Physical Quantities: Quantities and Units

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Where are we going with this? This page will assist in understanding the relationship between units and the quantities they measure.

When looking at the world around, people pretty much automatically attend to physical quantities, pretty much without thinking about them. The youngest child intuitively can judge differences with some skill.

By the time the child reaches school age, the following conversation would seem drastically out of place:

Teacher: "Mary, how far can you run?"
Mary: "I can run six pounds! Very heavy!"

Almost all children would know that far and heavy describe different physical quantities.

As most readers of this would know, one of the jobs of science is to complicate things. No… not that… 

to clarify things by assigning specific words with specific meanings to specific ideas, objects and concepts in order to make discussion precise and accurate (which are both words surrounded by confusion, ironically).

Therefore, with regard to physical quantities, a certain set of words are used to describe specific things.

Included in those sets of words are two sides to every quantity. 

One side is the concept of what is being described. Mary, in the dialog above, gave an answer about one quantity with words used for a different. 

The other side of the concept is, for every quantity, specific ways that it is measured (which are called units). Pounds do not go with a "how far" question!

First what are the things being described?

Types of Physical Quantities

The following list is certainly not all-inclusive. It includes a few very common and familiar quantities that are measured.

Distance: Distance is a measure of… okay, distance is so common it actually has different meanings, so in science, there are different words that are used so in order to be more specific.

Distance (Most General): The measure of how far two points are from each other, as in, "The tip of the antenna was 12 meters from the the surface of the window."

Distance (A specific case) Think Distance Traveled: The length of a path that a moving body takes, as in, "The dog ran from tree to tree through the park until it was at the tree next to the one where it started, covering a distance of 250 yards." Compare with Displacement, next!

Displacement: The measure in a straight line between where a moving body begins and ends, as in, "The dog ran from tree to tree through the park until it was at the tree next to the one where it started, resulting in a displacement of only 4 yards."

Length, Width, Height: Pretty much what you expect, on this! The measure of how far specific points of an object are from other points on the object.

Other things that would be included in the distance concept are circumference, radius, perimeter, range, altitude, depth… You can probably think of others. But will you?

In general, distance is the measure of how far apart two things are.


Volume: (Not talking about sound, which borrowed from this concept for its own purposes!) This also is pretty much what you expect!

Volume is the space in three dimensions that an object or substance takes up (or holds). If a tank holds 25 gallons of gasoline, then the volume of the gasoline that filled the tank would be (duh) 25 gallons. A bottle of soda holds 20 ounces. A different bottle holds 2 liters.

Mass: Mass is a physical quantity that is the result of how many protons, electrons, and neutrons are all in a given space. It is actually not directly observable, though there are devices (scales and balances) that combine with gravity (or other acceleration) and other laws of physics so that it can be measured. Mass is a measure of the amount of matter an object contains.

If two things with the same volume have different masses, one would feel heavier than the other.

Good news: There are scales that measure mass, so you can just plop things down and get a number!

Weight: Weight is a measure of mass under a particular condition. Many people have heard things like, "Well, on the moon, I'd only weigh 12 pounds."

Weight is a basic concept that people are very familiar with. Weight is the degree of heaviness something has.

Time: This is something people very intuitively understand, but which is actually a very abstract concept. Stop reading and write down a definition of time. Not how time is measured! Try a definition of time that does not use seconds, minutes, hours, days, etc. and see what you come up with!

According to the Oxford dictionary, time is the indefinite continued progress of existence and events in the past, present, and future regarded as a whole. 

Time is a very basic concept in science, and fortunately, our intuitive understanding is enough for us to use it.

How do you measure these quantities?

The list below will connect the quantities above with the SOME OF THE units used to measure them. Common units in science are in bold and "normal" abbreviations for the common units are in parenthesis..

NOTE: Italics units are from the English system

Distance: inches, feet, yards, miles, meters (m), kilometers (km), centimeters (cm), light-years

Volume: gallon, ounce, cup, teaspoon, liters (l or lt), milliliters (ml)

Mass: slugs, grams (g or gr), kilograms (kg)

Time: hours (hr), minutes (min), seconds (s)

A few more…

Some units are derived or combined from the basic units but are so common they are worth noting here.

Force: Newtons (n or N), Pounds

Weight (which is a force) is also measured in pounds, newtons

Speed or Velocity: MPH, m/s, cm/s, km/hr
Speed or Velocity is a distance unit divided by a time unit.

Acceleration: (m/s)/s, MPH/s, (km/s)/s, cm/s²  m/s²

Acceleration is a velocity unit divided by a time unit.

