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".

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

Name	Symbol	Power
giga	G,B	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,B	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 milliseconds into ___ seconds 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

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

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.

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

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.

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

practice problems (35)

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!

printout.pdf show topics and mark harder problems

Example: A trail sign says the lake is 3.4 km away. How far is that in meters?

solution

$$\mathrm{k} = 1000$$ $$3.4\,\mathrm{km} = 3.4(1000)\,\mathrm{m}$$ $$3.4\,\mathrm{km} = 3400\,\mathrm{m}$$

Example: Convert 0.075 km to meters.

solution

$$\mathrm{k} = 1000$$ $$0.075\,\mathrm{km} = 0.075(1000)\,\mathrm{m}$$ $$0.075\,\mathrm{km} = 75\,\mathrm{m}$$

Multiplying by 1000 moves the decimal three places to the right.

Example: A lab table is listed as 180 cm long in an old inventory sheet. How long is it in meters?

solution

$$\mathrm{c} = 0.01$$ $$180\,\mathrm{cm} = 180(0.01)\,\mathrm{m}$$ $$180\,\mathrm{cm} = 1.8\,\mathrm{m}$$

That's about as tall as a person, which is a reasonable length for a lab table.

Example: A AA battery is about 50 mm long. How long is it in meters?

solution

$$\mathrm{m} = 0.001$$ $$50\,\mathrm{mm} = 50(0.001)\,\mathrm{m}$$ $$50\,\mathrm{mm} = 0.05\,\mathrm{m}$$

Example: In a dim room, a phone camera keeps its shutter open for 12 ms to take a photo. A blink takes about 100 ms. How long is the shutter open in seconds?

solution

$$\mathrm{m} = 0.001$$ $$12\,\mathrm{ms} = 12(0.001)\,\mathrm{s}$$ $$12\,\mathrm{ms} = 0.012\,\mathrm{s}$$

The shutter is open for less than a blink, but long enough that a shaky hand can blur the photo.

Example: Convert 350 μs to seconds.

solution

$$\mathrm{\mu} = 10^{-6}$$ $$350\,\mathrm{\mu s} = 350\left(10^{-6}\right)\mathrm{s}$$ $$350\,\mathrm{\mu s} = 0.000\,35\,\mathrm{s}$$

Moving the decimal six places to the left takes 350 to 0.000 350.

Example: Red light from a small laser pointer has a wavelength of about 650 nm. How long is that in meters?

solution

$$\mathrm{n} = 10^{-9}$$ $$650\,\mathrm{nm} = 650\left(10^{-9}\right)\mathrm{m}$$ $$650\,\mathrm{nm} = 0.000\,000\,65\,\mathrm{m}$$

In scientific notation this is 6.5 × 10^-7 m. It's a very small distance, so a negative power of ten makes sense.

Example: In a straight line, New York is about 3.9 Mm from Los Angeles. How far is that in meters?

solution

$$\mathrm{M} = 10^{6}$$ $$3.9\,\mathrm{Mm} = 3.9\left(10^{6}\right)\mathrm{m}$$ $$3.9\,\mathrm{Mm} = 3\,900\,000\,\mathrm{m}$$

That's 3900 km. Megameters are rarely used, but they fit distances across a country or planet well.

Example: Convert 0.006 m to millimeters.

solution

$$\mathrm{m} = 0.001$$ $$0.006\,\mathrm{m} = 6(0.001)\,\mathrm{m}$$ $$0.006\,\mathrm{m} = 6\,\mathrm{mm}$$

Example: The 1500 m is a common track race. How many kilometers is it?

solution

$$\mathrm{k} = 1000$$ $$1500\,\mathrm{m} = 1.5(1000)\,\mathrm{m}$$ $$1500\,\mathrm{m} = 1.5\,\mathrm{km}$$

A mile is 1609 m, so the 1500 m is a little shorter than a mile.

