TA name ___________________________
Name _________________________
Lab section __________________________
UW Student ID # _________________________
Date ______________________________
TA Initials (on completion)_____________
Lab Partner(s) _________________________
_________________________
EXPERIMENT 1: ONE-DIMENSIONAL KINEMATICS
MOTIONS WITH CONSTANT ACCELERATION
117 Textbook Reference: Walker, Chapter 2
117 LABS - GENERAL INFORMATION
The lab exercises in this manual are all concerned with measurements of the motion and interaction of
material bodies. The frame of reference for measurements of motion is the laboratory. Within the
precision of the measurements possible with our equipment, the laboratory is an inertial reference
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precision of measurement should always accompany the measurement. This is usually done by
writing the uncertainty of a measurement as well as the value of the measurement. For example, a
velocity could be expressed as 1.25 ± 0.05 m/s. The uncertainty is 0.05 m/s and is 4% of the value of
the velocity, so the measurement can be described as a measurement with a 4% uncertainty. This is a
shorter way of saying that the precision of the measurement is 4%. An uncertainty expressed in this
way is usually a statistical uncertainty. In the case of a velocity measurement that involves measuring
a distance interval and a time interval, small differences are likely to occur in repeated measurements
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first or last point of the interval, or your estimate of the distance interval varied because you had to
estimate distance to 1/10 mm on a meter stick with marks every millimeter. The differences in these
measurements tend to be random and center on some average value. For most measurements, the
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average value as the number of measurements approaches infinity). A detailed mathematical treatment
of such random uncertainties exists and should be part of the background of all scientists.
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will use simpler estimates of
uncertainties that will not differ greatly from the more formal ones. A good part of the reason for this
is that repeated measurements must be made to determine formal uncertainties and there is simply not
enough time for much repetition in our 3-hour labs.
A detailed description of uncertainties as we will use them is contained in the eight-page document
that is part of the introduction to the lab entitled Uncertainties. You are encouraged to read this at
some time early in the quarter and to bring it to the lab as a reference. For this first lab, the relevant
descriptions of uncertainties will be introduced as required (this writeup is longer than usual!).
SYNOPSIS - LAB 1
In this lab, you will study the motion of an object with nearly zero, positive and negative
acceleration. You will measure the position of a cart on a track and of a ball moving under the
influence of the gravitational force (and therefore a constant acceleration). You will approximate
the instantaneous velocity, v(t), from the position versus time, x(t), graph and the instantaneous
Physics 117
© Copyright 2007, Dept. of Physics, U. of Washington
Winter 2007
acceleration, a(t), from the velocity versus time, v(t), graphs. For the case of uniformly
accelerated motion, a(t) = constant, you should learn: (1) how the shape of the position versus
time, x(t), graph depends on the acceleration for three cases, a > 0, a = 0 and a < 0 and (2) how
the slope of the velocity versus time, v(t), graph depends on the acceleration for three cases, a >
0, a = 0 and a < 0.
APPARATUS
You will use a computerized motion detection system. The motion sensor is an ultrasonic
detector and is used to measure the position of a cart or ball. The detector (see figures below)
sends out busts of 49kHz ultrasound waves and detects the echo reflected back by any objects in
the path of the ultrasound beam. The pulse rate for this lab is set at 50 pulses per second. The
detector measures the time difference between the time it sends out a pulse and the arrival time
of the echo. Since the speed of sound in air is known (~344 m/s), the distance between detector
and the object can be determined. The minimum of the range of our motion detector is about 15
cm. It does not give the right value of distance if the object is closer to the detector.
Figure 1-1 Motion Sensor
Note there is a switch on top that allows you to choose narrow beam or wide beam. Always set
this switch to narrow beam when you measure the distance of a cart.
The velocity of a moving target is obtained from the difference in distance between pulses (the
so-called first difference of the distance data) and the acceleration is obtained from the difference
in velocity between pulses (the second-difference of the distance data).
The DataStudio program measures positions directly, but it computes velocities by taking the
difference of positions at small time intervals and computes the accelerations from the
differences in velocities. Because such small differences have relatively large uncertainties, the
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PROCEDURE
Horizontal (or nearly so) track
Add two 500g bars to the cart. This will reduce the importance of the frictional forces acting
