Mathematical Toolkit
Objectives








State fundamental SI units for time, length, and mass.
Demonstrate an ability to use scientific notation.
Identify and use common metric prefixes.
Be able to perform arithmetic operations using factor-label and scientific notation.
Recognize that all measured quantities have uncertainties.
Distinguish between independent and dependent variables.
Be able to graph data points.
Recognize direct (linear), inverse (hyperbolic), and quadratic (parabolic) relationships.
Key Terms
metric system
SI
fundamental units
second
meter
kilogram
factor-label
scientific notation
prefix
significant digit
percent error
independent variable
dependent variable
linear relationship
slope
y-intercept
R2 value
direct relationship
indirect relationship
inverse relationship
quadratic relationship
interpolation
extrapolation
Summary Review
The Metric System, Factor-Label Method and Scientific Notation, and The Prefixes of SI

In science, the International System of Units (SI) are the standard units used worldwide
based on the metric system.

The fundamental units of SI are meter, second, and kilogram.

The Factor-Label Method is a process used to convert units.

Scientific notation is used to express numbers. The proper form is M  10n, where
1  M < 10 and n is an integer.

Prefixes are used to simplify scientific notation.

The proper form of a prefix is M up, where 1  M < 1000, p is a prefix, and u are the units.
Significant Digits in Measurement and Percent Error

A significant digit is a reliable number reported in measurement.

The device used to measure a quantity determines the number of significant digits.

Percent error is used to determine the variations of measurements compared to a known
value.
Graphing and Interpreting Data

The independent variable is plotted along the x-axis. The dependent variable is plotted
along the y-axis.

The coefficient of determination (R2 value) is a reliability range from 0 to 1 that indicates
how well the data falls on a line of best fit.

A linear relationship fits the equation y  mx  b .
m

An inverse relationship fits the equation y  .
x

A quadratic relationship fits the equation y  mx2  b .

Interpolation is reading a graph between data points. Extrapolation is reading a graph
beyond the limits of the collected data.
The Metric System
Demonstration and Notes

Hand out several objects (book, cup, marker, paper, etc.). Determine the dimensions of the
various items. Observe the units used to make measurements.
Americans are used to the English system of measurement. These units include the foot, pounds,
miles per hour, and gallon, just to name a few. To convert from one unit to another, various
numbers must be know. For example, 12 inches equal 1 foot, 3 feet equal 1 yard, and so on.
The standards for specific units of measurement used to vary from year to year. Often a king or
other ruler would define standard units. For example, one foot was defined as the actual length
of the king’s foot. Today, standards are set by the Department of Weights and Measures.
In science, especially physics, the English system is not used. The metric system is always
preferred when making measurements. This is due to the unique relationship that exists between
units of different size: Powers of 10.
metric system – system of measurement that uses base 10.
International System of Units (SI) – standard units used worldwide based on the metric system.
fundamental units – most basic units of measurement in the SI; include time, length, and mass.
second (s, sec) – standard unit of time; frequency of radiation emitted by a cesium-133 atom.
1
meter (m) – standard unit of length; distance light travels in
second in a vacuum.
299,792,458
kilogram (kg) – standard unit of mass of an object; only unit not defined in terms of properties of
atoms; the mass of a platinum-iridium metal cylinder stored near Paris, France.
There are actually seven principal units in SI. They are: ampere (A) electric current, Kelvin (K)
temperature, mole (mol) amount of a substance, and candela (cd) luminous intensity.
Name
second
meter
kilogram
ampere
Kelvin
mole
candela
Abbreviation
s, sec
m
kg
A
K
mol
cd
Category
time
length
mass
electric current
temperature
amount of a substance
luminous intensity
The Factor-Label Method and Scientific Notation
Demonstration and Notes

Determine how many seconds are in one year. Observe the amount of space the answer
requires. There are 31,536,000 seconds in one year.
1 year 
365 days 24 hours 60 min 60 sec



