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Some of the most common base units are meters, kilograms, and seconds. Apart from these, are there more units that are used in daily life? The answer is yes! Area, volume, and speed are some examples of quantities that can be found by combining base units.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Combining Base Units

The combination of base units creates new units of measure. Some of these combinations have their own names. Using the applet below, units like can be created following these two steps.

  • Click on and hold the click.
  • Drag it to the denominator of the fraction and release the click.

To have a clean start, press the Reset button. Any of the units inside the boxes can be drag either to the numerator or denominator of the fraction at the right. Build new units and find out if they have a particular name.

Applet allowing different combination of base units
Explore

Distance to the Lightning

One rainy morning, Jordan saw lightning from her bedroom window. After a few seconds of time had passed, she heard the sound of thunder, which shook the photos on her desk. Could Jordan know the distance from her house to the lightning struck? What information does she need?

House, rain, lightning
Challenge

Voyager Arrival Time

Voyager is a space probe launched by NASA in Currently, it is the most distant human-made object from Earth. It takes more than hours for a radio signal to get from Earth to Voyager — traveling at the speed of light! The nearest star to Earth (apart from the Sun) is Proxima Centauri, which is about light years, or kilometers away.
Sun and Proxima Centauri with concentric circles around the sun
Assume Voyager travels toward Proxima Centauri at a constant speed of kilometers per hour. At days per year, how many years will it take Voyager to reach the star? Round the answer to the nearest integer.
Discussion

Combining Length and Time Units

In the sentence Voyager travels at kilometers per hour, both a unit of time and a unit of length are involved. In this case, the unit of length is distance traveled, and the unit of time is the amount of time it takes for the item to travel the given distance. Combining these units leads to a new type of unit that describes how fast an object is moving.

Concept

Speed

The rate or speed of a moving object, denoted by measures how fast the object is moving — that is, the rate of motion of the object. It is calculated by dividing the distance traveled by the amount of time spent traveling.

Speed

From the above, three formulas can be derived relating speed, distance, and time. The triangular diagram below helps to set each formula.

Triangle with distance, speed, and time
 The above formulas can be used in different daily life situations. For example, if the speed of a train and the time it takes to travel between two stations are known, the distance between those train stations can be calculated.
Example

Distance Between Train Stations

A train from New York departed at and arrived in Boston at
Train Traveling from New York to Boston
Assuming the train makes no stops between the cities and travels at a constant speed of miles per hour, how far apart are the cities?

Hint

Multiply the speed of the train by the time it took to go from one city to the other.

Solution
Since the train left at and arrived at it took hours to travel from one city to other.
The speed of the train is miles per hour. When the combined unit of miles per hour is multiplied by the base unit hours, the result will be in terms of miles. Therefore, to find the distance between the cities, multiply the speed by the time.
▼
Substitute values and simplify
SubstituteII

,

Rewrite

Rewrite as

CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

DivByOne

Multiply

Multiply

In conclusion, the cities are miles apart.
Discussion

Converting Units of Measure

Sometimes, two quantities can be given using different units of measure. However, to operate with the quantities, they need to have the same unit. Consider the diagram below, where is given in meters, is given in miles, and the total distance is required to be calculated.

Two houses and a stadium in between. d1=300 meters and d2=0.25 miles

In such cases, one of the quantities needs to be converted into an equivalent quantity using the other quantity's unit of measure. For example, can be converted from miles to meters. In such instances, the conversion factor plays an important role.

Concept

Conversion Factor

A conversion factor is a fraction where the numerator and denominator represent the same quantity with different units.
Recall that hour and minutes represent the same quantity. Multiplying a quantity by a conversion factor changes the quantity to an equivalent quantity in different units. Examine how to convert hours to minutes using the above conversion factor.
Given Quantity Conversion Result

Although the final result is in minutes, both quantities represent the same amount of time. Note that the opposite conversion, from minutes to hours, has a conversion factor of If the task was to convert minutes to hours, minutes would be multiplied by this conversion factor.

Given Quantity Conversion Result

As shown in the examples above, the process of including units of measurement as factors is called dimensional analysis. Dimensional analysis can also be used when deciding which conversion factor will produce the desired units. In the table, some common conversion factors are used to convert the given measures.

Given Quantity Conversion Result

Some common conversions involve distance, mass, area, volume, time, and temperature.