between the cart and the track.
1. Level the track as well as you can with the small pocket level. [The tabletops are not completely
level, so you should not change the position of the track after you have leveled it.] When the track
is level, use the cart to check the leveling. Describe how you used the cart to check the leveling.
Physics 117
1-2
Autumn 2006
Attach the motion detector to one end of the track so that the transducer disk is aimed down the track.
Give the cart a push so it moves away from the detector to the opposite end of the track. Do not allow
the cart to collide with the motion detector or with the floor. Click the Open Activity option of
DataStudio and the Lab 1-One Dimensional Kinematics folder (Located in My Documents folder
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)
. Open the file Lab 1- Cart Practice taking data with the
motion detector. Click Start to begin data taking, push the cart and then click Stop at the end of the
cart motion. Note that the motion detector reading becomes more positive as the object moves away
from it, i.e., the positive x direction is away from the detector. Remember that the data are inaccurate
if the cart is closer than 15cm to the detector. Adjust your push so that the velocity is in the range 0.30
m/s to 0.50m/s.
2. When you have a good run, maximize the graph of position vs time, and then use the zoom
select feature to fill the screen with a portion of the graph that is nearly a straight line. The zoom
select icon is the magnifying glass fourth from the left. Click on it, then click-drag the mouse icon
(a magnifying glass) to create a rectangle about the region of interest. The selected area will
automatically expand. Print a copy for both partners. Write your name on the graph and a
description of the data. The description can be a reference to the lab instructions. For example,
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d be done for all graphs in 117.
3. Examine the x(t) graph you have just printed. Does the velocity of the cart increase with time,
decrease or remain the same? Explain how you know.
4. Determine the velocity and the uncertainty in the velocity from the graph of velocity vs time for a
time t0 that you choose near the center of the graph. Use seven data points centered on t0. Find the
velocity at t0 by averaging the velocity for times symmetric about t0. For example, if t3 immediately
precedes t0 and t4 immediately follows t0, the velocity at t0 is approximated by v(t0) = (1/2)[v(t3) +
v(t4)], the average of velocities just before t0 and just after. Calculate v(t0) by means of the three
pairs of velocities symmetric about t0 and the velocity at t0. Label the values 1, 2 and 3 in the order
of increasing time difference between pairs. You can read the values of the velocity and time
directly from the graph, but a simpler method is to use a feature of Data Studio called Smart Tool.
th
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from the left that
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locks onto the data point at t0. The screen will show values for v(t0) and t0. They will probably not
be as precise as you need. Double-click on the Run#1 icon under Velocity, Ch 1&2 (m/s) in the
Data column to the left of the screen. In the Data Properties window that appears, click the
Numeric tab. In the new window in the Style box select Fixed Decimals and set Digits to the
right of the decimal to 4. Click OK. The time and velocity values in the Smart Tool display
should now have 4 decimal places. Lock onto and record the values for seven data points you have
chosen. These data should be taken in pairs, as described above, with t1 some interval below t0 and
t2 the same interval above t0. Then t3 is a larger interval below t0 and t4 is that same interval above
t0, and so on. Thus each pair gives a value for v(t0) and you will compare these values.
Physics 117
1-3
Autumn 2006
t1 =
s
v(t1) =
m/s
t2 =
s
v(t2) =
m/s
t3 =
s
v(t3) =
m/s
t4 =
s
v(t4) =
m/s
t5 =
s
v(t5) =
m/s
t6 =
s
v(t6) =
m/s
t0 =
s
v(t0) =
m/s
5. From your measurements in 4, find four values for the velocity at t0.
(0)
v(t0) =
m/s
(1)
v(t0) =
m/s
(2)
v(t0) =
m/s
(3)
v(t0) =
m/s
Find the average of the four measurements and from the spread in them find the uncertainty vi (t0 ) in
an individual measurement, Equations U1 and U2 as described in Appendix B: Uncertainties.
v (t 0 )  v(t0)ave =
v (t ) =
i
0
m/s
m/s
The best estimate of the value of a quantity for which there is a set of measurements is the average
value of the quantity from the set. The uncertainty of an individual measurement, such as vi (t0 )
above, answers the question, if I were to make another measurement of v(t0), how closely would it
agree with the average? The question you are more likely to ask is, if I make another set of
measurements, how closely do I expect the average of the second set to agree with the average of the
first set? You likely feel instinctively that the averages would agree with each other more closely than
an individual measurement would agree with the average. This is indeed the case. Part 9 in Appendix
B: Experimental Uncertainties provide the following result for cases in which the uncertainty, xi , is
the same for all measurements. x i n where n is the number of measurements in the set and