 31,536,000 seconds
1 year
1 day
1 hour 1 min
factor-label – process used to convert units.
Extremely large and small numbers are often written in scientific notation.
scientific notation – exponent notation typically used to express large and small numbers; written
in the form M  10n, where 1  M < 10 (with a maximum of two decimal places) and n is an
integer.
When moving the decimal point to the right, you reduce the exponent when using scientific
notation (Right – REDUCE). When moving the decimal point to the left, you make the exponent
larger when using scientific notation (LEFT – LARGER). Common powers of ten include 100 =
1, 101 = 10, 102 = 100, etc.
Using scientific notation, there are 3.15  107 seconds in one year.
You can also enter answers in scientific notion in Moodle. Use either of the following formats:
Mx10^n or Men. (For example, 8.34 x 10 -4 would be entered as 8.34x10^-4 or 8.34e-4)
The Prefixes of SI
Demonstration and Notes

Recall how many seconds are in one year. There are 3.15  107 seconds in one year.
When single units are used, scientific notation can be replaced with a prefix. For example, one
year has 3.15  107 sec = 31.54  106 sec = 31.54 Msec.
prefix – letters placed before units used to simplify scientific notation; written in the form M pu,
where 1  M < 1000 (with a maximum of two decimal places), p is a prefix, and u are the units.
Prefix
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deca
Symbol
Y
Z
E
P
T
G
M
k
h
da
Multiple 
24
10
1021
1018
1015
1012
109
106
103
102
101
Example
yottameter (Ym)
zettagram (Zg)
exasecond (Es)
petameter (Pm)
teragram (Tg)
gigasecond (Gs)
megameter (Mm)
kilogram (kg)
hectosecond (hs)
decameter (dam)
Standard Unit – 100
n
p
f
a
z
y
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24
decimeter (dm)
centisecond (cs)
milligram (mg)
micrometer (m)
nanosecond (ns)
picogram (pg)
femtometer (fm)
attosecond (as)
zeptogram (zg)
yoctometer (ym)
Symbol
Fraction 
Example
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
d
c
m
Prefix

Prefixes are used in place of the power of ten (10n). For example, 5  10 –5 m can be written as
50  10-6 m = 50 µm (micrometers). Your calculator can also help with prefixes. Using ENG
Mode makes the calculator automatically choose the proper exponent, corresponding to the
appropriate prefix.
Scientific Notation, Prefixes, and Factor Label
Convert each of the given quantities to scientific notation.
1.
1,000,000 =
6.
0.6024 =
2.
0.0000000000564 =
7.
28,000 =
3.
67,800,000 =
8.
0.000004 =
4.
0.00379 =
9.
435,499,000,000,000 =
5.
4.13 =
10. 0 =
Convert each of the given measurements to the appropriate prefix.
11. 0.0034 m =
16. 45,000,000,000 g =
12. 8,340,000 g =
17. 0.0984 L =
13. 0.0004452 L =
18. 14,840,000,000,000 g =
14. 573 L =
19. 0.378 m =
15. 0.0000000000281 m =
20. 57 L =
Convert each of the given measurements using the factor-label method.
21. 33 km/h  m/s
24. 15 minutes  seconds
22. 17 years  seconds
25. 517 m/h  m/s
23. 299 km/min  m/s
26. 12 months  seconds
The Prefixes of SI
Prefix
yotta
zetta
exa
peta
tera
giga
mega
kilo
hecto
deca
Symbol
Y
Z
E
P
T
G
M
k
h
da
Multiple
24
10
1021
1018
1015
1012
109
106
103
102
101
Prefix
Symbol
Multiple
deci
centi
milli
micro
nano
pico
femto
atto
zepto
yocto
d
c
m
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
10-21
10-24

n
p
f
a
z
y
Unit Conversion Factors
LENGTH
1 km = 0.6214 mi
1 m = 3.281 ft
1 in = 2.540 cm
1 mi = 5,280 ft = 1.609 km
TIME
1 min = 60 s
1 hr = 3,600 s = 60 min
1 yr = 365.25 days
SPEED
1 m/s = 3.281 ft/s
1 km/hr = 0.6214 mi/hr
1 mi/hr = 1.609 km/hr
MASS
1 kg has a weight of 2.205 lb when g = -9.8
m/s2
ANGLE
1 revolution = 360 = 2 rad
FORCE
1 N = 0.2248 lb
1 lb = 4.448 N
POWER
1 hp = 746 W
ENERGY
1 ftlb = 1.356 J
Significant Digits in Measurement and Percent Error
Demonstrations and Notes

