One-Dimensional Kinematics
Adapted from Real Time Physics, Labs 1 and 2
© 1993-94 D. Sokoloff, P. Laws, and R. Thornton
GOALS
•
•
•
•
•
To understand how motion looks as a position-time graph.
To understand how motion looks as a velocity-time graph.
To understand how motion looks as an acceleration-time graph.
To understand the relationship between the three graphs.
To gain experience using the equipment and software used in future experiments.
INTRODUCTION
In this lab, you will use a motion detector to plot position, velocity, and acceleration-time graphs
of a cart moving along a track in one dimension. Graphical representations of data are very
important in physics since they allow us to model mathematically the physical system being
studied. This lab should help you develop an understanding of how a graph is related to reality.
You should also begin to see the mathematical relationships between position, velocity, and
acceleration. The study of these aspects of motion and their mathematical and graphical
representations is called kinematics.
As you work through the activities in this lab, you will be asked to make predictions, answer
questions, and make sketches of your graphs. Please record all of these (neatly) in your lab
notebook. Remember that your notebook is your only record of the results of this lab.
PART I: POSITION-TIME AND VELOCITY-TIME GRAPHS OF MOTION
One way to represent motion during an interval of time is by graphing the position with respect
to some reference point. This is a position-time graph. Another way to represent motion during a
time interval is with a graph describing how fast and in what direction you are moving. This is
called a velocity-time graph.
Both position and velocity are vector quantities. Position not only describes how far an object is
from a reference point, but in what direction. Velocity describes both the speed and direction of
the motion. You will look at the motion of an object (your hand) along a line. The reference
point, or origin, in this case is a motion detector. The position will always be positive in this setup. The velocity of the object may be positive or negative depending on whether the velocity is
in the positive (away from the detector) or negative (toward the detector) direction.
You will use an ultrasonic motion detector and the LoggerPro software on your computer to
relate graphs of position and velocity as a function of time to the motions they represent. Your
instructor will tell you a bit about how it works. You will move a cart in front of the detector
with your hand. LoggerPro will record the position and velocity as a function of time.
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One-Dimensional Kinematics
Start the LoggerPro software and set it up to use the motion detector. (Look in the LoggerPro
Reference in the front of your lab manual if you need help.) Make sure to display two graphs:
one position-time and the other velocity-time. This means that the time is on the horizontal axis.
Analyzing the graphs is easiest if they are about the same size and stacked one on top of the
other.
1. When you are ready to start graphing, one group member clicks once on the “Start” button,
and the other starts to move the cart. All group members should take turns in front of the motion
detector.
use cart or
hand
at least 50-cm
Figure 1. Track With Motion Detector and Cart.
You will notice that the velocity-time graphs are “bumpier” than the position graphs. They are
more sensitive to small variations in your motion. (Do you have any ideas about why this is?) To
make a smoother graph walk back and forth as you move the cart rather than just moving your
arm. There will be fewer bumps and wiggles in your graphs. You may also need to adjust the
position of the motion detector. You can adjust the tilt as well as moving it side to side. Keep
adjusting until you have very smooth position-time graphs and the detector “sees” the cart for the
entire length of the track
2. Predict a velocity graph for the position-time graph shown below. Sketch this position-time
graph and your prediction of the velocity-time graph in your lab notebook. Use dashed lines for
your predictions.
2
1
0
1
2
3
Time (s)
4
5
Figure 2. A sample position-time graph.
Set the experiment length to 5 seconds, and test your prediction. When you have made a good
duplicate of the position graph, sketch your actual graph over the existing position-time graph.
Use a solid line to draw the actual velocity graph on the same graph with your v-t prediction. (Do
not erase your prediction!)
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3. Now, you will find the average velocity of your motion using first the velocity graph and then
the position graph.
Select “Examine” in the Analyze menu. (There are at least two shortcuts to this feature. Can you
find them?) Notice as you move the line across the graphs by moving the mouse, the x, v, and t
values are displayed at the bottom of the screen. If the line does not move across both graphs
simultaneously, go to the Page menu and select “Group Graphs (x-Axes)”.
Find the average velocity in a relatively flat (but non-zero!) region of the v-t graph. Select the
region by clicking and dragging the mouse. Choose “statistics” from the Analyze menu. The
mean is the average value. Notice that you are given the standard deviation, σv, and the number
of points. Use these to calculate the standard error, σ<v>.
As you know, the average velocity during an interval of time is defined as
Δx
change in position
<v>=
=
change in time
Δt .
Notice that, by definition, this is also the slope of the position-time graph. Select the same time
range on your x-t graph as you did for the v-t graph. Perform a linear fit to the selected data.
Record the slope and uncertainty. (If the uncertainty does not appear in the results box, doubleclick on the box and check the “display uncertainty” option.)
