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.
Imperial 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 not fully adopted the metric system. The US uses metric for science and medicine, but not for everyday life.
Metric is the default for this site, but we will sometimes have to convert into metric at the start of some problems.
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".
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
$$10 \, \mathrm{\mu s} = 0{\color{DodgerBlue}.}\overleftarrow{\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{0}\undergroup{1}\undergroup{0}}.0\, \mathrm{s}= 0.000\,01 \, \mathrm{s}$$
Example: Watch this entire video.
After the video is over figure out how many views it has?
Also how many subscribers, and how many thumbs up / likes?
solution
In the summer of 2026 it had 17B views.
Both B and G stand for 1 000 000 000.
$$17 \times 10^9 = 17\,000\,000\,000\,\mathrm{views}$$
It had 84.4M subs and 46M likes.
M stands for 1 000 000.
$$84.4 \times 10^6 = 84\,400\,000\,\mathrm{subs}$$
$$46 \times 10^6 = 46\,000\,000\,\mathrm{likes}$$
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.
Example: A marathon is 26.2 miles. How far is a marathon in kilometers?
(1 mile = 1.6 kilometers)
solution
$$26.2 \, \textcolor{DeepPink}{ \mathrm{mile}} \left( \frac{1.6 \, \mathrm{km}}{1 \,\textcolor{DeepPink}{ \mathrm{mile}}} \right) = 41.92\, \mathrm{km}$$
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 trigfunctions (SOH-CAH-TOA) to find how the vector components relate to the angle of the vector.
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
In case you wanted more practice I used AI to make some more problems. The rest of the site I made by hand, but generating endless problems seemed safe. I did find mistakes in the AI generated problems, and there are probably some I didn't find. Let me know if something could be fixed. I also added a practice problem on each page with no solution. That's intentional. Have fun!
Example: A trail sign says the lake is 3.4 km away. Convert 3.4 km to meters.
solution
$$3.4\,\mathrm{km}\left(\frac{1000\,\mathrm{m}}{1\,\mathrm{km}}\right)=3400\,\mathrm{m}$$
Example: Convert 0.075 km to meters.
solution
$$0.075\,\mathrm{km}\left(\frac{1000\,\mathrm{m}}{1\,\mathrm{km}}\right)=75\,\mathrm{m}$$
Example: A lab table is listed as 820 cm long in an old inventory sheet. Convert 820 cm to meters.
solution
$$820\,\mathrm{cm}\left(\frac{1\,\mathrm{m}}{100\,\mathrm{cm}}\right)=8.2\,\mathrm{m}$$
Example: Convert 45 mm to meters.
solution
$$45\,\mathrm{mm}\left(\frac{0.001\,\mathrm{m}}{1\,\mathrm{mm}}\right)=0.045\,\mathrm{m}$$
Example: A camera flash lasts 12 ms, which is much shorter than a blink. Convert 12 ms to seconds.
solution
$$12\,\mathrm{ms}\left(\frac{0.001\,\mathrm{s}}{1\,\mathrm{ms}}\right)=0.012\,\mathrm{s}$$
Example: Convert 350 μs to seconds.
solution
$$350\,\mathrm{\mu s}\left(\frac{10^{-6}\,\mathrm{s}}{1\,\mathrm{\mu s}}\right)=0.00035\,\mathrm{s}$$
Example: Red light from a small laser pointer has wavelength about 650 nm. Convert 650 nm to meters.
solution
$$650\,\mathrm{nm}\left(\frac{10^{-9}\,\mathrm{m}}{1\,\mathrm{nm}}\right)=6.5 \times 10^{-7}\,\mathrm{m}$$
This is a very small distance, so a negative power of ten makes sense.
Example: Convert 4.2 Mm to meters.
solution
$$4.2\,\mathrm{Mm}\left(\frac{10^6\,\mathrm{m}}{1\,\mathrm{Mm}}\right)=4.2 \times 10^6\,\mathrm{m}$$
Example: Convert 0.006 m to millimeters.
solution
$$0.006\,\mathrm{m}\left(\frac{1000\,\mathrm{mm}}{1\,\mathrm{m}}\right)=6\,\mathrm{mm}$$
Example: Convert 1500 m to kilometers.
solution
$$1500\,\mathrm{m}\left(\frac{1\,\mathrm{km}}{1000\,\mathrm{m}}\right)=1.5\,\mathrm{km}$$
Example: Convert 0.0045 s to milliseconds.
solution
$$0.0045\,\mathrm{s}\left(\frac{1000\,\mathrm{ms}}{1\,\mathrm{s}}\right)=4.5\,\mathrm{ms}$$
Example: Convert 0.000 0032 s to microseconds.
solution
$$0.0000032\,\mathrm{s}\left(\frac{1\,\mathrm{\mu s}}{10^{-6}\,\mathrm{s}}\right)=3.2\,\mathrm{\mu s}$$
Example: A sensor records 18 counts in 3.0 s. Convert the reading to meters.
answer
This cannot be solved from the information given. The numbers are given, but counts are not a length unit. You would need a conversion factor from counts to meters.