Hand out several objects of known length. Determine the length of the various items using
measuring devices with various divisions. Observe the number of significant digits used in
making the measurements.
When making measurements, the device used to actually measure the object is important. Every
measuring device has a specific number of significant digits it can measure.
significant digit – reliable number reported in measurement.
When you read any scale, you always record the measurement by reading the smallest division
on the scale and then guessing or estimating the tenth of the smallest division. The number of
significant digits is dependant upon the number of base 10 divisions on the measuring device.
Determine the correct measurements below.
a. 13.50, b. 11.09, c. 14.35, d. 12.26, e. 12.62

Using the known lengths, determine the percent error of the measurements made above.
percent error – used to determine the variations of measurements compared to a known value.
Percent Error 
measured value  accepted value
100
accepted value
Graphing and Interpreting Data
Demonstrations and Notes

View a data table showing original speed, reaction distance, braking distance, and total
distance needed to stop. Interpret the data. As speed increases, total distance increases.
View a distance versus speed graph showing the distance needed to stop at various speeds.
Interpret the graph. As speed increases, total distance increases more with the speed.
Determine which variable is controlled. Speed.
Reaction and Braking Distances vs. Speed
Original
Speed
(m/s)
Reaction
Distance
(m)
Braking
Distance
(m)
Total
Distance
(m)
11
16
20
25
29
8
12
15
18
22
10
20
34
50
70
18
32
49
68
92
Distance vs. Speed
Distance (m)
100
90
Reaction Distance
80
Braking Distance
70
Total Distance
60
50
40
30
20
10
0
0
5
10
15
20
25
30
Speed (m/s)
independent variable – variable that is manipulated or changed in experiment; always the x-axis.
dependant variable – variable that responds to change of manipulated variable; always the y-axis.
The relationship between the independent and dependant variable may not be obvious from
simply looking at the written data. Plotting data and viewing it in graphical format often helps
interpretation. When plotting a graph, take the following steps.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Identify the independent and dependent variables. The independent variable is plotted on
the horizontal (x-axis). The dependent variable is plotted on the vertical (y-axis).
Determine the range of the independent variable to be plotted.
Decide if the origin (0,0) is a valid data point. Spread out the data as much as possible.
Let each space on the graph stand for a convenient unit.
Number and label the horizontal axis.
Determine the range of the dependent variable to be plotted.
Spread out the data as much as possible. Let each space on the graph paper stand for a
convenient unit.
Number and label the vertical axis.
Plot your data points on the graph.
Draw the best straight line or smooth curve that passes through as many data points as
possible. Do not use a series of straight line segments that connect the dots.
Give the graph a title that clearly tells what the graph represents. The standard format
consists of the dependent variable versus the independent variable (y-axis vs. x-axis).
Data can be viewed in various graphical formats. Some include linear, bar, pie, and scatter.
Distance vs. Speed
100
90
Braking Distance
80
Reaction Distance
Distance (m)
70
60
50
40
30
20
10
0
0
11
16
20
25
29
Speed (m/s)