Why

 The Reason That the Quantities Are Equivalent

The numerator and denominator of the conversion factor represent the same quantity. That means their quotient equals Then, by the Identity Property of Multiplication, the amount of the given quantity does not change when multiplied by the conversion factor.

Given Amount * 1 = Given Amount

When converting from one unit to another, the desired unit needs to be in the numerator of the conversion factor while the given unit needs to be in the denominator. That way when the quantity is multiplied by the conversion factor, the given unit will cancel out and the desired unit will remain.

X Given Unit * (Y Desired Unit)/(Z Given Unit) = (XY/Z) Desired Unit

Keep in mind that, despite the given quantity and the new quantity have different values, they represent the same amount.

To convert from miles to meters, use the fact that mile is the same as meters.
It has been shown that miles is the same as meters. Now that both and have the same unit of measure, can be found.
Example

Fastball Time to Plate

Tearrik is studying about the theory of baseball during study hall. He reads that on a baseball field, the distance from the pitcher's mound to home plate is meters.

Error loading image.

At his fastest, Tearrik can throw a fastball at per hour. How many seconds does it take for the ball to reach home plate? Round the answer to two decimal places.

Hint

Use the fact that mile is and is

Solution

From the given information, speed and distance are given and time is required.

Error loading image.
Note that the speed of the ball is given in miles per hour, but the distance from the pitcher's mound to home plate is in meters. This suggests that a conversion from miles to meters is needed.
Multiplying the given speed by the conversion factor will convert the speed from miles per hour to meters per hour.
Mult

Multiply by

▼
Simplify
WriteAsFrac

Write as a fraction

CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

Multiply

Multiply

Because the answer is required to be in seconds, a conversion from hours to seconds has to be made. To do so, the conversion factor needs to have a quantity in hours in the numerator and the equivalent quantity in seconds in the denominator.
Using the fact that that hour is equivalent to seconds, the conversion factor can be set.
Having the conversion factor set, the conversion can be performed.
Mult

Multiply by

▼
Simplify
WriteAsFrac

Write as a fraction

CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

CalcQuot

Calculate quotient

RoundDec

Round to decimal place(s)

When units of distance are divided by units of speed, the units of distance cancel out, giving a result in units of time.
Therefore, to compute the time it takes the ball to reach the home plate, the distance must be divided by the speed.
▼
Substitute values and simplify
SubstituteII

,

DivByFracD

CrossCommonFac

Cross out common factors

CalcQuot

Calculate quotient

RoundDec

Round to decimal place(s)

The ball reaches home plate in about seconds, around one-third of a second. This is the same amount of time it takes to blink!
Discussion

Defining New Units

As these examples show, speed combines two different base units, distance and time, in a quotient. Similarly, different combinations of base units can lead to new units, which are called derived units.

Concept

Derived Units

Derived units are units of measurement formed by multiplying or dividing base units in different combinations.

Quantity (Symbol) Expression Base Units Unit Symbol (Name)
Area length width
Volume area height
Speed
Acceleration
Density
Force mass acceleration N (newton)
Pressure Pa (pascal)
Energy force distance J (joule)
Power W (watt)
Frequency Hz (hertz)
The derived units shown in this table are only some of the many possible combinations of base units.

One common example of a derived unit is the acceleration of an object, found by dividing the change in speed by the change in time. The change in speed is measured as the difference between the final and initial speeds.

Acceleration

Note that if the final speed is less than the initial speed, the acceleration is negative and the object is therefore decelerating. When the speed is constant, the acceleration is
Example

Acceleration of a Formula Car

When crossing the starting line, the speed of a Formula car is meters per second. After seconds, its speed is meters per second. Find the acceleration rate, in meters per second squared, of the car in that time interval.

Hint

The acceleration rate equals the change in speed divided by the time taken. The change in speed is the final speed minus the initial speed.

Solution
It is given that the initial speed of the Formula car is meters per second and the speed after seconds is meters per second. Since the final speed is greater, it means the car is accelerating.
SubstituteII

,

SubTerms

Subtract terms

Now that the change in speed is known, the acceleration rate of the Formula car can be computed.
SubstituteII

,

WriteAsFrac

Write as a fraction

DivFracD

Multiply

Multiply

CalcQuot

Calculate quotient

In conclusion, the acceleration rate of the car during those seconds was meters per second squared.
Sometimes information is not provided explicitly, but it can be inferred from context.
Example

Impact Force

During baseball practice, a ball hit by Tearrik traveled seconds before hitting the center field wall at a speed of meters per second. Tearrik wondered about the force with which the ball impacted the wall. Since force is mass times acceleration, to help him, the coach told Tearrik that a baseball has a mass of grams.