Example: Convert 0.75 L to cubic meters. Use 1 L = 0.001 m³.
solution
$$0.75\,\mathrm{L}\left(\frac{0.001\,\mathrm{m^3}}{1\,\mathrm{L}}\right)=0.00075\,\mathrm{m^3}$$
Example: Convert 12 kg m/s² into the special unit abbreviation.
solution
$$1\,\mathrm{N}=1\,\mathrm{kg\,m/s^2}$$
$$12\,\mathrm{kg\,m/s^2}=12\,\mathrm{N}$$
Example: Convert 18 minutes to seconds.
solution
$$18\,\mathrm{min}\left(\frac{60\,\mathrm{s}}{1\,\mathrm{min}}\right)=1080\,\mathrm{s}$$
Example: Convert 2.5 hours to seconds.
solution
$$2.5\,\mathrm{hr}\left(\frac{60\,\mathrm{min}}{1\,\mathrm{hr}}\right)\left(\frac{60\,\mathrm{s}}{1\,\mathrm{min}}\right)=9000\,\mathrm{s}$$
Example: Convert 5400 s to hours.
solution
$$5400\,\mathrm{s}\left(\frac{1\,\mathrm{min}}{60\,\mathrm{s}}\right)\left(\frac{1\,\mathrm{hr}}{60\,\mathrm{min}}\right)=1.5\,\mathrm{hr}$$
Example: A car is moving 72 km/hr on a road with a 55 mph speed limit sign nearby. Convert 72 km/hr to m/s.
solution
$$72\,\mathrm{km/hr}\left(\frac{1000\,\mathrm{m}}{1\,\mathrm{km}}\right)\left(\frac{1\,\mathrm{hr}}{3600\,\mathrm{s}}\right)=20\,\mathrm{m/s}$$
Example: Convert 25 m/s to km/hr.
solution
$$25\,\mathrm{m/s}\left(\frac{1\,\mathrm{km}}{1000\,\mathrm{m}}\right)\left(\frac{3600\,\mathrm{s}}{1\,\mathrm{hr}}\right)=90\,\mathrm{km/hr}$$
Example: Convert 55 miles per hour to meters per second. Use 1 mile = 1.6 km.
solution
$$55\,\mathrm{mi/hr}\left(\frac{1.6\,\mathrm{km}}{1\,\mathrm{mi}}\right)\left(\frac{1000\,\mathrm{m}}{1\,\mathrm{km}}\right)\left(\frac{1\,\mathrm{hr}}{3600\,\mathrm{s}}\right)=24.4\,\mathrm{m/s}$$
Example: A sprinter runs 100 m in 9.8 s. What is the sprinter's average speed in m/s and km/hr?
solution
$$v = \frac{\Delta x}{\Delta t}$$
$$v = \frac{100\,\mathrm{m}}{9.8\,\mathrm{s}}$$
$$v = 10.2\,\mathrm{m/s}$$
$$10.2\,\mathrm{m/s}\left(\frac{1\,\mathrm{km}}{1000\,\mathrm{m}}\right)\left(\frac{3600\,\mathrm{s}}{1\,\mathrm{hr}}\right)=36.7\,\mathrm{km/hr}$$
Example: Convert 6 ft to meters. Use 1 ft = 0.3048 m.
solution
$$6\,\mathrm{ft}\left(\frac{0.3048\,\mathrm{m}}{1\,\mathrm{ft}}\right)=1.83\,\mathrm{m}$$
Example: Convert 30 inches to centimeters. Use 1 inch = 2.54 cm.
solution
$$30\,\mathrm{in}\left(\frac{2.54\,\mathrm{cm}}{1\,\mathrm{in}}\right)=76.2\,\mathrm{cm}$$
Example: Convert 4.5 miles to kilometers. Use 1 mile = 1.6 km.