Observe different graphs showing various relationships. Analyze each graph, focusing on
how the x-axis and y-axis relate to one another.
linear relationship – as the x-axis changes, the y-axis changes at a constant rate.
y  mx  b
slope (m) – rate of change; rise divided by run; constant value.
y-intercept (b) – where the curve crosses the y-axis; where x = 0.
R2 value – coefficient of determination; reliability range from 0 to 1 that indicates how well the
data falls on a line of best fit (trend line); >0.95 linear, 0.9 to 0.95 possibly linear, <0.9 not linear.
There are two types of linear relationships: direct and indirect.
direct relationship – the x-axis and y-axis are both increasing at a constant rate creating a positive
slope, y   mx  b .
Direct Relationship
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
indirect relationship – either the x-axis increases as the y-axis decreases at a constant rate
creating a negative slope or the x-axis decreases as the y-axis increases at a constant rate creating
a negative slope, y   mx  b .
Indirect Relationship
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
inverse relationship – as the x-axis increases, the y-axis decreases at a varying rate or as the xaxis decreases, the y-axis increases at a varying rate.
m
x
y
Inverse Relationship
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
quadratic relationship – as the x-axis increases, the y-axis increases at an parabolic rate.
y  mx 2  b
Quadratic Relationship
300
250
200
150
100
50
0
0
1
2
3
4
5
6
7
8
9
10

Observe a linear graph. Determine a data point not included in the raw data. Determine a
data point beyond the raw data.
Direct Relationship
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
interpolation – reading the graph between data points.
extrapolation – reading the graph beyond the limits of the collected data points.
10
Measurement, Conversions, and Percent Error
Measure the following items, using proper units as well as the proper number of decimal places
based on your measuring device. Convert the measurements to complete the blanks.
DO NOT MAKE ANY MARKS ON THE ITEMS LISTED BELOW WHILE MEASURING!
Using a stick marked off in 10 cm increments
1.
Length of line =
cm =
m
2.
Length of longest side of white board eraser =
cm =
m
Using a stick marked off in 5 cm increments
3.
Length of textbook spine =
cm =
m
Using a stick marked off in 1 cm increments
4.
Length of line =
cm =
mm
5.
Length of longest side of white board eraser =
cm =
mm
Using a standard ruler or meter stick
6.
Length of unsharpened pencil =
cm =
7.
Height from floor to bottom rim of front chalk board =
8.
The width (longer side of table top) of the lab desk =
Using a standard protractor
9.
θ=

