In this lab, you'll learn how to measure the
acceleration of a uniformlyaccelerating object. However, the method
is general enough to be applied to cases of nonuniform acceleration.

To measure the acceleration of a glider on an air track using the finite
difference method

Read the Introduction on this page in order to learn what the finite
difference method is and how to use it.

Do the WebAssign assessment L109PL.
You'll practice using the finite difference method.
How does one measure acceleration in the laboratory? One way is
to use a ticker tape timer. A long strip of paper is taped to an
object. As the object moves across a table, for example, the paper
is pulled through a device similar to a doorbell ringer, which strikes
the tape many times a second. A piece of carbon paper below the tape puts
a dot on the tape for each strike. If the object is moving with
uniform velocity, the dots are spaced uniformly on the strip as shown
below. (Click here for additional information to help with visualing this device. Here's a photo of a ticker tape timer.)
If your task were to measure the velocity of the object, you would do
something like this:
 Lay the strip out on a table with a ruler parallel to the line of
dots.
 Arbitrarily designate the first dot as corresponding to an elapsed time of 0 s.
 Measure the position of each successive dot on the ruler.
 Plot a graph of position vs. time.
 Draw the best straight line through the points on the graph and find
its slope.
While a quicker method would have been simply to measure the distance
between two adjacent dots and divide by the time interval, that method
is less accurate than drawing the graph. Moreover, by drawing the
graph, you can verify whether the velocity is constant. If the line
bent gradually downward, for example, you might surmise that the object
was being slowed by friction.
Now suppose the object is accelerating (speeding up to the right).
In that case, the spacing of the dots will increase as shown below.
In order to verify that the acceleration is constant and determine a
value for it, you would measure position as a function of time as before. However, a graph of position vs. time would have an increasing slope and,
in fact, would be parabolic if the acceleration were uniform. There's
a mathematical technique for finding the acceleration from a position
vs. time plot. It's called a leastsquares quadratic fit.
However, we're not ready to do that, since it obscures the physics. The method that you'll use in the present lab will make use of the physics
that you know. It's called a finite difference method. It
works like this:
Once you have the position vs. time data, select successive pairs of
points. Calculate the average velocity, Δx/Δt,
for each pair. If the intervals between the points are
small enough, the average velocities will be good approximations to
instantaneous velocities. Knowing that, plot a graph of the average
velocities vs. time. If the acceleration is uniform, then the
points should lie in a straight line to within the error inherent in
the measurements. The slope of that line is the acceleration.
It actually turns out that when the acceleration is uniform, the average
velocity over any time interval is equal to the instantaneous velocity
at the midpoint of the time interval. This is only true for uniform
acceleration. There will be more about this in the theory section.
Here are the specifics on how to carry out the finitedifference
method.
 Use a data table labeled like the following one but with enough rows
for the number of dots.
Index
n 
Elapsed Time, t_{n}
(s) 
Position, x_{n}
(m) 
Change in Position, Δx_{n}
(m) 
Change in time, Δt_{n}
(s) 
Average Velocity, Δx_{n}/Δt_{n}
(m/s) 
0 
0 




1 





2 





3 






The indices are consecutive numbers that mark successive positions
of the object starting with the first recorded position. This first
position has an index of 0 and an Elapsed Time of 0 s.

The Elapsed Time for Index, n, is the total time that passes
from t = 0 to the nth position of the object.

Position is measured according to whatever ruler scale is used for this.

Calculate the Δx's as x_{n+1}  x_{n1}.
That means if you're on row n in the table, you take the
value of position for the (n+1) row and subtract from that
the position for the (n1) row. Thus, you associate
the difference x_{n+1}  x_{n1} with time
t_{n}_{.} (Avoid the temptation to
calculate Δx as the difference x_{n+1}
 x_{n}. That would mean subtracting the position
of one point from the position of the very next point. If you
used this method, you wouldn't be able to associate the time with
either t_{n} or t_{n+1}. There wouldn't
be a corresponding row in the table.)

Calculating the Δt's is much simpler. This is
simply the time difference corresponding to each Δx. That is, Δt is the time it takes to go from position
x_{n1} to position x_{n+1}. These
should be the same for every value of n.

The Average Velocity for each n is simply the ratio of Δx_{n}
to Δt_{n}.

Once the data table is complete, plot a graph of Average Velocity
vs. Elapsed Time. Find the slope of the line to determine the acceleration.
Rather than using the ticker tape timer described above, the equipment
used in this lab will be a glider sliding with very little friction down
an inclined air track. A photo of the glider resting on the track is shown
below.
Note the thin wire that is used as a marker. In order to record the motion
of the glider, strobe photos were taken of the glider as it slid down
the track. The stroboscope flashed at a rate of 10.1 Hz, and recorded
a number of overlapping images of the glider on the same frame of film.
An example photo is shown below. The glider was sliding from right to
left. A very shallow incline was produced by placing a riser block under
the right end (not shown) of the track. While the individual images of
the glider can't easily be distinguished, the marker wire images can be
seen. Click either here
or on the photo below to display an enlarged version in a new window.
Examine that photo while continuing the reading.
Note the following about the enlarged photo:

The photo has been annotated to point out the first 4 positions of
the wire. Yellow arrows point to these, and the 4 positions are marked
0 to 4.