is the uncertainty in an individual measurement.
i 
6. Copy the best estimate of the value of v(t0) that you calculated above, and Calculate the
uncertainty in the best value.
v(t0)best =
±
m/s
v (t ) = ______________________________
0
v(t0)best and its uncertainty must b
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(See Part 5 in Appendix B:
Experimental Uncertainties.)
Warning: In this lab and future labs, any result and/or uncertainty not written according to
the two rules above will be marked WRONG, independent of their values.
Physics 117
1-4
Autumn 2006
7. With this warning in mind, rewrite your values of vi (t0 ) 

v(t0)best , and v (t0 ) , below.
v (t ) =
i
0
v(t0)best =
±
m/s
Inclined track
Having completed measurements with the level track, a ≈
0
,
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a > 0 and a < 0 by tilting the track. In each of the cases below, release the cart so that it will roll
downhill. DO NOT LET THE CART RUN INTO THE MOTION DETECTOR!
8. For a positive acceleration, a > 0, is the end of the track closest the motion detector higher or lower
than the other end? Explain.
9. Place one block under the leveling screws at one end of the track so that a positive acceleration is
produced. Start the data collection, and release the cart. Sketch below the relevant portions of the
resulting x(t), v(t), and a(t) graphs below. Provide appropriate scales. Be sure to label the graphs.
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data. If you have questions about this, discuss it with your TA.]
x(t)
Physics 117
v(t)
1-5
a(t)
Autumn 2006
10. Maximize the x(t) graph and select a time, t0, near the middle of the graph. Zoom-select the data
so that you can obtain a precise reading of the time and distance near t0. Print a copy for both
partners. Label your graph with your name and a title that describes the subject of the graph. For
example, this graph could be l
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Provide labels for all graphs you print in the 117 lab. Fortunately, the printer provides time and
date as well as the lab name. From the graph, determine the slope 
x
t of the curve at t0. To
approximate the slope at t0, requires four data values, two times and two distances. Write below
the data values from which you determine the slope, their uncertainties and their units. Data
Studio states that the accuracy of a distance measurement is 1.0 mm. Using a sampling rate of
50/s, the time between samples is 0. 02s, so using a value of 0.0004s for the time uncertainty is
reasonable. Use Smart Tool to read the data points.
t0=
s
datum 1 time =
±
datum 2 time =
±
s datum 1 position =
±
m
s datum 2 position =
±
m
11 From your data in 10, calculate the slope at t0 and its uncertainty. To find the slope, you will need
a time difference 
t and a distance difference 
x. You will also need the uncertainty t in 
t and
the uncertainty x in 
x in order to calculate the uncertainty in the slope. For the uncertainty in
the distance interval and the time interval, you will need the result that the uncertainty in the sum
or difference of two independent quantities x and y, Z Ax By , is
2
2
Z  
Ax 
By 









t=
±
s

x=
±
m
[U7]
12. The slope can now be calculated as the quotient of 
x and 
t and the uncertainty in the slope as
the uncertainty in the quotient.
The uncertainty in the quotient of two independent quantities Z = x/y is calculated using [U6b]
from the Appendix B: Uncertainties
2
by 
ax  


