Measurement Considerations In Kinesiology Imperial And Metric English Language Essay

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Did you know that if you buy a gallon of gasoline in Ireland or England you actually get more gasoline? That's because the system of measurement is difference. However, it's unlikely that you can actually buy a gallon in these coutries anymore as they now use the metric system which means you have to buy a liter. If you go to Canada, you'll notice distance on the highways is posted in kilometers as are car speedometers and not miles as in the United States. So around the world we basically have two measurement systems. Ye Olde English system (also called imperial) and the newer metric system (also called the international system). These two measurement units cover all aspects of measurement from weight, to distance, to volume and even time.

The English System (Imperial)

In the United States, the predominant system used for measurements is actually known as the English System (ES or imperial system). This system of weights and measures was originally developed in England but is commonly used in the United States today. Over the course of several centuries the English System was mainly used for the purpose of commercial land measurement. It arose from the creative way that people measured themselves. Familiar objects and parts of the body were used as measuring devices. For example, King Henry I originally defined a yard as the distance from the end of his nose to the thumb of his extended arm. People also measured distances on the ground with their feet. They measured longer distances by their paces and capacities with common household items such as cups, pails, and baskets. The word gallon comes from an old name for a pail. Perhaps you have heard the term of a 'gallon pail' or a 'five gallon pail'. Given the wide use and variability in measurements a standard was eventually set so that all measurements represented the same amount for everyone. Given today's use of the metric system the English System of measurement displays little logic and so you actually have to memorize the units or write them down. There are 12 inches to the foot, 3 feet to the yard, 5280 feet to the mile, 16 ounces to the pound, and 2000 pounds to the tonne.

Traditional Units

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The English System has several units, each one containing specific units for their properties (i.e. length, volume, weight and time). The units of the English System are defined in an arbitrary way and are presented in Table 1. You will also notice that there is a vague progression of units, for example, inches become feet, which become yards, which becomes miles, etc.

Table 3.1

Length: inch (in), foot (ft), yard (yd), mile (mi)

12 in = 1 ft

5280 ft = 1 mi

3 ft = 1 yd

1760 yd = 1 mi

Volume: fluid ounce (oz), cup (c), pint (pt), quart (qt), gallon (gal)

1 c = 8oz

1 pt = 16oz

2 c = 1 pt

32 oz = 1 qt

2 pt = 1 qt

4 qt = 1 gal

Weight: ounce (oz), pound (lb), tonne

16 oz = 1 lb

2000 lb = 1 tonne

Time: second (s), minute (min), hour (h), day (d), year (y)

60 s = 1 min

24 h = 1 d

60 min = 1 h

365 d = 1 y

Length (Inch, Foot, Yard, Mile)

In most systems of measurement, length is a fundamental unit, from which other units are derived. Length is the long dimension of any object. The length of an object is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth, which is the distance from side to side, measuring across the object at right angles to the length. In the physical sciences and engineering, the word "length" is typically used synonymously with "distance", and is denoted by the symbol L.

In the traditional measuring system (i.e. ES), short distances are originally based on the dimensions of the human body. The inch represents the width of a thumb (in many languages, the word for "inch" is also the word for "thumb"). The foot (12 inches) was originally the length of the human foot, although it has evolved to be longer than most people's feet. The yard (3 feet) also originated in England as the name of a 3-foot measuring stick, but it is also understood to be the distance from the tip of the nose to the end of the middle finger of the outstretched hand. Finally, if you stretch your arms out to the sides as far as possible, your total "arm span," from one fingertip to the other, is a fathom (6 feet).

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In Anglo-Saxon England (before the Norman conquest of 1066) short distances appear to have been measured in several ways. The inch was defined to be the length of 3 barleycorns, which is actually very close to its modern length. Several foot units were also in use, including a foot equal to 12 inches, a foot equal to 2 shaftments (13 inches) and the "natural foot" (pes naturalis, an actual foot length, about 9.8 inches)

When the Normans arrived, they brought back to England the Roman tradition of a 12-inch foot. It was during the Reign of Henry I that the 12-inch foot really became official. It was Henry I that also ordered construction of 3-foot standards, which were called "yards", thus establishing that unit for the first time in England.

