcolor{DeepPink}{ \mathrm{mile}}} \right) = 41.92\, \mathrm{km}$$

Example: I'm 6 feet and 1 inch tall. How many meters tall am I?

solution

$$\mathrm{ft \to inches}$$ $$6\, \mathrm{ft} + 1\, \mathrm{in}$$ $$6\, \textcolor{DeepPink}{\mathrm{ft}}\left( \frac{12\,\mathrm{in}}{1\, \textcolor{DeepPink}{\mathrm{ft}}}\right) + 1\, \mathrm{in}$$ $$72\, \mathrm{in} + 1\, \mathrm{in}$$ $$73\, \mathrm{in}$$
$$\mathrm{inches \to meters}$$ $$\text {a google search returns: } 1 \, \mathrm{m} = 39.37 \, \mathrm{in}$$ $$73\,\textcolor{DeepPink}{ \mathrm{in}} \left( \frac{1\, \mathrm{m}}{39.37 \, \textcolor{DeepPink}{ \mathrm{in}}}\right) $$ $$73 \left( \frac{1\, \mathrm{m}}{39.37}\right) $$ $$1.85\, \mathrm{m}$$

Space and Time

Our universe has three dimensions of space and one dimension of time. The three different spacial dimensions make communicating information about space ambiguous.

Information about the three dimensions of space needs a direction in addition to the magnitude. The pairing of direction and magnitude is called a vector.

If I say "bike 10 km to get to the store", you still can't find the store because you don't know the direction. I should say "bike 10 km west".

Scalars

A scalar is a variable that has a magnitude, but not a direction in space. Temperature is a scalar. You couldn't say it is 50°C to the left.

Variable	Magnitude
air pressure	101.3 kPa
temperature	21° C
price	$50
speed	10 m/s
distance	3000 m

Speed and distance seem like they could have a direction, but they are defined as only the magnitude of velocity and displacement vectors.

Vectors →

A vector is a variable that has a magnitude and direction in space. Velocity is a vector. You could say a velocity is 10 m/s in the west direction.

Variable	Magnitude	Direction
displacement	10 m	west
displacement	5.5 m	up
velocity	20 m/s	20° above the x-axis
velocity	3 m/s	10 left
acceleration	9.8 m/s²	↓

Think of a vector like the hypotenuse of a right triangle. This makes the other two sides of the triangle the components of the vector. Often the components are horizontal and vertical, but they don't have to be.

We can use the pythagorean theorem to solve for the magnitude of the sides.

$$a^{2} + b^{2} = c^{2}$$ Example: You walk 3 miles north and then 4 miles west. How far away are you from your starting location?

solution

$$a^{2}+b^{2} = c^{2}$$ $$3^{2}+4^{2} = c^{2}$$ $$25 = c^{2}$$ $$5 \, \mathrm{miles} = c$$

Vectors and Angles

We can use trig functions (SOH-CAH-TOA) to find how the vector components relate to the angle of the vector.

$$ \sin \theta = \frac{\mathrm{opp}}{\mathrm{hyp}} $$ $$ \cos \theta = \frac{\mathrm{adj}}{\mathrm{hyp}} $$ $$ \tan \theta = \frac{\mathrm{opp}}{\mathrm{adj}} $$

$$ \mathrm{adj} = (\mathrm{hyp}) \cos \theta \quad \quad \mathrm{opp} = (\mathrm{hyp}) \sin \theta $$

For a velocity vector the hypotenuse is v. The adjacent and opposite sides of the triangle are the x and y part of v.

$$ v_{x} = v\,\cos \theta$$ $$ v_{y} = v\,\sin \theta $$

For displacement the equations are the same

$$ x = d\,\cos \theta $$ $$ y = d\,\sin \theta $$

In the simulation below position the mouse to around magnitude 250 and angle 14°. What are the x and y components of the vector?

At what angles are the magnitude and the x-component equal?

What does a negative sign mean for a vector?

Example: A plane is taking off at an angle of 14° above the horizon. If the plane is moving at 250 m/s how fast is it moving in only the vertical direction?

solution

$$ v_{y} = (v) \sin \theta $$ $$ v_{y} = (250 \, \mathrm{\tfrac{m}{s}}) \sin(14 \degree) $$ $$ v_{y} = 60 \mathrm{\tfrac{m}{s}} $$

Example: You want to walk to the closest pokémon gym. The compass on your phone says you have to walk northeast. You arrive at the gym after walking 75 meters. Sadly you find that you need to be level 5, but you are level 3. How far north did you walk?

solution

Northeast is an angle of 45 degrees to the north.

$$ d_\mathrm{north} = d \cos \theta $$ $$ d_\mathrm{north} = (75 \, \mathrm{m}) \cos(45 \degree) $$ $$ d_\mathrm{north} = 53 \, \mathrm{m} $$