Describing position and motion with graphs
Equipment needed: meter sticks, graph paper, other equipment described later
In groups - discuss all the questions and checks……
Individually - write them up in your notebook;
In groups - discuss checks with an instructor. Final one is
turned in without instructor discussion.
To study motion and the relationship between force and motion, we often use graphs to describe the
the position and changing positions of an object. Motion in general is in three dimensions. Think of
a bird flying through the air; the motion involves up and down, left and right and forward and
backward. Three quantities must be specified. As you probably know in math and science to
represent position in three dimensions we usually use the three coordinates x, y and z instead of updown etc. Sometimes 2 dimensional representations of our world, are useful, such as in our usual
flat maps. In science and math we usually use an x,y grid. In all cases, we use a reference point,
called an origin, and accurately locate points relative to this origin that we have specified. With this
x-y system any point on your grid can be described as an ordered pair (x,y). Both x and y can take
on positive and negative values.
One dimensional motion, position, displacement and distance
a) For most of our activities on force and motion we will be concentrating on motion that occurs in a
straight line. We call this one-dimensional motion. We almost always use x as the coordinate for
representing position along a straight line. If we are describing any kind of motion in a straight
line, we always put our x axis along the line of motion. Even if you move along a diagonal line an
and x-y grid, for convenience we change our choice of axes so that the x axis is along the line of
motion.
b) As an exercise each group should imagine an ant moving in a straight line across the table in front
of you. One of the group can move his or her hand simulating the imagined motion. Now, choose
an origin; someone hold a finger there until a meter stick can be placed. Then choose a direction
for your x-axis, and place the meter stick along this line with 0 cm at the origin. (This of course is
along the line of motion of the imagined ant.) This corresponds to positive values of x, increasing
as one moves away from the chosen origin. The opposite of this direction corresponds to negative
values of x. Are positions corresponding to negative values of x valid as positions of objects? As
as the ant moves, is it valid motion if it moves from negative values of x, through the origin and
toward positive values of x? We call this moving toward increasing values of x.
c) Now move the ant toward decreasing values of x going from positive values, through the origin to
negative values. Is this motion toward decreasing values of x valid? Is this object always going to
decreasing values of x?
d) Imagine that the ant starts from an initial point about half way along the positive x-axis, i.e. at +50
cm. Does an object always have to start motion from the origin? From this initial point move it
several 10 cm or so in the positive x direction then move it fewer number of cm in the negative x
direction. What is the final position? For the change from the initial point to the final point what is
the distance traveled? Note that distance traveled is always a positive number. Think of a way to
describe distance traveled in general and for one dimensional motion. The change in position
from the initial point to the final point is called the displacement and can be calculated by final
position (x) minus initial position (x) or x(final)-x(initial) or xf – xi also called delta x or ∆x . It can
be positive or negative. What is the displacement for the small object in this problem? Is it
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positive or negative? What does the sign tell you? Is the displacement different from the
distance traveled in this problem? The change in position is in general different from distance
traveled. For example, if you drive from here to Louisville and back, what is the distance
traveled and what is the displacement?
e) Again take your ant and start it from the same initial x point. From this initial point move it several
tens of centimeters in the negative x direction then move it a fewer tens of centimeters in the
positive x direction. For the change from the initial point to the final point what is the distance
traveled? Recall that distance traveled is always a positive number. What is the displacement in
this case? Is it positive or negative? What does the sign tell you?
f) Two students conduct an experiment in which an object is moved in a straight line from an initial
position xi to a final position xf. Student 1 and student 2 each make different choices of their origin
location. Since each student has chosen a different origin, we say they are using different
coordinate systems. A direct way to do this is to imagine that the line of motion of the object is
between two meter sticks. The meter sticks both are along the line of motion, but one has its origin
or x=0cm at on place and the other has its origin at a different place. Describe the positions that
students 1 and 2 would have for xi and xf in each case and describe the displacement in each
case. Do this with symbols without use of numbers, and then do it using numbers for two
different sets of xi and xf.
In your notebook answer the following questions on the basis of your results above. Explain your
reasoning.
 Does the measurement of position depend on the choice of a coordinate system?
 Does the measurement of displacement depend on the choice of coordinate system?
Time
The word time has several meanings in ordinary language. For example: What time did he arrive? How
much time does that job take? We had a fine time. Time out. Our time has come. In physics, the word
time must be used carefully because it is used to name several quantities. The exact meaning of the
word in physics is often told in the accompanying words, for example, at that time, for a long time. We
will be concerned with two uses of the word corresponding to two different quantities: one called the
time, meaning a clock reading, and the other called duration, time interval, or amount of time.
These quantities are discussed below.
In describing position, we imagine a meter stick that continues forever in both directions. When
considering time, we imagine a clock that always has been running and will continue to run forever.
Instead of an ordinary clock, we imagine a clock that just keeps count of seconds. It does not start over
after 12 hours as our usual clocks do. The clock we imagine keeps counting up to millions and billions
of seconds. We use the symbol t to represent the clock's reading.