Longer distances in England are traditionally measured in miles. The mile is also a Roman unit, and was originally defined to be the length of 1000 paces of a Roman legion. A "pace" here means two steps, right and left, or about 5 feet, so the mile is a unit of roughly 5000 feet. In medieval England, various mile units were used. Eventually, what made the most sense to people was that a mile should equal eight furlongs, since the furlong was an English unit roughly equivalent to the Roman stadium and the Romans had set their mile equal to eight stadia. This correspondence is not exact; the furlong is 660 English feet and the stadium is only 625 Roman feet.

In 1592, the British Parliament settled this discrepancy by setting the length of the mile at 8 furlongs, which works out to 1760 yards or 5280 feet. If you follow horse racing you will know that this is the unit of measurement that is used for many horse racing distances. The old quarter mile race is actually two furlongs. This Parliament decision completed the English distance/length system. Since this occurred just before the American colonies were settled, British and American distance units have always been the same.

Volume (Fluid Ounce, Cup, Pint, Quart, Gallon)

The traditional names of volume units are named after the names of the standard containers in which the product was stored or sold. Before the eighteenth century, it was incredibly difficult to accurately measure a containers capacity in cubic units, so instead the standard containers were defined by specifying the weight of a particular substance, like wheat or beer. A gallon, which is the basic English unit of volume, was originally the volume of eight pounds of wheat. Gallons are always divided into four quarts, which are then divided even further into two pints each.

On both sides of the Atlantic, the smaller volumes of liquid are traditionally measured in fluid ounces, which are at least roughly equal to the volume of one ounce of water. In different systems, the smaller U.S pint is divided into sixteen fluid ounces, and the larger British pint is divided into twenty fluid ounces.

A traditional unit of volume used in recipes in the United States is the cup. One cup equals ½ of a pint of liquid or eight fluid ounces. Americans use the same size cup for measuring both liquid and dry substances. These volumes are summarized in Table 3.2 and 3.3.

Table 3.2

Name                                                Abbreviation                                    Fluid Ounces

Fluid Ounce                                         Fl. oz.                                                   1

Cup                                                     C                                                         8

Pint                                                      Pt.                                                     16

Quart                                                   Qt.                                                     32

Gallon                                                  Gal.                                                  128

  Table 3.3

1 gallon = 4 qt = 8 pt = 16 c = 128 fl oz

1 quart = 2 pt = 4 c = 32 fl. oz

1 pint = 2 c = 16 fl. oz

1 cup = 8 fl. oz.

Weight (Ounce, Pound, Ton)

The basic traditional unit of weight, the pound, originated as a Roman unit as was used throughout the Roman Empire. The Roman pound was divided into twelve ounces, but many European merchants preferred to use a larger pound of sixteen ounces, because of the ease at which a sixteen-ounce pound is divided into halves, quarters, or eighths. During the middle ages there were many different pound standards in use. The use of these weight units naturally followed trade routes around the world since merchants had to be familiar with the units used at both ends of the trip. Eventually the 16oz pound was widely adopted and is still in use today.

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The oldest English weight system has been used since Saxon kings time (@500AD). It is based on the twelve-ounce troy point, which also provided the basis of which coins were minted and gold and silver were weighed.

The ounce, another traditional unit of weight, is commonly used in the United States and is 1/16 pound. The word ounce is from the Latin uncia, meaning a 1/12 part, because the Roman pound was divided into twelve ounces. The symbol oz is from the old Italian word onza, meaning ounce.

The ton measure, originated as a unit to measure wine. It was defined to equal approximately 2240 pounds. It was during the nineteenth century that a disagreement arose between the British and Americans concerning the larger weight units. The British "long" ton remained at 2240 pounds while the American "short" ton became exactly 2000 pounds. Today, most international shipments are reckoned in metric tons, which are rather close in weight to the British long ton.

Table 3.4

Weight:

16 ounces (oz) = 1 pound (lb)

2000 lb = 1 ton

Time (Second, Minute, Hour, Day, Year)

You will of course know that a day has 24 hours, and each hour has 60 minutes, and each minute has 60 seconds. These constitute the global units of time. This the main component of the measuring system that is used to sequence events, compare duration of events and the intervals between them, and to quantify the motions of objects is Time. The base unit for time is the second. From the second, larger units are defined. Common units like minute, hour, day and year are further derived from the second. The length of a day is based on the relationship of the earth's rotation within itself and around the sun. While a day is always 24 hours, the effect of solar light is highly variable as the earth is also tilted on an axis. This explains why we have more light and heat in our summer months versus our winter months. Table 3.5 summarizes the common units of time.

