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One-Dimensional Motion (1D Kinematics)
Reading Assignment:
HRW Chapter 2, Sections 2-10
Knight Chapter 2, Sections 1-7
Introduction:
Kinematics is the description of objects’ motion (dynamics, which we will study soon in this
course, is the explanation of objects’ motion). Kinematics focuses on the relationships, both
mathematical and graphical, between position, displacement, velocity, speed, acceleration as a
function of time or during intervals of time.
Motion can be a thrilling part of our everyday
experience. Much of the excitement in riding a
roller coaster arises from the high speeds and
accelerations experienced during the ride.
The Steel Phantom at Kennywood
Amusement Park
http://www.kennywood.com/
Position (x), velocity (v) and acceleration (a) are
vectors, which means that they have both a
magnitude (numerical value and units) and a
direction. For one-dimensional motion that
direction shows up as a + or – sign in front of the
numbers (+ is implied if no sign is given). Even
when motion occurs in more than one dimension,
as on a roller coaster, motion in each dimension
can be considered separately.
One important special case of one-dimensional motion is constant acceleration motion, which
we will see later occurs any time that an object is subject to a constant force. This case also
includes constant velocity motion (i.e., the constant acceleration has a value of zero), which itself
includes the case of an object at rest (zero velocity and zero acceleration). For constant nonzero
acceleration motion, at any instant of time the velocity could be zero (but for only an instant), in
the same direction or in the opposite direction of the acceleration.
Constant acceleration
a=0
Constant velocity
v=0
At “rest”
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Conceptual Understanding Goals: By the end of this laboratory, you should be able to:
(1) Articulate the relationships between position, velocity and acceleration in onedimensional motion.
(2) Draw and recognize x(t), v(t), and a(t) graphs corresponding to constant acceleration
motion.
(3) State the conditions on velocity and acceleration for which speed (|v|) increases or
decreases.
Laboratory Skill Goals: By the end of this laboratory, you should be able:
(1) Explain the function, operation, and limitation of the ultrasonic motion sensor.
(2) Collect data using DataStudioTM software and a motion sensor, tools that you will use for
many of the laboratories in this course.
(3) Visualize and analyze data within DataStudio – e.g., create graphs, select regions of data,
find slopes, and do linear and quadratic fits to data.
(4) Draw reasonable conclusions about the motion of an object based upon data.
Equipment List:
1.2 meter Low Friction Track (end bracket on left of track)
Dynamics Cart with plunger at one end
Ultrasonic Motion Detector
¾”-Thick Wooden Block (to fit under legs of track)
Computer with DataStudio™ and ScienceWorkshop 750 Interface
Angle Measuring Device (near right end of the track)
You will be doing three activities:
Activity 1. Learning to use the Motion Sensor and DataStudio software
Activity 2. Exploring your own motion
Activity 3. Exploring the constant acceleration model
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Physics 211R: Lab Report Template
Motion in One Dimension (1D Kinematics)
PRINT THESE PAGES (p. 3-4) WHEN YOU BEGIN AND INCLUDE IN YOUR REPORT
Warm-up: Show the relationships between x, v, and a; also, list the X and V eqns for constant a
x
v
a
X:
V:
Q3. (Draw a large dot for the origin and an arrow pointing in the positive direction.)
Q7. What would the velocity graph look like because of this spike in the position data?
v
x
t
t
Q9.
v
x
t
t
4
Q15.
x
t
v
t
a
t
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Physics 211R: Lab Report Template
Motion in One Dimension (1D Kinematics)
(Type in this document and print these pages at the end of the laboratory)
Notes:
 A maximum of three students will be allowed per group.
o In the event that a group of four students must be formed, you need to check with your
laboratory instructor before starting to work on the activity.
 All the members of the group must participate in the activity. If a student is not participating
(even when present) s/he may receive a score of zero in the activity.
 Students arriving 10 minutes or more past start will not be admitted.