10. φ =

Using a large field protractor
11. The angle between your rear chair leg and the floor =
km
cm =
cm =
m
m
Using a stopwatch
12. Time it takes a toy car to travel 5 floor tiles =
sec =
hrs
13. Time for 15 pendulum to swing back and forth 5 times =
sec =
min
Using a photogate timer
14. The time it takes the car to pass between the gates =
sec =
ms
Using a hand balance
15. The mass of a tennis ball =
g=
16. The mass of a coffee mug =
g=
kg
kg
Using an electronic balance
17. The mass of a golf ball =
g=
kg
Using a force spring scale
18. The weight of a collision cart =
N=
lbs
When you have collected all of your measurements, sit down and begin the conversions.
Accepted values will be given once everyone is complete.
Calculate the % error for each accepted value given. Show a complete sample calculation for
the first answer. Round each % error to the same number of decimal places as your
measurement.
#8
Accepted
% error =
#10
Accepted
% error =
#15
Accepted
% error =
MICROSOFT EXCEL BASICS
Scatter Plots
1. Click on the Microsoft Excel icon found under the START button, PROGRAMS,
MICROSOFT OFFICE. Excel allows you to manipulate data and create graphs.
2. You will see a spreadsheet. This is where you enter all of your data. Each box of the
spreadsheet is called a cell. Cells are defined by their column and row location in the
spreadsheet. Data found in the fifth row of the third column is said to be in cell C5.
3. In column A, type the values of the independent axis (x-axis) starting at row 1 (A1).
4. In column B, type the values of the dependent axis (y-axis) starting at row 1 (B1). Columns
A and B represent your ordered pairs (A,B).
5. Format the cells to represent the proper number of decimal places. Highlight the column of
data you want to manipulate by right clicking on the column letter and select FORMAT
CELLS from the menu list.
6. Select the NUMBER tab and select the
proper number of decimal places.
7. Highlight the cells you want to include in
your graph. This should include all the
data entered under columns A and B. To
highlight these cells, click on cell A1 and
while holding the mouse button down,
drag the mouse pointer over the other cells
you wish to include. The highlighted cells
will change color when they are selected.
Highlight only the cells you want to
include in your graph. There can be no
empty cells.
8. Now create a chart by clicking on the
INSERT tab. Click the SCATTER button
to choose the type of graph you wish to
create. The chart with only data points will ALWAYS be used.
9. Your graph will appear imbedded in your data sheet.
10. From the DESIGN tab, select the Move Chart button.
11. Click on AS NEW SHEET to create
a separate screen for your chart.
12. Select the LAYOUT tab. This area
allows you to title the graph, label
the x-axis, label the y-axis, include
major/minor gridlines, and include
a legend if multiple sets of data are
displayed.
13. Click on Chart Title and select to place the title Above Chart.
14. Click Axis Titles and select Primary Horizontal Axis Title. Place the primary horizontal axis
title Below Axis.
15. Click Axis Titles and select Primary Vertical Axis Title. Place the primary vertical axis title
as a Rotated Title.
16. Select GRIDLINES. Include Major Gridlines for the Primary Horizontal and Primary
Vertical axes.
17. Select LEGEND. If only one set of data is used, choose None. If multiple sets of data are
displayed, choose Show Legend at Right.
18. After all of your preferences have been set, click in each text box to properly label each
preference with the appropriate heading(s).
19. Click once on the title box and add your name below the title.
20. Draw a line of best-fit by clicking on TRENDLINE. Select More Trendline Options.
21. Select a LINEAR Trend/Regression Type. Include the equation and R-squared value on the
graph by checking the appropriate boxes.
22. Make the graph fill up the paper. Eliminate white space above and below the graph. Right
click on any x-axis value and select FORMAT AXIS from the menu list.
23. Select the SCALE tab and choose an appropriate range for the maximum and minimum
values. Choose major units. Minor units are typically half of the major units. Click OK to
see your graph.
24. Right click on any y-axis value and repeat the above steps to set the scale for the y-axis.
GRAPHING AND INTERPRETING DATA LAB
Introduction – What’s Going On?
Displaying data graphically is often a valuable tool used to interpret laboratory results. Creating
a graph by hand can be tedious and time consuming. Luckily, there are numerous software
packages available that aid in plotting graphs. This lab contains two parts that allow for practice
of graphing data using Microsoft Excel.
If you are not familiar with Microsoft Excel, your teacher will provide step-by-step instructions
after all the data is collected. The graphs you will create for this activity are called scatter plots.
This type of graphing method is used when the data collected does not all lie in a perfectly
straight line. Scatter plots allow you to find lines of best fit or smooth curves that fit the data.
Part I – Time vs. People







Decide how many people will make up the chain (this number can never change) and who
will be the timekeeper.
The chain will form a circle of people all holding hands. The first person will squeeze the
hand of the person to their right. That person squeezes the hand of the person to their
right. This continues until the first person has their left hand squeezed.
Record the number of people in the chain and the time it takes to complete it.
Complete three trials of time data.
Allow the chain to be completed twice. This will double the number of people in the
chain. Record the new number of people in the chain and the time to complete it. Repeat
three times.
Continue to increase the number of times each person of the chain participates. Record
the number of people in each chain and the time it takes to complete the chain.
After collecting ten trials of data, graph a Time vs. People scatter plot. Use the average
time from each set of data.
Practice First.
Record data only after you are consistent!
Try not to anticipate the squeeze, wait for it to happen.
Close your eyes if necessary!
GRAPHING AND INTERPRETING DATA LAB
Part II – Height vs. Shoe Size