Z Z  


x  y 
2

[U6b]
  
 
x

, and the uncertainty is slope 
slope  x  t 
t
x  
t 

±
m/s
2
In our case, slope =
slope =
Physics 117
1-6
2
Autumn 2006
13. Compare the value of the slope from 12 with the velocity at t0 read from the v(t) graph. Note that
the times for the velocity data points are midway between those for the position points, so to get
v(t0) from the graph, take the values immediately above and below t0 and average them. Note:
whenever the word compare appears, a calculation must be done. For example, if D1 and D2 are
two measurements of the same quantity,
2
Criterion for agreement: D1 D2 2 
22
1 
[U8]
Use your value for vi (t0 ) from Section 8 above for the uncertainty in v(t0).
slope =
±
m/s


(from Section 13 above)
v(t0) =
±
m/s
(from v(t) graph)
14. Use the same method you used in 11 - 13 to find the slope of the v(t) graph at some time t0 for
which the graph is relatively smooth. Find the uncertainty in the slope. Zoom-select the data
around t0. Record and label the data necessary below. Show your work.
t0 =
s
point 1 t=
±
v=
±
point 2 t=
±
v=
±
slope =
±
m/s2
15. Compare the slope from 14 with the acceleration at t0 from the a(t) graph. In order to
determine a, place the horizontal axis of Smart Tool to so that it best fits the acceleration
data (the acceleration should be a constant, but will have large fluctuations. Use a =
±0.15 m/s2.
a(t0) =
±
m/s2
Consider the case of a<0.
16. Place two blocks under the leveling screws at one end of the track so that the acceleration will be
negative, a<0. Obtain x(t), v(t) and a(t) g
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Zoom-select the relevant part of the x(t) graph and print it for both partners. Determine the slope
and the uncertainty in the slope for a time t0 near the center of the graph. This time a method will
be used that is different than that used in 10 - 12. With the ruler provided, draw a line tangent to
the x(t) graph at t0. [Note: the line should extend across the graph so that an accurate value can be
found for the slope of the line]. Record the coordinates of two points, (xa,ta) and (xb,tb) near
opposite ends of the tangent line.
t0 =
s
xa =
ta =
slope =
Physics 117
xb =
tb =
(m/s)
1-7
Autumn 2006
17. To find the uncertainty in the slope you must draw two more lines. The first is the line tangent to
the x(t) graph at t0 with the most negative slope that you believe still fits the data. The second is
the line tangent to x(t) graph at t0 with the least negative slope that you believe still fits the data.
Find the two slopes as you did in 16.The uncertainty in the slope is then one-half the difference
between these two slopes.
max negative slope:
xc =
tc =
slope =
xd=
td=
xf =
tf =
m/s
min negative slope:
xe =
te =
slope =
m/s
slope(t0) =
±