Table 3.5: Common Units of Time

Unit

Size

Nanosecond

0.000 000 001 seconds

Microsecond

0.000 001 seconds

Millisecond

0.001 seconds

Second

SI BASE UNIT

Minute

60 seconds

Hour

60 minutes

Day

24 hours

Week

7 days

Month

28 to 31 days

Quarter

3 months

Year

12 months

Common Year

365 days

52 weeks + 1 day

Leap Year

366 days

52 weeks + 2 days

Decade

10 years

Score

20 years

Generation

20-30 years

Century

100 years

Millennium

1,000 years

The Metric System (Le Système International d'Unites)

The system of measurement that is presently used by every major country in the world

except the United States is Le Système International d'Unites (The International System

of Units), which is commonly known as the S.I. or The Metric System. The Metric System of Measurement was created about two hundred years ago by a group of French scientists to simplify measurement in response to a request from King Louis XVI to the French Academy of Sciences during the 1790's. In 1875 the 'Treaty of the Meter' was signed by 17 countries agreeing to adopt the use of the metric system.

Since that time the metric system has grown in popularity worldwide. The SI comprises three types of measurements; they are base units, supplementary units, and derived units. There is also another category referred to as specially named units because they contain units that don't really fit into any of the aforementioned categories. These measurement units are briefly described below and then the measurements within those units are described in greater detail later in the chapter.

Base Units

The base units are arbitrarily defined and comprise the units for our major measurements across all disciplines. They include the four main units of Length (meters [m]), Mass (kilogram [kg]), Time (Seconds [s]), and Temperature (Kelvin [k]). Then there are a secondary set of units that include electric current (ampere [A]), luminous intensity (candela [cd]), and the amount of a substance (mole [mol]).

Length (Meters)

The evolution of the meter has an interesting history. The meter was supposed to be one tenth-millionth the length of the distance from the North Pole to the Equator when measured on a straight line running along the surface of the earth through Paris. In 1793, during Napoleon's time, the French government adopted this new system of standards called the metric system, based on what they called the meter. With the meter now determined as the basis of the metric system, other linear units of the system were set up in decimal ratios with the meter. Compared to the yardstick, the meter is just a little longer: 39.37 inches long. With this system, all units are in multiples of ten: ten decimeters in a meter, a hundred centimeters in a meter, and a thousand millimeters in a meter. In the other direction, there are ten meters in a decameter, a hundred meters in a hectometer, and a thousand meters in a kilometer.

Mass (Kilograms)

An interesting fact about the kilogram is that it is the only SI base unit to incorporate a prefix. Why wasn't the gram used? Well it was believed that the gram was such a small amount of stuff that it would be easy to make a mistake in creating the standard or when measuring a substance. The use of a kilogram (1000 grams) provided a more accurate measure to work with a larger amount of material.

The definition of the kilogram was adopted in France in 1901 and is equal to the mass of the international prototype of the kilogram (IPK) which is close in weight to a liter of water. What this means is that there is a cylinder containing 90% platinum and 10% iridium in a vault and that total amount weighs exactly one kilogram. This IPK is stored in the town of Sevres, France.

The kilogram is the only unit still based on a single collection of material. All the other definitions can be reproduced with a high degree of accuracy in any laboratory with the proper equipment.

Time (Seconds)

Time, while an accepted SI unit of measurement does not actually conform to the metric system. The metric system defines the second as the unit variable for measurement. However, the 'day' itself is the most important natural time period, so most proposed metric time reforms choose a base unit that is a decimal subdivision of the day. Since time is a universal we are all familiar with the multiple denominations of day, hour, second etc, and so use them interchangeably.

Metric prefixes are often used for decimal sub-multiple units of time interval, such as milliseconds, microseconds and nanoseconds, but larger quantities are usually measured in non-decimal units such as minutes, hours, days, etc., instead of kiloseconds and megaseconds. For this reason, it is claimed that truly metric time does not exist, and since the second is not a decimal fraction of the day, it also cannot be used as the basis of a decimal time of day.