Example: Convert 0.0045 s to milliseconds.

solution

$$\mathrm{m} = 0.001$$ $$0.0045\,\mathrm{s} = 4.5(0.001)\,\mathrm{s}$$ $$0.0045\,\mathrm{s} = 4.5\,\mathrm{ms}$$

Example: Convert 0.000 003 2 s to microseconds.

solution

$$\mathrm{\mu} = 10^{-6}$$ $$0.000\,003\,2\,\mathrm{s} = 3.2\left(10^{-6}\right)\mathrm{s}$$ $$0.000\,003\,2\,\mathrm{s} = 3.2\,\mathrm{\mu s}$$

Example: A bike wheel turns 250 times during a 3 minute ride. How far did the bike travel in meters?

solution

This cannot be solved from the information given. Each turn of the wheel moves the bike forward one wheel circumference, the distance around the tire. We don't know the size of the wheel, so we can't convert turns into meters. The 3 minutes doesn't help either.

Example: A water bottle holds 0.75 L. How much is that in cubic meters? (1 L = 0.001 m³)

solution

$$0.75\,\textcolor{DeepPink}{\mathrm{L}}\left(\frac{0.001\,\mathrm{m^{3}}}{1\,\textcolor{DeepPink}{\mathrm{L}}}\right)$$ $$0.000\,75\,\mathrm{m^{3}}$$

A cubic meter is huge compared to a water bottle. It would hold about 1300 of these bottles.

Example: Write 12 kg m/s² using the special unit abbreviation.

solution

The newton is the special abbreviation for kg m/s².

$$1\,\mathrm{N} = 1\,\mathrm{kg\,\tfrac{m}{s^{2}}}$$ $$12\,\mathrm{kg\,\tfrac{m}{s^{2}}} = 12\,\mathrm{N}$$

The number doesn't change. N is just a shorter way to write the same unit.

Example: Convert 18 minutes to seconds.

solution

$$18\,\textcolor{DeepPink}{\mathrm{min}}\left(\frac{60\,\mathrm{s}}{1\,\textcolor{DeepPink}{\mathrm{min}}}\right)$$ $$1080\,\mathrm{s}$$

Example: A movie is 2.5 hours long. How many seconds is that?

solution

$$2.5\,\textcolor{DeepPink}{\mathrm{hr}}\left(\frac{60\,\textcolor{DodgerBlue}{\mathrm{min}}}{1\,\textcolor{DeepPink}{\mathrm{hr}}}\right)\left(\frac{60\,\mathrm{s}}{1\,\textcolor{DodgerBlue}{\mathrm{min}}}\right)$$ $$9000\,\mathrm{s}$$

Example: Convert 5400 s to hours.

solution

This time seconds are the old unit, so they go on the bottom of the fraction.

$$5400\,\textcolor{DeepPink}{\mathrm{s}}\left(\frac{1\,\textcolor{DodgerBlue}{\mathrm{min}}}{60\,\textcolor{DeepPink}{\mathrm{s}}}\right)\left(\frac{1\,\mathrm{hr}}{60\,\textcolor{DodgerBlue}{\mathrm{min}}}\right)$$ $$1.5\,\mathrm{hr}$$

Example: A car is moving 72 km/hr on a road with a 55 mph speed limit sign nearby. How fast is the car moving in m/s?

solution

$$72\,\mathrm{\tfrac{\textcolor{DeepPink}{km}}{\textcolor{DodgerBlue}{hr}}}\left(\frac{1000\,\mathrm{m}}{1\,\textcolor{DeepPink}{\mathrm{km}}}\right)\left(\frac{1\,\textcolor{DodgerBlue}{\mathrm{hr}}}{3600\,\mathrm{s}}\right)$$ $$20\,\mathrm{\tfrac{m}{s}}$$

72 km/hr is about 45 mph, so the car is under the speed limit.

Example: Convert 25 m/s to km/hr.

solution

$$25\,\mathrm{\tfrac{\textcolor{DeepPink}{m}}{\textcolor{DodgerBlue}{s}}}\left(\frac{1\,\mathrm{km}}{1000\,\textcolor{DeepPink}{\mathrm{m}}}\right)\left(\frac{3600\,\textcolor{DodgerBlue}{\mathrm{s}}}{1\,\mathrm{hr}}\right)$$ $$90\,\mathrm{\tfrac{km}{hr}}$$