At a glance…

System	Mass	Weight	Distance	Time	Acceleration Due to Gravity
English	Slug	Pound lb	Feet ft	Seconds s	32 f/s/s
Metric	Kilogram kg	Newton N	Meters m	Seconds s	9.8 m/s/s

And also…

Temperature: Temperature is, within the kinetic theory of matter model, defined as the average kinetic energy of the molecules in a system or substance. How much thermal energy is present? How hot is it? The absence of thermal energy is described as being cold. Temperature is measured in degrees. In science, officially in Kelvin degrees (°K), but often in Celsius degrees (°C). 

SI Units

With so many units, things could get confusing. So… Some smart people made a decision that, a a general best-practice science would be built on seven base units. They also gave it a fancy name:

The International System of Units

Then… they decided they would abbreviate it… from French 

Système international (d'unités)

The seven base SI units and some corresponding constants allow meaningful and uniform dialog about all matters of science. 

https://en.wikipedia.org/wiki/International_System_of_Units

Acceleration, Velocity, Distance, and Time

The study of motion with relationship to time considers primarily how the position or velocity of an object changes with respect to time. In exploring these changes, definitions and formulas emerge:

displacement: the magnitude of the change in position of an object.

velocity: the rate at which the position of an object changes with respect to time.

acceleration: the rate at which the velocity of an object changes with respect to time.

df = di + vt

where df is final (or total) displacement, di is initial displacement, v is average velocity, and t is elapsed time.

vf = vi + at

where vf is final velocity, vi is initial velocity, a is acceleration, and t is elapsed time.

v(ave) = (vf+vi)/2

where v(ave) is average velocity, vf is final velocity, and vi is initial velocity.

Using the formulas above, it is possible to directly solve problems that ask about any combination of displacements, velocity and time or that ask about velocities, acceleration and time. However, to solve problems that ask about displacements, acceleration, and time, three steps were needed, if only the above formulas were used.

To solve the problems that that ask about displacements, acceleration, and time directly, a fourth formula is needed. This formula can be derived from the other three, but to get started, here it is:

df = di + vit + 1/2at²          (t² means t•t or "t squared")

This equation is much more fun with di and vi are zero! But, to break it down as is first

df is the final, total displacement.

di is the initial displacement. How far from whatever point of reference is the object when the thing starts accelerating?

vit accounts for the motion of the object based on its starting velocity. It keeps covering distance at the initial rate.

The last term tells how much MORE distance is covered based on the acceleration. Since any di and vi are covered in the first term, the last term can be analyzed for the zero case, but first two examples.

EXAMPLE 1 (the zero case)

How far will a rocket travel while it accelerates from rest at a rate of 4 (m/s)/s for 5 seconds?

di = 0

vi = 0

vf = not given, not asked for

df = what you are looking for

a = 4

t = 5

df = di + vit + 1/2at²

df = 0 + 0 + 1/2 • 4 • 5•5

PEMDOS

df = 1/2 • 4 • 25

df = 2 • 25

df = 50

EXAMPLE 2

How far will a rocket travel while it accelerates from 20 m/s at a rate of 6 (m/s)/s for 3 seconds?

di = 0

vi = 20

vf = not given, not asked for

df = what you are looking for

a = 6

t = 3

df = di + vit + 1/2at²

df = 0 + 20•3 + 1/2•6 • 3•3

PEMDOS

df = 60 + 1/2 • 6 • 9

df = 60 + 3 • 9

df = 60 + 27

df = 87

CONCLUSION:

The above examples can be checked using the three-step method. The equation used above can be used to solve for any of the variables, but does require dealing with squares and square roots. However, if that level of math does not present a problem, 

df = di + vit + 1/2at²

becomes the only equation needed for all cases of motion related to displacement. Combined with

vf = vi + at

all of the motion problems can be solved.

_____________________________________

Example 3 (Find a when given distance and time.)

Working through the three-step method to find acceleration for the case of di and vi = 0 results in a quick and easy calculation method that requires

Step 1: Divide df by t

Step 2: Double it

Step 3: Divide by t again.

Using the new equation, this emerges like this:

df = 1/2 a t²

df = 1/2 a • t • t

2 • df = a • t • t

2df/t = a • t

(2df/t)/t = a

(Same as the 3 step method! Wooo!)

To more correctly write the above equation results in:

2•(df/t²) = a

And that is called physics!

_____________________________________

WHO LOVES SOME MATH?

Okay, don't answer that. 

It is not hard to come up with the df = di + vit + 1/2at² formula. Starting with the basic, simple case formulas, combining them quickly results in

df = 1/2at²

How is that done? Easily! Begin with

df = vt 

where v is the average velocity.

Remember that 

vf = at 

where vf is the final velocity. In the case of vi = 0, the formula for average velocity easily reduces:

v(ave) = (vf + vi)/2

v(ave) = (vf + 0)/2

v(ave) = (vf)/2

Now, back to vf = at! Plug it in!