Error loading image.

If the speed of the ball when Tearrik hit it was meters per second, find the impact force with which the ball hit the wall. Round the answer to two decimal places.

Hint

From the derived units table, the base units for newtons are kilograms multiplied by meters per second squared. Therefore, the mass of the ball needs to be converted from grams to kilograms. Acceleration equals the change in speed divided by the change in time.

Solution
According to the derived units table, force is obtained by multiplying the mass of the object, in kilograms, by its acceleration in meters per second squared.
First, convert the mass of the baseball from grams to kilograms using the fact that kilogram is grams.
Mult

Multiply by

▼
Simplify
CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

Multiply

Multiply

CalcQuot

Calculate quotient

It is given that the ball hit the center field wall at a speed of meters per second. This is the final speed of the ball. Also, it is said that the speed of the ball at the moment it was hit — the initial speed — was at meters per second.
Using this speed and the fact that the ball traveled seconds before hitting the wall, its acceleration can be found.
▼
Substitute values and evaluate
SubstituteII

,

WriteAsFrac

Write as a fraction

DivFracD

Multiply

Multiply

CalcQuot

Calculate quotient

Finally, the force with which the ball impacted the center field wall can be calculated.
▼
Substitute values and simplify
SubstituteII

,

Multiply

Multiply

RoundDec

Round to decimal place(s)

Substitute

The ball impacted the wall with about of force.
Graphs can also be used to represent the relationship between units in a given situation. When time is involved, it usually is placed on the horizontal axis.
Example

Time vs. Speed Graph

Saturday morning before baseball practice, Tearrik went out for a jog beginning at He was meters away from his starting location at Over the next minutes he jogged another meters. Tearrik then jogged another meters before finally stopping at Make a graph representing the time and Tearrik's speed during his jog.

Answer
Time vs Speed graph

Hint

Place time on the horizontal axis and speed on the vertical axis. Find the speed in each time interval by dividing the distance covered by the time interval.

Solution

Start by organizing the given information in a table. Note that information for three time intervals is given.

Start Time End Time Time Interval (minutes) Distance Covered (meters)

To find the speed in each interval, divide the distance by the time interval.

Time Interval (minutes) Distance Covered (meters) Speed (meters per minute)

Next, place time on the horizontal axis and speed on the vertical axis. Each number to the right of the origin corresponds to the minutes after Based on the table, it is convenient to divide the horizontal axis into intervals and the vertical axis into intervals of meters per minute.

Time vs Speed graph

Next, translate the information from the table to the graph.

  • The first minutes, Tearrik's speed was at meters per minute. This information is represented by a horizontal line from to
  • The next minutes, Tearrik's speed was at meters per minute. A line from to represents this information.
  • The last minutes, Tearrik's speed was meters per minute. This is represented by a line from to
Consequently, the graph representing the time and speed during Tearrik's jog is represented as follows.
Time vs Speed graph
Closure

Voyager 1 Arrival Year

With all this new information learned, the time it takes Voyager to reach Proxima Centauri can now be calculated. Recall that Voyager travels at a constant speed of kilometers per hour and that the distance from Earth to Proxima Centauri is light years, which is the same as kilometers.
To determine when Voyager will reach Proxima Centauri, use the formula below.

Next, substitute the values and simplify. Remember, stands for the speed (rate).
▼
Substitute values and simplify
SubstituteII

,

DivByFracD

CrossCommonFac

Cross out common factors

CancelCommonFac

Cancel out common factors

Since the time is asked in years, a unit conversion is needed. To do this conversion, use the fact that year has days and each day has hours.
Mult

Multiply by

▼
Simplify
CrossCommonFac

Cross out common factors

MultFrac

Multiply fractions

Multiply

Multiply

CalcQuot

Calculate quotient

RoundInt

Round to nearest integer

Consequently, Voyager will take about years to reach Proxima Centauri.
Trip of Voyager 1 from Earth to Proxima Centauri


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