 This activity must be returned at the end of the lab period. All the students completing the
activity must be present when handing this to the laboratory instructor; a student not present
at this time may not get credit for the activity.
o Writing the name of a person not present is not permissible and may result in a
potential academic integrity violation being processed.
 After you receive the graded report back, you should make a copy of the front page (this
page) and keep it for you records. This will serve as evidence of your grade for this activity.
 You are responsible for checking your grade (in the course website) and report any mistakes
to your TA within two weeks after the activity.
Date: ________________
Enter your name as it appears in your PSU registration, no nicknames please.
Name:
Section #
Name:
Section #
Name:
Section #
Clean Up Check:
After you finish working and completing the lab report, you need to clean and organize your
working area. Then call one of your laboratory instructors who will check your area, initialize
below and take the lab report. All the members of the group must be present at that time. If you
leave the lab before your laboratory instructor performs the check up, you will be deducted 5
points from your score for this lab report.
Laboratory Instructor Initials: _______
Score: _______
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Activity 1. Exploring the operation of the Motion Sensor and DataStudioTM
This first activity is to familiarize yourself with the motion sensor and DataStudio software so
that you are able to design experiments in subsequent laboratories using this equipment. Many of
the laboratories in this course will use this equipment so what you learn here will aid you the
entire semester. Understanding a piece of equipment also requires understanding its limitations
and what settings are best for a given context.
Set up the Motion Sensor and DataStudio software as specified in the supplemental document on
the course website laboratory page and then explore how the motion sensor and software works
so that you can give complete answers to these guiding questions.
For Activities 1 & 2, you will need: the position graph, velocity graph, and you can use the
default sampling rate of 10 Hz.
Guiding Questions:
(1) How does the motion sensor work and what are its limitations?
(2) What implied coordinate system is the motion sensor employing?
(3) What does noise in the data look like, what affect does it have, and how can it be
minimized?
Briefly answer the questions below (based on the guiding questions) as you explore this
equipment (note: the motion sensor can be slid off the track). Each of these questions should take
only a minute or two to answer so do not get bogged down in this first section of the laboratory.
1. What is the origin of the coordinate system (x = 0) for the motion sensor?
The origin is calibrated to the right most side of the low friction track.
2. What is the positive direction in this coordinate system – towards or away from the motion
sensor? Is positive velocity moving towards or away from the motion sensor?
Away. Away.
3. If the motion sensor is on the right, as it will be in Activity 3, what is the x-axis? (Draw a large
dot for the origin and an arrow pointing in the positive direction.) [Draw on your printed sheet.]
4. Is there a minimum distance from the motion sensor that be can measured effectively? What
complications arise with measuring large distances?
About .15 meters, it also has some tolerance problems (+- .5 mm). There is an initial position
that must be factored as a constant into x(t).
5. With the motion sensor clipped on the right end of the track, you may discover that it will not
“see” the cart on the track when the sensor is set in wide mode. Why is that happening?
The motion detector picks up motion outside the track that would prove erroneous to readings of
a sole object on the track.
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6. When you see a sudden spike in your position data (e.g., for an object at rest, seven
consecutive data points might be 1.0 m, 1.0 m, 1.0 m, 3.1 m, 1.0 m, 1.0 m, 1.0 m). What do you
think might have happened at that middle data point (when you observed the object did not
suddenly jump to 3.1 meters and then back again)?
It picks up motion from outside the track or for that data point, the cart failed to reflect a sonic
wave at its current position and instead hit something outside the experiment.
7. What would the velocity graph look like because of this spike in the position data? [Draw on
your printed page]
8. Are the velocities being calculated from the position data “instantaneous velocities at an
instant of time” or “average velocities over an interval of time”? With that in mind, why are the
velocity data points midway between the times of the position data, not at the same times as the
position data points?
The average velocities between the position points, but the points are on such small time
intervals they can nearly be treated as instantaneous graphs.