Do the two values for <v> agree within uncertainty?
4. Now try predicting the appearance of a position-time graph from a velocity-time graph. Study
the velocity graph shown below, and sketch it and your prediction of the corresponding
position graph in your notebook. Use a dashed line for your prediction.
+1
0
2
4
6
8
10
Time (s)
-1
Figure 3. A sample velocity-time graph.
Test your prediction. You will first need to change the experiment length to 10 seconds. Do your
best to duplicate the velocity-time graph. When you have a good result, draw your actual results
for velocity and position over your prediction graphs, using solid lines.
PART II: INTRODUCTION TO ACCELERATION
Acceleration is the third quantity used to describe motion. It is defined as the change in velocity
with respect to time.
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One-Dimensional Kinematics
Set up LoggerPro to display three graphs, one on top of the other. Make sure you have position,
velocity, and acceleration on the y-axes, respectively. If you need to change an axis label, move
the cursor over the label, hold down the mouse button, and select a label from the pop-up menu
that appears.
1. How must you move the cart to make a curved position-time graph? Try to make each of the
graphs shown below. Comment on your results, describing the kind of motion you had to
perform. (What direction did you move? Did you speed up or slow down? Etc…)
Figure 4. Position-time graphs for a cart starting 0.25-m from motion detector and moving away
from detector for 5 seconds, ending 1.5-m away.
2. For the final set of data, use the low friction cart with a mass attached to the cart by a string
hanging over a pulley. Start with about 20-g of mass.
Figure 5. Set-Up for Constant Acceleration Motion.
LoggerPro should still display three graphs. Set the range of the position axis to 0 to 2.0-m,
velocity to -1.0 to 1.0-m/s, acceleration to -1.0 to 1.0-m/s2, and the time interval to 5.0-s. A
short-cut method of setting the axis limits is to click on the number you wish to change and type
in the desired value
You will start the cart at the far end of the track from the motion detector and give it a push
toward the detector. It will slow down, reverse direction, then speed up in the opposite direction.
One-Dimensional Kinematics
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For each part of the motion, predict if the velocity and acceleration are positive, zero, or
negative. You may want to make a table like the one below in your notebook. (Only consider the
motion of the cart after it leaves your hand.)
Moving Toward
Detector
At the Turning Point
Moving Away
From Detector
Velocity
Acceleration
As before, sketch your predictions of the velocity-time and acceleration-time graphs of this
motion.
Now, test your predictions. Start taking data and, when you hear the clicks from the motion
detector, give the cart a gentle push so that it travels at least one meter before reversing
direction.
Change the limits on the graphs if necessary (you may find the “Autoscale” option useful) to best
display the data. Were your predictions correct?
3. Find the average acceleration of the motion in two ways. First, select “Examine” from the
Analyze menu and select a region of the a-t graph with fairly constant non-zero acceleration.
Find <a> and its standard error as you did for the average velocity in Part I.
Next, find the average acceleration (and uncertainty) by fitting a line to the same time range of
the velocity-time graph, since the slope should equal the average acceleration:
Δv
change in velocity
<a> =
=
change in time
Δt .
Do the two values of <a> agree within uncertainty?
4. Sketch or print your results, and label all graphs to indicate
a. when the push started.
b. when the push ended (when your hand left the cart.)
c. when the cart reached its turning point.
d. when you stopped the cart.
PRE-LAB QUESTION:
A person stands 1-m away from a motion detector, walks away from the detector slowly and
steadily for 5 seconds, stops for 5 seconds, and then walks toward the detector quickly and
steadily for 5 seconds. Sketch the position-time and velocity-time graphs representing this
motion. (Time should be on the horizontal axis of both graphs.)
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One-Dimensional Kinematics
YOUR LAB NOTEBOOK SHOULD INCLUDE:
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All sections listed in the “Lab Format and Grading” reference at the beginning of the lab
manual.
Sketches or printouts of predictions.
Sketches or printouts of experimental results.
Things to include in the Discussion:
o Note any surprises or difficulties encountered in predicting or matching the
motion graphs. Explain any disagreements between predictions and results.
o How is the direction of motion reflected in a position vs. time graph? How about
in a velocity vs. time graph?
o How is the speed of motion reflected in a position vs. time graph? In a velocity
vs. time graph?
o How can you tell from a velocity-time graph that the moving object has changed
direction? What is the velocity at the moment the direction changes? Is it possible
to actually move your body (or any object) to make vertical lines on a velocitytime graph? Why or why not?
o During the time that the cart is speeding up due to the falling mass, is the
acceleration positive or negative? How does speeding up while moving away
from the detector result in this sign of acceleration?
o Explain the observed signs of the acceleration (moving toward the detector, at the
turning point, and moving away from the detector) in the last exercise.