Table 3.7: Units accepted for use with the SI

Name

Symbol

Value in SI Units

Minute

min

1min = 60s

Hour

h

1hr = 60min = 3600s

Day

d

1d = 24hr = 1440min = 86400s

Temperature (Kelvin)

Three temperature scales are in common use in science and industry. Two of those scales are SI metric (Celsius and Kelvin).

The degree Celsius (°C) scale was devised by dividing the range of temperature between the freezing and boiling temperatures of pure water at standard atmospheric conditions (sea level pressure) into 100 equal parts. Temperatures on this scale were at one time known as degrees centigrade; however that terminology is rarely used nowadays.

The kelvin (K) temperature scale is an extension of the degree Celsius scale in that it goes down to absolute zero, a hypothetical temperature characterized by a complete absence of heat energy and where everything freezes. Temperatures on this scale are called kelvins, NOT degrees Kelvin. Kelvin is not capitalized and the symbol (capital K) stands alone with no degree symbol. The official name was changed to "kelvin" and the symbol "K" adopted by the 13th General Conference on Weights and Measures (CGPM) in 1967.

The degree Fahrenheit (°F) non-metric temperature scale was devised and evolved over time so that the freezing and boiling temperatures of water are whole numbers, but not round numbers as in the Celsius temperature scale. This scale is used in the United States but not really anywhere else in the world.

Table 3.8: Some baseline temperatures in the three temperature scales

temperature

Kelvin

degree Celsius

degree Fahrenheit

symbol

K

°C

°F

boiling point of water

373.15

100.

212.

melting point of ice

273.15

0.

32.

absolute zero

0.

-273.15

-459.67

Table 3.9: Temperature Conversions between temperature scales

Kelvin = degree Celsius + 273.15

degree Celsius = Kelvin - 273.15

Problem:

What is the Kelvin equivalent of 38°C?

Answer:

°C = K - 273.15

K = °C + 273. 15

K = 38°C + 273.15

K = 311.15

Supplementary Units

These units are also arbitrarily defined and relate to angles and angular speed or movement. They include the radian (rad), which is the unit for a plane angle, and the 'degree' which is more general use. The radian is a unit of plane angle, equal to 180/Π degrees, or about 57.2958 degrees. It is the SI standard unit of angular measurement in all areas of mathematics. The radian is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.2 rad" or "1.2c" (the second symbol can be mistaken for a degree so we need to be careful: "1.2°"). However, the radian is mathematically considered a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol ° is used.

The radian was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.

Radian (Unit for Plane Angle)

Usually, a person first learns how to measure the size of an angle using degrees. We all know there are 360 degrees all the way around a circle and it is certainly a measurement unit that most of us are more familiar with using. The method of using a radian is a less arbitrary way dividing a circle into parts. The only difference with radians is that they don't divide the circle into equal parts. For example a circle has approximately 6.2 rad.

Derived Units

These units were established from base units in accordance with fundamental physical principles and are derived to allow us to measure other specific, but related parameters. As such, they are expressed in terms of their respective base unit. For example area is defined in square meters (m2), speed is defined in meters per second (m/s), and acceleration in meters per second squared (m/s2). Units expressed algebraically in terms of their base units or other derived units are called Derived Units. The symbols for derived units are collected through mathematical operations of multiplication and division. Examples of the derived units and their symbols are presented in Table 3.1.

Table 3.10: Examples of SI derived units expressed in terms of SI base units

Derived quantity

Name

Symbol

area

square meter

m2

volume

cubic meter

M3

speed, velocity

meter per second

m/s

acceleration

meter per second squared  

m/s2

wave number

reciprocal meter

m-1

mass density (density)

kilogram per cubic meter

kg/m3

specific volume

cubic meter per kilogram

M3/kg

current density

ampere per square meter

A/m2

magnetic field strength

ampere per meter

A/m

amount-of-substance concentration (concentration)  

mole per cubic meter

mol/m3

luminance

candela per square meter

cd/m2

Area (Square Meters)

The square meter, also called the meter squared, is the SI unit of area. The symbol for square meter is m². Less formally, square meter is sometimes abbreviated sq m.