So…

v(ave) = (vf)/2 becomes

v(ave) = (at)/2

Now, back to df = vt! Plug in (at)/2 for v:

df = (at)/2 • t

Clean up time:

df =  a • 1/2 • t • t

df = 1/2 • a • t • t

df = 1/2at²

Bam! That just happened!

Displacement and Acceleration

If the relationship between displacement, acceleration, and elapsed time is extended to the most general case, the following equations need to be combined.

1.)

df = di + vt

where df is the final displacement from the fixed point, di is the initial displacement from the fixed point, v is rate of change in position, (average velocity), and t is elapsed time.

2.)

Vf = Vi + at

where Vf is final velocity, Vi is initial velocity, a is the rate of acceleration, and t is the elapsed time.

3.)

Vave = (Vf  + Vi) / 2

where Vf is final velocity, Vi is initial velocity, and Vave is the average velocity.

To begin, start with the general displacement equation:

df = di + vt

Understanding that v is average velocity, the equation becomes:

df = di + ((Vf  + Vi) / 2) • t

And since Vf can be found in relationship to acceleration, the following emerges:

df = di + (((Vi + at) + Vi) / 2) • t

Solving the problem in steps generally is more easily understood. To find displacement when acceleration is present, do the following.

Step 1.) Find the final velocity: Vf = Vi + at

Step 2.) Find the average velocity: Vave = (Vf  + Vi) / 2

Step 3.) Find the displacement: df = di + vt

EXAMPLE

A model car is 3 meters from the starting line on a model car race track and it is moving at 2 m/s. It accelerates at a rate of 5 (m/s)/s for 4 seconds. How far does it end up from the starting line?

Step 0.) Collect the data!

di = 3 m

vi = 2 m/s

a = 5 (m/s)/s

t = 4

Step 1.) Find final velocity.

Vf = vi + at

vf = 2 + 5•4

vf = 2 + 20

vf = 22

Step 2.) Find average velocity.

v(ave) = (vf + vi)/2

v(ave) = (22 + 2)/2

v(ave) = 24/2

v(ave) = 12 m/s

Step 3.) Find the total displacement.

df = di + vt

(remember v is average velocity and t is the same elapsed time as for the acceleration)

df = 3 + 12•4

df = 3 + 48

df = 51 meters 

So, the model car ends up 51 meters from the starting line.

Elapsed Time

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Elapsed Time

How long did it take? How much time passed?

The answer to these types of questions is answered by finding the elapsed time.

Let's start with a definition:

elapsed time: the difference between the final time and the initial time.

To find out how much time passes, we can subtract the clock or stopwatch reading at the beginning of the event from the clock or stopwatch reading at the end. With that in mind, 

initial time is the clock or stopwatch reading at the beginning of an event. For a stopwatch, it could often be zero.

final time is the clock or stopwatch reading at the end of an event.

For example, a track coach starts a stopwatch when the gun fires and stops it when the runner crosses the finish line. The initial time reading was zero and the final reading on the stopwatch was 10.5 seconds. Thus, the time that the runner ran was 10.5 seconds.

Elapsed time will always be expressed with time units, such as seconds, minutes, hours, etc.

Calculating elapsed time is generally easy to understand. If your movie starts at 7:00 and ends at 9:15, how long does it last? Or, in other words:

What is the elapsed time of a movie that begins at 7:00 and ends at 9:15?

Most people will almost instantly say 2 hours and 15 minutes. But wait! There is actually a math way to do this with math!

If elapsed time is t and initial time is ti and final time is tf, then:

t = tf - ti

Of course, having a formula spoils the simplicity of the concept, but it also allows for the concept to be applied in non-intuitive situations. But, for now, the example above:

t = what we are looking for

tf = 9:15 o'clock

ti = 7:00 o'clock

   9:15

- 7:00

______

  2:15 hours

The process becomes more complex when you have to "borrow" because, time. A minute is 60 seconds. An hour is 60 minutes. So when you borrow, you have to borrow 60 of the other thing.

Thus, the problem goes like this:

t = what we are looking for

tf = 9:15 o'clock

ti = 7:30 o'clock

   9:15

- 7:30

______

????

When you borrow 60 minutes so that you can subtract, you get this:

   8:75

- 7:30

______

  1:45 = 1 hour and 45 minutes.


For examples worked out and more…

	Scan the QR code to open movie.
Finding Elapsed "Clock Time"	Scan the QR code to open movie.

This can be extended to days, as well, though with less clarity. (What do you mean by "day"? Is it 24 hours or is it the actual amount of time that it takes for the earth to rotate once, which is more than 24 hours?)

When reading a stopwatch, you will see the readings written like this:

HH:MM:SS.fs

Where HH is hours, MM is minutes, SS is seconds and fs is fractions of a second.

The math works the same—if you borrow, you have to borrow in 60s.