Measure the height in centimeters of each person in the class. Record the heights.
Record the shoe size of each person in the class.
After all the heights and shoe sizes are found, graph a Height vs. Shoe Size scatter plot.
You must convert women’s shoe sizes to reflect men shoe sizes.
This is done by subtracting 1½ from the known shoe size.
For example, if Amy’s shoe size is 8, the converted size would be 6½.
Part III – Wrap Up


Displaying Data – Making sure you follow all of the guidelines listed in the Microsoft
Excel Basics handout, create your graphs using Microsoft Excel and save them, along
with your data to a single file called Graphing and Interpreting Data Lab. Include a
trendline, equation, and R-squared value for each graph.
Report Your Findings – As a separate (fifth) tab in Microsoft Excel, write a minimum of
one paragraph for each part of this lab (two parts total). Explain your results and any
conclusions you can draw from what you experienced. Determine the relationship, if any,
that exists between the data for each graph . If any of your observations contradict what
you believe should have happened, explain the reason for the error. In addition to the
paragraphs, include the following:
o Using extrapolation, determine how long would it take for 300 people to complete
the chain. Show your work. Is this a legitimate approximation? Explain.
o Using interpolation, determine the height of someone with a shoe size not
recorded. Show your work. Is this a legitimate approximation? Explain.
o What does the slope of the Time vs. People graph represent?
USE THE TABS IN MICROSOFT EXCEL!
Enter data for each graph in its own sheet.
Rename the sheets to easily identify the data.
Data Table
Part I
Number of People
in Chain
Time
(seconds)
Data Table
Part II
Height
(centimeters)
Shoe Size
(converted to men’s size)
The Handy Physics Answer Book
What are the standards for measurement in physics?
The International System of Units, officially known as SI, was adopted by the eleventh
General Conference on Weights and Measures in Paris, 1960. Basic units were to be
based on the meter-kilogram-second (MKS) system. This is commonly known as the
metric system.
Why doesn’t it seem as though the United States uses the SI system?
Although the American scientific community uses SI, the general American public still
uses the English system of measurement. In an effort to change over to the metric
system, the United States government instituted the Metric Conversion Act in 1975.
Although the act committed the United States to increasing the use of the metric system,
it was on a voluntary basis only. The Omnibus Trade and Competitiveness Act of 1988
required all federal agencies to adopt the metric system in their business dealings by
1992. Therefore, all companies that held government contracts had to convert to metric.
Although approximately 60% of American corporations manufacture metric products, the
English system still seems to be the predominant system of measurement in the United
States.
Who defined or developed the meter?
In 1798, French scientists determined that the meter would be measured as 1/10,000,000
the distance from the North Pole to the equator. After calculating this distance, the
scientists made a platinum-iridium bar that measured that exact distance This standard
was used until 1960, when newer and more accurate methods of measuring the meter
evolved.
The Handy Physics Answer Book
How is the length of a meter determined?
Today the meter is the standard for both the SI and metric system. The meter was
originally defined as the distance between two lines on a bar of platinum-iridium alloy,
1/10,000,000 the distance from the North Pole to the equator. However, the bar was
deemed inaccurate because it would expand and contract depending on its temperature.
In 1960, scientists determined that light would be a better method of measuring a meter.
As a result, the meter was defined as the distance of 1,650,763.73 wavelengths of
reddish-orange light emitted by the krypton-86 isotope. The standard for measuring the
meter was changed again in 1983 as the distance light travels in a vacuum in
1/299,792,458 of a second.
What is the standard unit for measuring mass?
The kilogram is the standard unit for mass in the metric system and SI. The kilogram was
originally defined as the mass of one cubic decimeter of pure water at 4 Celsius. A
platinum cylinder, the same mass as the cubic decimeter of water, was the standard until
1889 when the platinum cylinder was replaced by a platinum-iridium cylinder. This
cylinder, whose mass is close to the original platinum cylinder, is permanently kept near
Paris.
How is a second measured?
Atomic clocks are the most precise devices to measure time. Atomic clocks such as
rubidium, hydrogen, ammonia, and cesium clocks are used by scientists and engineers