While the glider was released from rest, we can't assume that the
glider's velocity for the first image was 0. The flash most
likely went off a short time after the glider was released.

The horizontal doubleended arrow below the track indicates the distance
between support posts. This is used for scaling distances measured
on the photo to actual distances. There are actually 4 support posts
shown. They come in pairs, one behind the other. If you viewed a pair
of posts head on, you would just see the front post, since the rear
post would be hidden behind the front one. The reason you see both
posts in a photo is due to optical distortion. This results because
the posts are different distances from the camera and are viewed from
different angles. To see this for yourself, line up the forefinger
of each hand in front of your right eye while keeping your left eye
closed. With one finger behind the other, you only see the front finger.
Now close your right eye and open your left eye. You'll see both fingers.
As you might guess, the presence of this kind of distortion in the
photo introduces uncertainty in the calculation of distances.
 Print a copy of this
table for your data page. This will be your original data page. Follow the usual datarecording procedures. Note that the data table (shown below) has one additional column from the one given in the Introduction. Change
in Position has two columns, one for the changes as determined
directly from the photo and one for the changes scaled to actual
distances.
Image
Index
n 
Elapsed Time t_{n}
(s) 
Position x_{n}
(photo m) 
Change in Position Δx_{n}
(photo m) 
Change in Position Δx_{n}
(actual m) 
Change in Time Δt_{n}
(s) 
Average Velocity Δx_{n}/Δt_{n}
(m/s) 
0 






1 






2 






3 






 Click on the Photo ID below to open in a new window the photo below
that corresponds to your position in the alphabet. Print the photo in portrait orientation. The entire photo must print
on a single page to be usable, so you may need to set your print options to less than 100% size. Click here for an example of how a photo fits on the printed page. In this case, the print size was 50% of original, but that percentage will depend on your printer. Note that you may print in grayscale as color isn't necessary for the analysis.
Use this photo if your
last name begins with... 
Photo ID 
Riser block
height (m) 
Frequency
of strobe (Hz) 
A  E 
168 
0.0254 
10.1 
F  J 
169 
0.0254 
10.1 
K  O 
171 
0.0381 
10.1 
P  R 
173 
0.0381 
10.1 
S  Z 
175 
0.0127 
10.1 

In the Additional Data section of the data page,
record the Photo ID, the frequency of the strobe, and the riser block
height for your photo. When you record a value, begin with a phrase
describing the number that you're going to write, for example,
Height of riser block = _________ m. The phrase tells
what the object is as well as what property of the object that you're
recording.
 Three other items of data to record in the Additional Data
section are the following.

the actual distance between the air track support posts. This
is 0.3050 ± 0.0005 m

the distance between the air track support posts that you measure
on your photograph

the scale factorThis is the ratio of item 4a to item
4b

Now starting with 0, number the images of the marker wire on your
photo consecutively. If your photo has more than 15 marker images, you may stop at 15.

Lay a ruler beside the images of the marker wire. There's no need
to align the 0 mark of the ruler with spark 0 but you may do so if
you wish. Tape the ruler down to the table to keep it from shifting
while taking measurements.

Read the position of each marker wire image to the nearest 0.0001 m (0.1 mm) using a consistent sighting
technique. Record your measurements in the Position column. Record
the corresponding Elapsed Time as well. In recording the latter, you'll
need to decide how many zeros to place behind the last nonzero number
(for example, 0.1 or 0.10 or 0.100). In order to make this decision,
use the fact that the stroboscope is accurate and precise to the nearest
0.1 Hz.

Calculate the changes in position from the position measurements.
Then apply your scale factor to convert the values in photo m
to actual m.
 Complete the Change in Time and Average Velocity columns.
Scan your original data page and the photo that you analyzed. Save
them to a single file that you upload to WebAssign by the date given
in the schedule. Name the file L109Dlastnamefirstinitial.pdf. You'll upload your final report at a later date.
See the Laboratory Recording and Reporting Guide for more information.
 Your report will be scored based on appearance and format in addition to content.
 The composition of the report must be original. Don't copy and paste from the lab instructions into your report. Exception: Use the tabular formats provided for you.