slope =
m/s
m/s
18. Compare the slope at t0 to the velocity at t0 obtained from the v(t) graph. For the uncertainty in
the velocity, use your value from 8.
19. How, if at all, do the graphs in 10 change when an additional block is placed under the same end
of the track as in 10? Sketch your answers below.
x(t)
Physics 117
v(t)
a(t)
1-8
Autumn 2006
20. Draw similar sketches corresponding to one block placed under the opposite end of the track.
x(t)
v(t)
a(t)
21. Identify that inclined track set-up (one block or two, at same end as the motion detector or at the
opposite end), which produces each of the accelerations below.
Smaller negative acceleration
Larger negative acceleration
Smaller positive acceleration
Larger positive acceleration
Vertical Motion
Click the Open Activity option of DataStudio or the file menu at the top of the page and open the
appropriate file, Lab 1-Ball.
22. Place the motion detector on the lab table facing upward and practice throwing the Nerf ball
upward above it. Try not to detect your hand while the Nerf ball is in flight. After throwing the
ball, move your hand out from above the detector and catch the ball on its way down. You will
find it e
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the ball too high. 0.3m to 0.8m is fine. Hold the ball over the detector, start the data collection and
toss the ball upward. If you move deliberately, most of your data may not be useful. This is not a
p
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data. Zoom-select the good part of the x(t) graph. Maximize the graph and print it.
Physics 117
1-9
Autumn 2006
23. Describe the shape of the x(t) graph.
24 Zoom-select, maximize and print the portion of the v(t) g
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portion of the x(t) graph.
25. Indicate on the v(t) graph when the ball is moving upward, when it is moving downward and
when it is at the top of it trajectory.
26. Describe the motion of the ball when it is on its way up, on its way down and when it is at the top
of its trajectory in terms of its position, velocity and acceleration.
up:
down:
top:
27. Determine the slope of the v(t) graph when the ball is on the way up, when it is on the way down
and when it is at the top. Use a method similar to the one you used in 4. Select and record the
t
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bottom of the trajectory. Use Smart Tool to measure the time and velocity (four digits after the
decimal) for two data points, one just before the chosen time, vb,tb, and one just after, va,ta.
Record the values below and use them to calculate the slopes.
tup =
s
up:
s
vb =
m/s
tb =
s
vb =
m/s
tb =
s
vb =
m/s
tb =
s
vb =
m/s
tb =
s
vb =
m/s
tb =
s
vb =
m/s
m/s2
slopeup =
ttop =
tb =
s
top:
m/s2
slopetop =
tdown =
down:
slopedown =
Physics 117
s
m/s
2
1-10
Autumn 2006
28. Compute the average of the three slopes, the uncertainty in the individual values of the slope and
the uncertainty in the average. As described in Section 5 above, the uncertainty in the average of n
measurements is less than the uncertainty in any individual measurement. If each value of slope
has the same uncertainty, ai , then the uncertainty in the average slope is a ai
with n = 3 from the three measurements: a =
n
m/s2
and the final result of the measurement is aave =
±
m/s2.
If the three values of “
a”
are considered to be independent measurements, then the uncertainty in their
mean is smaller than the uncertainty in an individual measurement by 1/
3.
The v(t) graph should be a nearly straight line with a negative slope. The slope could be determined
by hand fitting a straight line to the data, but in this case we will let the computer do the work.
29. Zoom-select the straight-line portion of the v(t) graph. In the toolbar above the graph, select Fit
and from the Fit menu, select linear. A straight line will appear through the data with the slope of
the line and its uncertainty in a box. Both should have 4 digits past the decimal. If not, adjust the
number of digits by double clicking on the linear fit icon in the Data column on the left of the
screen, selecting the Numeric tab and adjusting the data to four digits as you did in 4. Drag the fit
box so it does not cover the data. Print the graph and record the value of a and its uncertainty
below. Compare the value to local g = 9.80 ± 0.01 m/s2.
afit =
±
m/s2
REFERENCE
Salviati: ... Very well; up to this point you have explained to me the events of
motion upon two different planes. On the downward inclined plane, the heavy
moving body spontaneously descends and continually accelerates, and to keep it at
rest requires the use of force. On the upward slope, force is needed to thrust it
along or even to hold it still, and motion which is impressed upon it continually
diminishes until it is entirely annihilated ... Now tell me what would happen to the
same movable body placed upon a surface with no slope upward or downward.
from Dialog
by Galileo Galilei (1632)
Galileo realized that, although he could not directly measure the position versus time of a freely
falling object accurately because the full acceleration of gravity quickly produced velocities too
large for him to measure with the clocks available to him (his pulse and a Chinese-style water
clock). He could measure the uniformly accelerated motion of a ball rolling down a very smooth
track and found that all uniformly accelerated motion behaves in the same way! By using an
inclined track, he was able to reduce the acceleration sufficiently that he could make more
reliable position versus time measurements. In this lab you have used a modern version of
Galileo's method to study motion for two different cases: 1) almost uniform velocity and
2) uniform acceleration.
Physics 117
1-11
Autumn 2006