Example: Convert 55 miles per hour to meters per second. (1 mile = 1.6 km)

solution

$$55\,\mathrm{\tfrac{\textcolor{DeepPink}{mile}}{\textcolor{DodgerBlue}{hr}}}\left(\frac{1.6\,\textcolor{ForestGreen}{\mathrm{km}}}{1\,\textcolor{DeepPink}{\mathrm{mile}}}\right)\left(\frac{1000\,\mathrm{m}}{1\,\textcolor{ForestGreen}{\mathrm{km}}}\right)\left(\frac{1\,\textcolor{DodgerBlue}{\mathrm{hr}}}{3600\,\mathrm{s}}\right)$$ $$24.4\,\mathrm{\tfrac{m}{s}}$$

Example: In 2009, Usain Bolt set the 100 m world record with a time of 9.58 s. What was his average speed in m/s and in km/hr?

solution

Use the average velocity equation from the motion page.

$$v = \frac{\Delta x}{\Delta t}$$ $$v = \frac{100}{9.58}$$ $$v = 10.44\,\mathrm{\tfrac{m}{s}}$$
$$10.44\,\mathrm{\tfrac{\textcolor{DeepPink}{m}}{\textcolor{DodgerBlue}{s}}}\left(\frac{1\,\mathrm{km}}{1000\,\textcolor{DeepPink}{\mathrm{m}}}\right)\left(\frac{3600\,\textcolor{DodgerBlue}{\mathrm{s}}}{1\,\mathrm{hr}}\right)$$ $$37.6\,\mathrm{\tfrac{km}{hr}}$$

That's an average. His top speed during the race was even faster.

Example: A highway speed limit sign in Canada says 100 km/hr. Is that faster or slower than a 65 mph speed limit in the US? (1 mile = 1.6 km)

solution

Convert the Canadian speed limit into mph so the two numbers have the same units.

$$100\,\mathrm{\tfrac{\textcolor{DeepPink}{km}}{hr}}\left(\frac{1\,\mathrm{mile}}{1.6\,\textcolor{DeepPink}{\mathrm{km}}}\right)$$ $$62.5\,\mathrm{\tfrac{mile}{hr}}$$

100 km/hr is about 62.5 mph, so the Canadian limit is a little slower. The bigger number doesn't mean faster when the units are different.

Example: A 5 km race is also called a "5K." How many miles is it? (1 mile = 1.6 km)

solution

Kilometers are the old unit, so km goes on the bottom of the fraction.

$$5\,\textcolor{DeepPink}{\mathrm{km}}\left(\frac{1\,\mathrm{mile}}{1.6\,\textcolor{DeepPink}{\mathrm{km}}}\right)$$ $$3.1\,\mathrm{miles}$$

A mile is longer than a kilometer, so there should be fewer miles than kilometers.

Example: A student is 5 feet 4 inches tall. How tall are they in meters? (1 inch = 2.54 cm)

solution

$$\mathrm{ft \to inches}$$ $$5\,\textcolor{DeepPink}{\mathrm{ft}}\left(\frac{12\,\mathrm{in}}{1\,\textcolor{DeepPink}{\mathrm{ft}}}\right) + 4\,\mathrm{in}$$ $$60\,\mathrm{in} + 4\,\mathrm{in}$$ $$64\,\mathrm{in}$$
$$\mathrm{inches \to meters}$$ $$64\,\textcolor{DeepPink}{\mathrm{in}}\left(\frac{2.54\,\textcolor{DodgerBlue}{\mathrm{cm}}}{1\,\textcolor{DeepPink}{\mathrm{in}}}\right)\left(\frac{0.01\,\mathrm{m}}{1\,\textcolor{DodgerBlue}{\mathrm{cm}}}\right)$$ $$1.63\,\mathrm{m}$$

Example: TV screens are measured diagonally. A TV is advertised as 55 inches. How big is the screen in centimeters? (1 inch = 2.54 cm)

solution

$$55\,\textcolor{DeepPink}{\mathrm{in}}\left(\frac{2.54\,\mathrm{cm}}{1\,\textcolor{DeepPink}{\mathrm{in}}}\right)$$ $$139.7\,\mathrm{cm}$$

That's about 1.4 m from one corner of the screen to the opposite corner.

Example: An American football field is 100 yards long between the goal lines. How long is that in meters? (1 yard = 0.914 m)

solution

$$100\,\textcolor{DeepPink}{\mathrm{yd}}\left(\frac{0.914\,\mathrm{m}}{1\,\textcolor{DeepPink}{\mathrm{yd}}}\right)$$ $$91.4\,\mathrm{m}$$

Question: A weather app shows a temperature of 18°C, wind of 15 km/hr from the west, air pressure of 101 kPa, and 2 mm of rain today. Which of these are vectors, and which are scalars?

answer

Only the wind is a vector. It has a magnitude, 15 km/hr, and a direction, from the west.