You may prepare either a handwritten or a wordprocessed report.
 If you prepare a handwritten report, write in ink on typing or copy paper. It's much easier to read on paper without lines.
 If you prepare a wordprocessed report, including drawings and equations requires special attention. Of course, you could use Word drawing tools and an equation editor. Another method is to leave spaces for diagrams and equations as you type the report. Then print the file and add the equations and diagrams by hand. (This is what people did before the days of word processors.) Finally, scan the completed report to a pdf.
Here is the rubric that the teacher will use to evaluate your report. Whether you handwrite or word process your report, make sure to label all sections and present your work clearly. Here is a complete list of the sections to include in your report in the order in which they should appear.
 Heading and title
 Goal
 Theory
 Method
 Data table: This should be a neater version of your original data table.
 Graphical analysis
 Error analysis
 Conclusion

When an object has uniform acceleration,
the instantaneous velocity at the middle of any time interval is
identical to the average velocity over that time interval. 
It's well worth your while to take the time to understand this point,
as such an understanding will add a powerful tool to your problemsolving
toolkit.
Consider this:
Suppose a uniformlyaccelerating object undergoes an increase
of velocity, Δv, in a time, Δt. Then in
a time Δt/2, which brings one to the middle of the time
interval, the velocity will have increased by an amount, Δv/2.
In the latter half of the time interval, there is an equal increase of
Δv/2 in the velocity. This is the result of uniform
acceleration, for which an object undergoes equal increases of velocity
in equal times. The average velocity, which is an average over time, must
therefore coincide with the middle velocity.
To see that the above is not true for an object with increasing acceleration,
let's follow a similar line of reasoning. Over the time interval, Δt,
the first increase of Δv/2 in the velocity takes more time
than the next increase of Δv/2. That's because the rate
at which the velocity increasesthat is to say, the accelerationincreases
with time. The object spends more time at lower velocities. As a result
the average velocity will be less than the middle velocity. It's like
the jogdrive problem of P101. In
that problem, you jogged and drove the same distance, but you spent much
more time jogging than you did driving. As a result, your average velocity
for the whole trip was much less than the velocity midway between the
jogging and driving velocities.
In the theory section of your report, do the following.

Draw a line on a v vs. t graph representing the
motion of a uniformlyaccelerating object. Make this graph fill half
a page for clarity. Next label three widelyseparated points on the
graph. These points will represent initial and final velocities over
a time interval as well as the middle velocity and the time at which
it occurs. What it comes down to is that you'll be plotting three
points (t_{i},v_{i} ), (t_{f},v_{f} ), and (t_{m},v_{m} ), where i, f, and m represent initial, final, and middle correspondingly.
Your graph must make clear the fact that the object increases in velocity
from v_{i} to v_{m} in the same
amount of time as it increases from v_{m} to v_{f}..

Repeat the previous problem for the case of an object undergoing
uniformlyincreasing acceleration. (By the way, the actual
physics name for the rate of change of acceleration is the jerk.
So this will be a case of uniform jerk.) In this case, though, your v vs. t graph must make it clear that the object
increases in velocity from v_{i} to v_{m} in more time than it takes to increase from v_{m} to v_{f}.
 Now do the following problem. If you apply the key point of this
lab, the problem is fairly simple to do. You may find it easier to describe
your method in words than to carry out an algebraic solution. Feel free
to do so.
Problem: An object falls from rest without air friction.
In the last 0.200 s of its fall, it falls a distance of 20.0 m. What
total time was the object falling?
Graphical Analysis
From here on, the analysis will parallel that
of L103. In that lab, you drew a graph
and best fit line, found the slope, constructed a matching table, and
wrote the equation of the fit. You'll carry out these same operations
next with your data. You'll draw this graph by hand like you did the one
for L103. Here's graph
paper if you need it.

Prepare your graph paper for a graph of Average Velocity vs. Elapsed
Time. If necessary, review the
Graphing Guidelines.

Draw your best fit line through the data points as best as you can
judge it. Select the two points that you'll use to calculate slope.

Show the calculation of the slope.

Complete the matching table for your line of best fit. This time,
we leave you to fill in all the blank cells, except for the Value
(expected). We don't expect you know how to determine the latter
at this point.

Write the equation of your best fit line.
Error Analysis
Complete the uncertainties table to determine
the absolute uncertainty in a representative value of the Average Velocity.
There are more entries than in L103 due to the fact that the scale factor
conversion introduces additional uncertainty. Base your estimate of the
absolute uncertainty in the Post separation (photo) on the discussion
in the Equipment Setup section. For the elapsed time, information was
given in the Method and Data section that should help with your estimate
of the absolute uncertainty.
Measurement 
Representative
Value of Measurement 
Absolute
Uncertainty 
Relative
Uncertainty 
Position (photo) 



Post separation (photo) 



Post separation (actual) 



Change in time 



Average Velocity 




Explain your estimate of the absolute uncertainty for Post separation
(photo).

Explain your estimate of the absolute uncertainty for Elapsed Time.
 Calculate the experimental error in your value of acceleration. Here are the accepted values for the 5 photos.
Photo ID 
Accepted value of acceleration (m/s^{2}) 
168 
0.249 
169 
0.249 
171 
0.373 
173 
0.373 
175 
0.124 
 Since the measured acceleration depends on Average Velocity and Elapsed Time, assume that the relative uncertainty in the acceleration is the same as that in the Average Velocity. Given that, calculate the absolute uncertainty in the acceleration. Then compare that result to the experimental error. Is your value of uncertainty
in the acceleration large enough to account for the experimental error? Tell what you learn by making this comparison.
State your results to make it clear that you achieved the goal of the
lab.
Before you submit your report, take time to review it to make sure it's
complete and properly presented. Once again, here's the
rubric.
Submit the scanned PDF file of your lab report by the due date.