The number on the clock is called the time. We describe when something happens by giving the time
shown on the clock (the clock reading) as the event occurs. A typical event, for example, would be the
passing of the 10 cm mark by a moving object. If the clock read 25 seconds as this happened, we would
say that x = 10 cm at t = 25 seconds.
Just as x = 0 is not necessarily a special place, t = 0 is not necessarily a special instant. In particular, t =
0 does not have to be the instant motion begins. For example, suppose two cars are making the same
trip but one of them has a five-minute head start. We might set our clock to read t = 0 when the first car
starts, but then the second car's motion would begin at t = 5 minutes.
Instants before t = 0 have negative time numbers, such as t = -5 seconds. This is similar to what we do
for years before the year one, such as 1026 B.C. Negative times are perfectly ordinary; they are just
earlier than positive times.
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a)
b)
c)
Suppose a ball was dropped at t = 2140.0 seconds and hit the floor at t = 2141.2 seconds. How
much time did it take the ball to fall?
The problem above illustrates the method of determining duration or amount of time between two
instants. We use the symbol Δt to mean tf - ti, the amount of time between tf and ti. Again, Δt
always means "final time minus initial time," and is read "delta t."
Two observers timed the motion of a car from one place to another. The first observer's clock read
262 seconds at the start and 375 seconds at the end of the car's motion. The second observer's
clock read -86 seconds at the start. What did it read at the end?
Tell whether the answer to each of the following questions would be a time or a duration:
A When will we go shopping?
B. How long will it be before the bus comes?
C. When did his term in office end?
D. How old is that building?
E. When the war ended, how long had you been overseas?
F. How much time does a job like that take?
Note that the term instant needs special attention because its meaning in physics is not quite the same
as it is in ordinary language. In physics, an instant is always a time, never a time interval. An instant
is when (like 2:15 pm), not how long.
CHECK#1: Write a description of what you have learned about the origin, position, displacement,
distance traveled and time, and check with an instructor.
Motion
When an object moves, we not only need to specify its position but we also need to specify the time it
has that position. Because of the way a graph is constructed, it allows us to accurately infer
information about the motion at points/times that are not measured. In this way we gain the benefits of
many measurements without having to do all the work! We can also use words and pictures like
diagrams of the motion to represent and describe motion in a straight line; these have advantages
especially for people beginning to learn about motion. However, the most accurate and most efficient
representation of motion is through graphs, but the user must have an excellent understanding of
graphs.
Let's begin by making simple graphs of the motions you have observed for which you have data,
namely the Kick-Dis (abbreviated KD) and the person on the skate board (abbreviated SB). This gentle
introduction to graphs will prepare you for the activities that follow.
Equipment needed:
 Graph paper
a)
b)
c)
Using the data recorded last time for the KD and the SB, think first about the selection of the
axis and origins for both time and position. We chose the most convenient and useful values for
these.
Think about how to make a graph describing the motion of each. Think about what you learned
in math about dependent and independent variables. In this case which is the independent
variable and which is the dependent variable? Which usually goes on the horizontal axis
of the graph and which goes on the vertical axis of the graph? Usually on an x-y plot we
have x on the horizontal axis and y on the vertical axis; however here since position is
considered the dependent variable, we put it on the vertical axis and put time, the independent
variable on the horizontal axis, giving us a position versus time graph also known as a x-t graph.
Make a graph for data that you have for the uniform motion of the “Kick Dis” Does the graph
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d)
e)
f)
seem reasonable? Save the paper graph, and later make copies for every member of your group.
Look at the graph of points on the x-t graph. Sight along the points. Is a straight or curved
line the best representation of the data? We know that the data for the KD has measurement
errors in both the time and position data. What effect does measurement errors have on the
graphs? Do you think simply connecting the dots with straight line segments is the best
representation of the data between the measured points? If you think that the data are
best represented by a straight line, draw the “best fit” straight line through them.
Roughly, a “best fit” line has a distance between the line and data points that are above the line
in the same amount as the line and the data points below the line. If we now want to use the
graph for determining anything about our measurements, is it better to use the actual
points or the graph which is a best fit? One will have a tendency to want to use our actual
data for determining the position at for example 2 seconds, but the best fit straight line is
actually better since some of the measurement error is averaged out.
If your data can be described as a straight line, look at the line that you drew that is the
best fit to the data. Explain how you can use your line to predict the position of the “Kick
Dis” at times not measured. We have data at the end of each second; times not measured
include times intermediate to these and at larger times not measured. Explain what this
means.
Looking at the plot line, how does the speed with which an object moves manifest itself on
the graph? If the line is steeper, is the object moving faster, slower or the same speed? If
the line is less steep, is the object moving faster, slower or the same speed?
CHECK #2 Write a check summarizing the important points you learned in this section about
describing uniform motion with graphs.
NOTE: If available, using a photocopy machine, make copies of the graphs with all the lines and notes
for each member of your group. These can be placed in your notebook.
a)
b)
d)
e)
Now make a graph for each of the points for SB, riding-on skate board motion. In all graph
making take advantage of as much of the graph paper as you can. Accuracy is lost if the graph
you make only uses a corner of the graph paper.