Tip: You can erase all past data in DataStudio (under the “Experiment” on the toolbar) to reduce
clutter and this is a good time to do it!
Activity 2. Exploring Your Own Motion
9. Predict what your position and velocities graphs would look like if you were to walk away
from and then towards (constant speed each direction) the motion sensor. [Draw on your printed
sheet.]
Now do the experiment and try to get a run with no large spikes (which setting works better for
this – narrow or wide?). Keep in mind the constraints on the motion sensor and be aware of your
surroundings (i.e., don’t back into anyone).
10. What do you notice about the value of your position and velocity at the point where you
switched from moving away to moving towards the motion sensor?
The position graph has a tangent that is approximately = 0 at the critical point, and constant
velocity that changes from + to -.
11. Using your graphs of position and velocity as data, make a case that the position-velocity
relationship you stated in the “warm-up” holds (or doesn’t hold) for your motion in this activity.
It does hold because we can treat the graph of position exactly as the warm up equation, for we
can split up the position equation at the critical time, and treat the constant velocity as two
separate V0. We can do this for the velocity function also because v(t) is the derivative of a piece
wise function x(t).
12. Highlight a section of your position data for going forwards or backwards and use the fit tool
(linear fit) in DataStudio to get the slope of the data. What is that slope and what value should it
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match? Does it match that value? (Note: you can average values on a graph using the statistics 
tool.)
Slope1: 0.267
Slope2: 0.6, No they don’t match.
13. Your instructor has drawn an x(t) graph on the board. Describe below how you would move
to match that graph (do this before attempting to match the graph).
The cart moves forward with constant positive velocity and zero acceleration starting at the
origin, next it remains at constant position, then it travels with a negative velocity for a short
time and with an acceleration of zero, and then stays at constant position before the position
functions domain is terminated.
14. Now see how close to that graph you can get (each group member may make one attempt).
(Clip one of your graphs – preferably your best one – and attach below.)
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Check with an instructor at this point (show off your graph!)
Activity 3. Testing the constant acceleration model.
Guiding Questions:
(1) How are the position, velocity, acceleration relationships reflected in their graphs?
(2) Under what circumstances can we use the constant acceleration motion model?
(3) How do the relative directions of velocity and acceleration affect the speed |v|?
1) For the rest of this lab, you will be using the carts on the track so attach the motion sensor to
the right end of the track. Your #1 rule for using this equipment is: Never crash the cart into the
Motion Sensor!
2) Tilt the track at a small angle (about 1 degree) by placing a block of wood under the track’s
legs near the motion sensor (the motion sensor should be at the top of the track).
3) Point the plunger away from the motion sensor so that you get a soft landing when the cart
bumps the lower end of the track.
4) Change the sampling rate to 25 Hz (why might we want a faster sampling rate?). Click on
“Setup” like you did when you first set up the motion sensor to change this setting.
5) Add the acceleration a(t) graph to your existing position and velocity graphs.
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The experiment you are going to do in this Activity is this: with the cart at the bottom of the
track, you will briefly push the cart so that it glides up the track, but does not get too close to the
motion sensor. You will want to practice, starting with slow speeds and increasing the speed as
needed, to be able to do a good push when you actually do your experimental run.
15. Before you take any data, predict what the position, velocity, and acceleration graphs will
look like if you launched the cart up the track, starting at the bottom, and it eventually returns
back to its release point (without hitting the motion sensor at the top of the track!) Start your
graph when you let go of the cart. [Draw on your printed sheet.]
Now do this experiment: briefly push the cart, initially at the bottom of the track, upwards but
not so fast that it gets too close to the motion sensor (remember what you learned in Activity 1).
Allow the cart to bounce off the end of the track a couple times before you stop collecting data.
Have a group member hold the track in place since it might move when the cart hits the end of
the track1. You should be able to get data without any large spikes in the position data (if you are
getting spikes, you may need to change the narrow-wide beam setting or the positioning or aim
of the motion sensor).