Temperature, air pressure, and rainfall are scalars. They have a size, but it wouldn't make sense to give them a direction.

If the app only showed "wind 15 km/hr," that would be a speed, which is a scalar.

Example: A rescue boat leaves the dock and travels 3.0 km north, then 1.5 km east. How far is the boat from the dock?

solution

The two legs of the trip are the sides of a right triangle. The distance from the dock is the hypotenuse.

$$a^{2}+b^{2} = c^{2}$$ $$3.0^{2}+1.5^{2} = c^{2}$$ $$11.25 = c^{2}$$ $$c = 3.35\,\mathrm{km}$$

The boat traveled 4.5 km, but it's only 3.35 km from the dock.

Example: A soccer ball is kicked at 20 m/s at 30° above the ground. How fast is it moving horizontally and vertically?

solution

Make sure your calculator is in degree mode.

$$v_{x} = v\cos\theta$$ $$v_{x} = (20)\cos(30\degree)$$ $$v_{x} = 17.3\,\mathrm{\tfrac{m}{s}}$$
$$v_{y} = v\sin\theta$$ $$v_{y} = (20)\sin(30\degree)$$ $$v_{y} = 10\,\mathrm{\tfrac{m}{s}}$$

The angle is less than 45°, so the horizontal part is bigger than the vertical part.

Example: A wheelchair ramp is 4.0 m long and slopes up at 5° above the horizontal. How high does the ramp rise?

solution

The ramp is the hypotenuse, and the rise is the side opposite the angle.

$$y = d\sin\theta$$ $$y = (4.0)\sin(5\degree)$$ $$y = 0.35\,\mathrm{m}$$

That's 35 cm, about the height of two stairs. Ramps have to be long because the angle is small.

Example: A displacement is 120 m at 60° above the x-axis. Find the x and y components.

solution

$$x = d\cos\theta$$ $$x = (120)\cos(60\degree)$$ $$x = 60\,\mathrm{m}$$
$$y = d\sin\theta$$ $$y = (120)\sin(60\degree)$$ $$y = 104\,\mathrm{m}$$

The angle is more than 45°, so the y component is bigger than the x component.

Example: A hiker walks 50 m at 30° north of west. How far west and how far north did the hiker go?

solution

The angle is measured from west, so west is the adjacent side and north is the opposite side.

$$d_\mathrm{west} = d\cos\theta$$ $$d_\mathrm{west} = (50)\cos(30\degree)$$ $$d_\mathrm{west} = 43.3\,\mathrm{m}$$
$$d_\mathrm{north} = d\sin\theta$$ $$d_\mathrm{north} = (50)\sin(30\degree)$$ $$d_\mathrm{north} = 25\,\mathrm{m}$$

Example: A vector has an x-component of 12 m and a y-component of 5 m. What are the vector's magnitude and angle above the x-axis?

solution

$$a^{2}+b^{2} = c^{2}$$ $$12^{2}+5^{2} = c^{2}$$ $$169 = c^{2}$$ $$c = 13\,\mathrm{m}$$

The y-component is opposite the angle and the x-component is adjacent.

$$\tan\theta = \frac{\mathrm{opp}}{\mathrm{adj}}$$ $$\tan\theta = \frac{5}{12}$$ $$\theta = \tan^{-1}\left(\frac{5}{12}\right)$$ $$\theta = 22.6\degree$$

Example: Earth's diameter is about 1.27 × 10⁷ m. How many megameters is that?

solution

$$\mathrm{M} = 10^{6}$$ $$1.27 \times 10^{7}\,\mathrm{m} = 12.7\left(10^{6}\right)\mathrm{m}$$ $$1.27 \times 10^{7}\,\mathrm{m} = 12.7\,\mathrm{Mm}$$

That's 12 700 000 m, or about 3 times the distance from Los Angeles to New York.

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.

Reading (8 minutes): Read Physics Owes a Lot to a Little-Loved Math Class by Rhett Allain from WIRED. Then answer these questions.

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.