Examine the graphs. Sight along the points. Can you draw straight lines with a good fit
through all of the data? If your data are not straight lines, can you draw a smooth curve
that adequately represents all the points? You may not go through all the points exactly, but
you should be able to draw a smooth curve, not sections of straight lines, that best fits the
observed data. What does this curve mean (as opposed to a straight line)? Can you use this
curve to predict the position of the object at times not measured (intermediate and beyond
the measured times)? Clearly explain how, and give examples.
Is there a special characteristic of the appearance on a position vs. time graph for motion
of objects that are either always speeding up?
Compare this graph with the graph for the motion of the Kick-Dis moving with uniform motion.
When you have straight line data, can you look at the plot and get a sense of the speed of
the object – is it slow or fast? With the curved data for speeding up and slowing down,
can you get a sense of how much the speed of the object is changing? How fast it is moving
at the beginning and how fast it is moving at the end?
CHECK #3 Write a check summarizing the important points you learned in this section about
describing speeding-up motion with graphs.
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Making graphs of position versus time using the Vernier Lab Quest
Equipment needed:
LabQuest
Motion Detector for LabQuest
Clear view of the wall for each group
Masking tape.
Your instructors will form all of the participants into groups of
two.
a) Obtain a LabQuest Device from an instructor. Read the startup guide and turn the device on.
b) Familiarize yourself with its operation.
c)
 Using the stylus select File.
 Choose View Lab Instructions
 scroll down using scroll bar on right, and choose Middle School Science with
Vernier and using your stylus click OK,
 scroll down and choose 33. graphing your motion_html and push OK
d) Read about the experiments we will be doing. Although the experiments describe position,
velocity and acceleration measurements, today we will start with position, and at a later time
look at velocity and acceleration. These instructions and other activities are described below.
1.
2.
3.
4.
5.
6.
7.
The instructions call for you to connect the Motion Detector, a device which you can get from
an instructor.
Find an open area at least 4 m long in front of a wall. Use masking tape on the floor to mark
distances of 1 m, 2m and 3 m from the wall. You will be measuring distances from the wall
using the Motion Detector which you will carry in your hand.
If your Motion Detector has a switch, set it to Normal. Connect the Motion Detector to DIG 1
of the LabQuest and choose New from the File Menu.
Open the hinge on the Motion Detector. When you collect data, hold the MD so the round,
metal detector is always pointed directly at the wall. Sometimes, you may have to walk
backwards.
Observe the position readings as you move back and forth and confirm that the values make
sense.
Make a graph of your motion when you walk away from the wall at a constant speed. To do
this, stand about 1 m from the wall and start data collection by pushing the start button in the
center of the LQ (just above the circle of buttons ). Walk backward, slowly, away from the wall
after data collection begins (listen for the “clicks” as well as seeing data on the screen. In
your notebook record a sketch of the graph, including axes labels and
units and title the graph according to the type of motion.
Sketch what you think the graph will look like if you walk in the same manner but faster.
Check your prediction. In your notebook record a sketch of the graph,
including axes labels and units and title the graph according to the
type of motion.
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f) Each person should do the motions in every case.
g) In a similar manner, make, label and title graphs of your motion for the following (If you can,
instead of carrying the MDLQ with you, set up the motion detector on the table and move in
front of it. Your partner can turn it off and on.):
 moving away speeding up
 moving away slowing down
 coming towards with constant velocity, slow
 coming towards with constant velocity, fast
 coming towards speeding up
 coming towards slowing down
 standing still
h) Look at the graphs you made for the KD and SB. Using your own motion and observing on the
screen, see if you can quantitatively replicate the KD motion graph and qualitatively replicate
the SB motion graph. (Again it is better if you can set up the motion detector on the table and
move in front of it, having your partner turn it on and off. Where was the origin for the KD
and SB experiments? Where is the origin for the motion detector experiments when you
move in front of it, leaving it stationary on the table?)
I) Now with graph on the LQ, choose Analyze, choose Motion Match>New Position Match from the
menu. A target graph will be displayed for you to match as you walk and carry the MD.
j) Carefully describe and write down your prediction of how you would walk to reproduce the target
graph.
k) To test your prediction, choose a starting position and stand at that point. Start data collection. Wait
a moment and then walk in such a way that the graph of your motion matches the target graph
on the screen.
l) If you were not successful, start data, and when you are ready begin walking. Repeat this process
until your motion closely matches the graph on the screen. Sketch the graph with your best
attempt.
m) Perform a second position or distance graph by choosing Motion Match>New Position. This will
generate a new target graph for you to match.
n) Make sure that everyone gets a chance to match two graphs.
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Final Check :
 to be done individually without assistance from other participants or instructors
 turned in
You will be given graph of x vs t . Describe carefully and fully in words the motion represented on the
graph. Assume that you are writing your description for someone who knows very little; when in doubt
write it out.
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