When calculating area, it is important to realize that area is proportional to the square of the linear dimension. Thus, if all linear dimensions are doubled, the area becomes four times (22) as great; if all linear dimensions are cut to 1/3, the area becomes 1/9 (1/32) as great. An area of 1 m2 is equal to 10,000 centimeters squared (104 cm2) or 1,000,000 millimeters squared (106 mm2). In the opposite sense, 1 m2 is equal to 0.000001 kilometer squared (10-6 km2).

When converting between square meters and non-SI units of area such as square inches (sq in) or square miles (sq mi), the linear-unit conversion factor must be squared. For example, one meter is approximately 39.37 inches (39.37 in); therefore 1 m2 = 39.372 = 1550 sq in (approximately). As another example, 1 meter is about 0.0006215 mile (6.215 x 10-4 mi); therefore 1 m2 = (6.215 x 10-4)2 = 3.863 x 10-7 sq mi (approximately).

Motion (Speed & Velocity)

Speed is defined as the rate of motion, or the rate of change in a position, and is expressed as the distance (d) traveled per unit of time (t). Motion uses two standard units to define itself. They are the meter (m) which measures the change in position, and the second (s) which is used for measuring time. Therefore motion is two dimensional. Two quantities are often used to describe the rate of change of position: velocity and speed. Velocity, which is a vector quantity, has measurements that contain magnitude, or numeric value and a direction. Speed only contains a numeric value.

Again, speed is the quotient of total distance traveled and time, while velocity is displacement (total change in position) divided by time. These are discussed in greater detail in later chapters and example of both was also presented in chapter 2.

Acceleration (Meters per Second Squared)

Acceleration is the rate of change of velocity (or speed). So, average acceleration is the rate of change of the rate of displacement. The standard units used for acceleration are "meters per second per second", or "meters per second squared" (m/s2).

Mathematically, this translates to:

Acceleration = ∆ v

∆ t

Where ∆ = change.

A mathematical example follows:

A car is traveling at 60 kilometers per hour (km/h).

The driver decides to enter the freeway, where the speed limit is 100 km/h (although, since direction of travel is relevant, perhaps it should be called the "velocity limit") .

The car must accelerate to avoid an accident.

In 5.0 seconds, the driver accelerates at a constant rate from 60 km/h to 100 km/h.

What was the car's average acceleration? (Average acceleration is expressed in terms of change in velocity and time elapsed):

Since we will be dividing by seconds, km/h can be converted to m/s for simpler calculations.

100 km / h Ã- 100 m / km Ã- 1 h / 3600 s = 27.2228 m / s

60 km / h Ã- 1000 m / km Ã- 1 h / 3600 s = 16.6667 m / s

Acceleration = 27.7778 m / s - 16.6667 m / s

5.0 s

Acceleration = 11.1111 m / s

5.0 s

Acceleration = 2 m / s²

Specially Named Units

The final category of units is referred to as specially name units. Certain SI Units have special signs and symbols. They are defined through the use of fundamental equations and defined SI base units. They allow us to develop measurement units for more non-standard quantities that are common but not as broadly used as the base units. The specially named units are given below.

Table 3.11: SI derived units with special names & symbols

Derived Quantity

Special Name

Special Symbol

Expression in terms of other SI units

Expression in terms of base units

Frequency

hertz

Hz

-

s-1

Force

newton

N

-

m · kg · s-2

Pressure, Stress

pascal

Pa

N/ m2

m-1 · kg · s-2

Energy, Work, Quantity of Heat

joule

J

N Ã- m

m2 · kg · s-2

Power, Radiant Flux

watt

W

J/s

m2 · kg · s-3

Electric Charge, Quantity of Electricity

coulomb

C

-

s Ã- A

Electric Potential, Potential Difference, Electromotive Force

volt

V

W/A

m2 · kg · s-3 · A-1

Example of unit measurement terminology: Newton's Second Law of Motion

Newton's Second Law of Motion states that when a body that is free to move is subjected to a force, it will experience acceleration proportional to that force and inversely proportional to is own mass:

Force = mass Ã- acceleration

1 Newton = 1 kg Ã- 1 m / s²

Problem:

How much force must be applied to move a box that weighs 12kg at a rate of 2.5 m / s²?