when computing distances with Global Positioning Systems (GPS) and measuring the
rotation of the Earth.
The most stable measurement clock that is used as the standard for the second is the
cesium-133 atomic clock. The measurement of the second is defined as the time it takes
for 9,192,631,770 periods of microwave radiation to transfer the cesium-133 atom from a
lower-energy state to a higher-energy state.
Metric Mishap Caused Loss of NASA Orbiter
by Robin Lloyd
http://www.cnn.com/TECH/space/9909/30/mars.metric.02, September 30, 1999
NASA lost a $125 million Mars orbiter because a Lockheed Martin engineering team used
English units of measurement while the agency’s team used the more conventional metric system for a
key spacecraft operation, according to a review finding released Thursday.
The units mismatch prevented navigation information from transferring between the Mars
Climate Orbiter spacecraft team in at Lockheed Martin in Denver and the flight team at NASA’s Jet
Propulsion Laboratory in Pasadena, California.
Lockheed Martin helped build, develop, and operate the spacecraft for NASA. Its engineers
provided navigation commands for Climate Orbiter’s thrusters in English units although NASA has been
using the metric system predominantly since at least 1990.
No one is pointing fingers at Lockheed Martin, said Tom Gavin, the JPL administrator to whom
all project managers report.
“This is an end-to-end process problem,” he said. “A single error like this should not have caused
the loss of Climate Orbiter. Something went wrong in our system processes in checks and balances that
we have that should have caught this and fixed it.”
The finding came from an internal review panel at JPL that reported the cause to Gavin on
Wednesday. The group included about 10 navigation specialists, many of whom recently retired from
JPL.
“They have been looking at this since Friday morning following the loss,” Gavin said.
The navigation mishap killed the mission on a day when engineers had expected to celebrate the
craft’s entry into Mars’ orbit.
After a 286-day journey, the probe fired its engine on September 23 to push itself into orbit.
The engine fired but the spacecraft came within 60 km (36 miles) of the planet – about 100 km
closer than planned and about 25 km (15 miles) beneath the level at which the it could function properly,
mission members said.
The latest findings show that the spacecraft’s propulsion system overheated and was disabled as
Climate Orbiter dipped deeply into the atmosphere, JPL spokesman Frank O’Donnell said.
That probably stopped the engine from completing its burn, so Climate Orbiter likely plowed
through the atmosphere, continued out beyond Mars and now could be orbiting the sun, he said.
Climate Orbiter was to relay data from an upcoming partner mission called Mars Polar Lander,
scheduled to set down on Mars in December. Now mission planners are working out how to relay its
data via its own radio and another orbiter now circling the red planet.
Climate Orbiter and Polar Lander were designed to help scientists understand Mars’ water
history and the potential for life in the planet’s past. There is strong evidence that Mars was once awash
with water, but scientists have no clear answers to where the water went and what drove it away.
NASA has convened two panels to look into what led to the loss of the orbiter, including the
internal peer review panel that released the Thursday finding. NASA also plans to form a third board –
an independent review panel – to look into the accident.
Metric system used by NASA for many years
A NASA document came out several years ago, when the Cassini mission to Saturn was under
development, establishing the metric system for all units of measurement, Gavin said.
The metric system is used for the Polar Lander mission, as well as upcoming missions to Mars,
he said.
That review panel’s findings now are being studied by a second group – a special review board
headed up by John Casani, which will search for the processes that failed to find the metric to English
mismatch. Casani retired from JPL two months ago from the position of chief engineer for the Lab.
“We’re going to look at how was the data transferred,” Gavin said. “How did it originally get
into system in English units? How was it transferred? When we were doing navigation and Doppler
(distance and speed) checks, how come we didn’t find it?”
“People make errors,” Gavin said. “The problem here was not the error. It was the failure of us
to look at it end-to-end and find it. It’s unfair to rely on any one person.”
Lockheed Martin, which failed to immediately return a telephone call for comment, is building
orbiters and landers for future Mars missions, including one set to launch in 2001 and a mission that will