16. Highlight just the data requested above (the initial fall, and two bounces from the end) and
zoom the graph on that data. Be sure to highlight the same time interval for all three graphs. Save
your three graphs and include in your report. (You can save all three separately or do a
screenshot with all three in view.)
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Remember this phenomenon when we study conservation of momentum!
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17. As a group, you are discussing when you can reasonably apply the constant acceleration
motion model to the cart as it goes up, back down, bounces off the bottom end, goes back up,
and falls down a second time.
Partner A says, “Since it’s accelerating the entire time, we can use the constant acceleration
model for the entire time interval.”
Partner B says, “We should consider three separate intervals within which we can use the
constant acceleration model: (1) the first trip up and down, (2) when the plunger is in
contact with the end of the track, and (3) the second trip back up and down again.”
Partner C says, “Since sometimes its velocity is upwards and sometimes downwards, we
need to use four separate intervals within which we can use the constant acceleration model:
(1) going up, (2) falling down, (3) going back up, and (4) falling back down the second
time.”
With which partner, if any, do you agree? Use evidence from your graphs to justify your position
in your response to your partners.
Partner A because the acceleration is constant the entire time.
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For this next section, use only the position, velocity, and acceleration data associated with the
cart’s motion after it was released but before it hit the bottom end (i.e., the first trip up and
down).
18. Consider the direction of the acceleration and the velocity during the cart’s motion. Part of
this time the velocity is in the same direction as the acceleration; part of the time the velocity is
in the opposite direction as the acceleration. How does the relative direction of velocity and
acceleration determine what happens to the speed |v| of the cart?
When the acceleration and velocity are in the same direction, the speed of the car is increasing,
but when the acceleration and velocity are in opposite directions, the speed is decreasing.
19. You and your partners are discussing the point in time when the cart reaches its highest point
on the ramp. What is the velocity at that point? Is the acceleration zero at that point?
The velocity is zero and the acceleration is a positive constant.
20. Does the acceleration look constant during this interval? (The acceleration data will be rather
noisy, so consider the average of several points not individual points.) From the acceleration
graph, what is the average acceleration during this time interval?
Yes it looks constant. It is approximately .29 m/s^2
21. What type of function – linear of quadratic – would best fit the velocity v(t)? Use the Fit
feature of DataStudio to fit that function to the velocity data. How good a fit is it? What fit
parameter tells you the average acceleration during this time interval and how does it compare to
the value you got from the acceleration graph?
Linear. It is a good fit. The A parameter tells us the average acceleration, it is similar.
22. What type of function – linear of quadratic – would best fit the position x(t)? Use the Fit
feature of DataStudio to fit that function to the position data. How good a fit is it? What fit
parameter tells you the average acceleration during this time interval and how does it compare to
the value you got from the acceleration graph?
Quadratic. It is a good fit. The slope, m, tells us the average acceleration during this time
interval, and it is very similar.
23. Calculate the area under the velocity graph for some time interval (hint: look at the options
for the statistics  function). What quantity should it tell you about that interval (and does it)?
0.01 m. It should tell us the net displacement over the whole interval.
24. Is the cart’s motion well described by the constant acceleration model during this first trip up
and down the ramp? Justify your statement based on your data.
Yes, the linear equation in the data tool had little slope for a(t).
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25. If you started the cart at a different location on the ramp and with a different initial speed,
how would that affect the acceleration values on your acceleration graph?
The acceleration would remain constant due to the constant of gravity.
26. Bouncing off the end of the track. What is the acceleration of the cart during the bounce off
the end at the bottom of the track? (You may only have a couple data points in this interval.)
It would remain constant in theory, however the data indicated it became zero.
27. Up and back down the ramp again. How would you expect the acceleration in the second
trip up and down the ramp to compare to the acceleration during the first trip up and down the
ramp? Determine the acceleration during this interval from the position and velocity graphs
(state how you did it) and comment on the two acceleration values.
It would still be the same. We looked at the graph to determine the constant value.