Answer:

Force = mass Ã- acceleration

Force = (12 kg) Ã- 2.5 m / s²)

Force = 30 Newtons

Conversion Factors in Measurement

Conversion factors are those equalities which allow for direct conversion from one system to another regardless of the original unit of measurement.

Table 3.12 Examples and Problems of Conversion Units

English to Metric

Length:

1 inch = 0.0254 m

1 foot = 0.3048 m

1 yard = 0.9144 m

1 mile = 1609 m

Problem: How many centimeters are in 4ft?

Answer: 4 feet = (4 ft) Ã- (0.3048 m)

= 1.2192 m

1 meter = 100 cm

(1.2192 m) Ã- (100 cm)

= 121.92 cm

Weight:

1 oz = 28.350 g

1 lb = 0.4563 kg

Problem: How many kilograms are in 5lbs?

Answer: 5 lbs = 5/2.2=2.27kg

2.27kg = 2270g = 80oz

Temperature:

°C to °F = (°C Ã- 1.8) + 32

°F to °C = (°F - 32) / 1.8

Problem: Convert 77°F to °C

Answer: 77 - 32 = 45

45 / 1.8 = 25°C

Prefixes

In the Metric System, the base units are converted into more appropriate descriptions based on the quantity being measured by adding what is called a prefix to the name of the base unit. They allow very large or very small numerical values to be avoided. A prefix attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit.

The prefixes and symbols listed below are commonly used to form names and symbols of the decimal multiples and sub multiples of the SI units.

Table 3.13: Metric System Prefixes

(http://www.wsdot.wa.gov/reference/metrics/factors.htm)

Names and Scientists

Throughout history many scientists have discovered different variables and properties. As a result many of our common measurement units are named for the scientists that discovered them. You will recognize many of these units even though you may not have known the origin of the unit terminology. The names for units are derived by using the names of scientists who made the greatest discovery in that field of study. Below is a chart of some of the greatest scientists and the discovered units they made throughout history.

Table 3.14

Symbol

Unit

Quantity

Scientist

A

ampere

Electric Current

Andre-Marie Amphere

°C

celsius

temp

Anders Celsius

Hz

hertz

frequency

Heinrich Rudolph Hertz

J

joule

energy

James Prescott Joule

N

newton

force

Sir Isaac Newton

Pa

pascal

pressure

Blaise Pascal

V

volt

Electrical Potential

Count Alessandro Volta

W

watt

power

James Watt

Understanding the Difference between ES and SI

By now you are familiar with inches, feet, miles, pounds, quarts. You know that a car will go a certain number of miles per hour. Gas costs a certain amount of cents per gallon. All these things are measurements in the English system.

But, there is another system of measurement that uses decimals. That system is called the metric system. The metric system is based on the number 10 or decimalization. There are, for example, 10 millimeters per centimeter. There are 1000 (10 times 10 times 10) meters per kilometer. (The prefix kilo- means one thousand times - so a kilometer is one thousand times the length of a meter.) There are one thousand milliliters in a liter. (The prefix milli- means "one thousandth of" - so one milliliter would be one one-thousandth of a liter.)

In chemistry, physics - in fact, all the sciences, the metric system is used. The basic advantage to the metric system is that it is so easy to go from one unit to another. You just multiply or divide by 10. The English system, on the other hand, has no real consistency between units.

Problems

How many meters in 1.5km?

How many meters in 1 mile?

How many kilometers in 1 mile?

A marathon is @ 26 miles, how many kms is this?

Identification of The International System of Units

Table 3.6

Force

kilograms times meters per second per second

(kg Ã- m/s2). 

Acceleration

meters per second squared (m / s²)

Speed

meters per second ( m / s)

Density

mass per unit volume ( kg / v)

Volume

liter

Area

square meters (m²)

Length

meters

Mass

kilograms

Time

seconds

Temperature

Kelvin (K) or Celsius (°C)

Plane Angle

ratio of two lengths (m/m = 1)

Power

watt

joule / second

Energy & Work

joule

kg Ã- m2 / s2

Pressure & Stress

force / area

Need to add additional review problems.