solution
$$4.5\,\mathrm{mi}\left(\frac{1.6\,\mathrm{km}}{1\,\mathrm{mi}}\right)=7.2\,\mathrm{km}$$
Example: Convert 5 yards to meters. Use 1 yard = 0.914 m.
solution
$$5\,\mathrm{yd}\left(\frac{0.914\,\mathrm{m}}{1\,\mathrm{yd}}\right)=4.57\,\mathrm{m}$$
Example: You walk 6 m east and then 8 m north. How far are you from where you started?
solution
$$a^2+b^2=c^2$$
$$(6)^2+(8)^2=c^2$$
$$100=c^2$$
$$c=10\,\mathrm{m}$$
Example: You walk 10 m north and then 10 m east. How far are you from where you started?
solution
$$a^2+b^2=c^2$$
$$(10)^2+(10)^2=c^2$$
$$200=c^2$$
$$c=14.1\,\mathrm{m}$$
Example: A plane is moving 20 m/s at 30° above the horizontal. How fast is it moving vertically?
solution
$$v_y = v\sin\theta$$
$$v_y = (20\,\mathrm{m/s})\sin(30\degree)$$
$$v_y = 10\,\mathrm{m/s}$$
Example: A plane is moving 20 m/s at 30° above the horizontal. How fast is it moving horizontally?
solution
$$v_x = v\cos\theta$$
$$v_x = (20\,\mathrm{m/s})\cos(30\degree)$$
$$v_x = 17.3\,\mathrm{m/s}$$
Example: You walk 50 m northeast. How far north and how far east did you walk?
solution
Northeast is 45° between north and east, so the components are equal.
Example: A displacement is 120 m at 60° above the x-axis. Find the x and y components.
solution
$$x=d\cos\theta$$
$$x=(120\,\mathrm{m})\cos(60\degree)$$
$$x=60\,\mathrm{m}$$
$$y=d\sin\theta$$
$$y=(120\,\mathrm{m})\sin(60\degree)$$
$$y=104\,\mathrm{m}$$
Example: A vector has an x-component of 12 m and a y-component of 5 m. What is the vector's magnitude?
solution
$$a^2+b^2=c^2$$
$$(12)^2+(5)^2=c^2$$
$$169=c^2$$
$$c=13\,\mathrm{m}$$
Example: A plane is moving 250 m/s at 40° above the horizontal. How fast is it moving vertically?
solution
$$v_y=v\sin\theta$$
$$v_y=(250\,\mathrm{m/s})\sin(40\degree)$$
$$v_y=161\,\mathrm{m/s}$$
Example: A long spacecraft tracking distance is written as 2.4 × 106 m. Convert it to megameters.
solution
$$2.4 \times 10^6\,\mathrm{m}\left(\frac{1\,\mathrm{Mm}}{10^6\,\mathrm{m}}\right)=2.4\,\mathrm{Mm}$$
Reading (8 minutes): Read On Being the Right Size by J. B. S. Haldane. Then answer these questions.
Haldane imagines a person made 10 times taller while keeping the same proportions. By what factor would the person's volume and weight increase? By what factor would the cross-sectional area of each leg increase?
answer
Volume, and therefore weight, would increase by a factor of 1000 because all three dimensions are 10 times larger. The cross-sectional area of a leg would increase by only a factor of 100.
Why does that comparison help explain why a giant person with ordinary human proportions would have trouble standing?
answer
The legs would need to support 1000 times as much weight, but their supporting cross-sectional area would be only 100 times larger. The stress on the leg material would therefore be much greater.
An animal is scaled so that its height, width, and length each double. What geometric change would help its legs support the larger body, and why do real large animals not simply look like enlarged small animals?
answer
Its legs need to become relatively thicker, giving them more cross-sectional area to support its weight. Real large animals change their proportions because weight grows faster with size than the area of similar supports.
Why do two right triangles with the same angle have the same sine and cosine ratios even when one triangle is much larger?
answer
The triangles have the same shape, so they are similar. Their corresponding side lengths grow by the same factor, leaving each side ratio unchanged.
How does trigonometry help turn a force or velocity vector into useful horizontal and vertical components?
answer
The vector and its two perpendicular components form a right triangle. Sine and cosine use the vector's angle and magnitude to find the sizes of those components.
The article lists force, velocity, acceleration, momentum, and fields as vectors. What information would be missing if one of these quantities were reported with only a number and no direction?
answer
Its magnitude would be known, but not how it acts in space. Direction is needed to combine vectors and predict the resulting motion or force.