return some Mars rocks to Earth a few years down the line.
It also has helped with the Polar Lander mission, set to land on Mars on December 3 and conduct
a 90-day mission studying Martian weather. It also is designed to extend a robotic arm that will dig into
the nearby Martian soil and search for signs of water.
NASA managers have said the Polar Lander mission will go on as planned and return answers to
the same scientific questions originally planned – even though the lander will have to relay its data to
Earth without help from Climate Orbiter.
Error points to nation’s conversion lag
Lorelle Young, president of the U.S. Metric Association, said the loss of Climate Orbiter brings
up the “untenable” position of the United States in relation to most other countries, which rely on the
metric system for measurement. She was not surprised at the error that arose.
“In this day and age when the metric system is the measurement language of all sophisticated
science, two measurements systems should not be used,” Young said.
“Only the metric system should be used because that is the system science uses,” she said.
She put blame at the feet of Congress that she said has squeezed NASA’s budget to the point that
it has no funds to completely convert its operations to metric.
“This should be a loud wake-up call to Congress that being first in technology requires funding,”
she said, “and it’s a very important area for the country.”
Rocket Scientists’ Metric Conversion Error
Doomed Mars Orbiter
by Andrew Pollack
The New York Times, October 1, 1999
Scientists lost a $125 million spacecraft as it approached Mars last week essentially because they
confused feet and pounds with meters and kilograms, NASA said yesterday.
An internal review team at NASA’s Jet Propulsion Laboratory, in a preliminary conclusion, said
engineers at Lockheed Martin Corp., which had built the spacecraft, specified certain measurements
about the spacecraft’s thrust in pounds, an English unit, but that NASA scientists thought the information
was a metric measurement known as newtons.
The resulting miscalculation, undetected for months as the craft was designed, built, and
launched, meant the Mars Climate Orbiter, as it was called, was off course by about 60 miles as it
approached Mars.
“This is going to be the cautionary tale that is going to be embedded into introductions to the
metric system in elementary school and high school and college physics till the end of time,” said John
Pike, director of space policy at the Federation of American Scientists in Washington.
Lockheed’s reaction was equally blunt.
“The reaction is disbelief,” said Noel Hinners, vice president for flight systems at Lockheed
Martin Astronautics in Denver. “It can’t be something that simple that could cause this to happen.”
The finding was a major embarrassment for NASA, which said it was investigating how such a
basic error could have gone through the space mission’s checks and balances.
“The real issue is not that the data was wrong,” said Edward C Stone, the director of Jet
Propulsion Laboratory in Pasadena, California, which was in charge of the mission. “The real issue is
that our process did not realize there was this discrepancy and correct for it.”
Some experts also wondered how something so basic could have gone undetected for so long.
“This is kind of the very first thing in Physics 101 or Engineering 101,” Pike said. “This is the
only significant program failure that any-one’s ever heard of that’s due to this.”
The failure is another black eye for Lockheed, the nation’s largest military contractor, which in
the last year or so has suffered several failures of rockets and missiles it has developed. These failures
led the company to restructure its operations and change management in its space business.
NASA officials said they are checking to make sure the same error does not occur in the Mars
Polar Lander, now en route to Mars and due to reach the planet December 3.
Two separate review committees have already been formed to investigate the loss of the Mars
Climate Orbiter – the internal Jet Propulsion Laboratory peer review team which made yesterday’s
preliminary findings, and a special review board of experts from the laboratory and elsewhere. An
independent NASA failure review board will be formed shortly.
It is not known with certainty what happened to the Mars Climate Orbiter. At first, there was
speculation that it crashed on Mars or burned up in the atmosphere.
But the review team now tends to think that the spacecraft is now orbiting the sun, a Jet
Propulsion Laboratory spokesman said. Under this theory, the spacecraft approached too close to Mars
and overheated, causing the engine that was being fired to bring the craft into orbit to stop functioning, so
the Orbiter continued on into space.