1. Introduction.
In this experiment, collisions between pucks moving on a nearly frictionless surface (provided by an air table) are studied to test the principle of conservation of linear momentum AS A VECTOR in two dimensions. You will study an example of an elastic collision. The moving pucks are imaged at a series of equally spaced time intervals using a CCD camera interfaced to a
Computer, and a movie is recorded. Measurements of the puck images are used to analyze the collisions and to test for conservation of linear momentum. The degree of conservation of kinetic energy in an approximately elastic collision will also be checked.
2. Required Equipment
A. An air table and pucks. Each puck has a distinctive design on the top to identify it. The air table has an adjustable launcher in one corner and a small brush mounted above the center to hold the target puck in position. The table must be level so that free pucks do not rapidly drift to one side when the air table is turned on.
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B. A camera mounted approximately above the table and connected to the video input of a computer. The computer records video images at 1/30 second intervals. The camera is capable of recording the video in room light. The movie will be recorded with Video
Point Capture.
C. A lab bench computer with which you can digitize your movie using Video Point
Fundamentals and then analyze the data using Graphical Analysis.
3. Procedure
Data Recording
Laboratory partners should work together. One controls the video recording; the other fires the puck. Be careful not to disturb the alignment of the video camera.
Specific steps are now indicated which should allow successful video data recording.
Click on the large orange “C” in the dock on the desktop of the computer used to control video recording. Click on Capture.
A recording window will be activated and a live video display will appear in this window. Move your hand in front of the video camera lens to verify that this is the case.
At this stage you are ready to record and test the movie.
To prepare for movie recording, turn on the air table. Locate the target puck near the middle of the field of view using the camel’s hair brush. Place the projectile puck in the launcher and hold down on the pressure key.
You will take one movie of an elastic collision. Make sure that the target puck is near the center of the field of view.
The computer operator starts the movie recording by click on the [REC] button. The computer operator tells the launcher to fire the projectile puck. The computer operator stops the movie (by clicking on the [Stop] button) after the collision has occurred and the pucks have reached the retaining wires at the edge of the table.
You can now replay the movie to see if it has provided acceptable video data. To do so, click on
Play, this will play the movie from start to finish. Watch your movie of the collision.
If the images look good and the elastic collision is good, reduce video size by use confirm edit
(move the little right triangles at each end of the bar near the bottom of the screen. Move the left triangle to the right until a puck begins to move. Then, move the small diamond-shaped figure all the way to the right. Next, move the right-side triangle to the left until you reach the point at which your movie should end. Remember, you have no interest in what happens after either puck hits a retaining wire. So, your movie should end the first time a puck hits a retaining wire.
After you have moved the right-side triangle to the left until you reached the point at which your movie should end, click on Confirm Edit).
Save the movie to the Movie desktop.
Once the movie has been saved on the desktop, copy it to your lab bench computer (by use a memory stick).
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4. Digitization of the Movies and Tabulation of Data
Since there is no absolute value (such as g) to find in this experiment, it is not necessary to convert to true space distances and hence we will use CCD pixel values directly. (In any case the picture does not include a grid or scale). Also note that the time interval of 1/30 sec between successive images is also fixed for all measurements, and we will use this feature to simplify our analysis considerably.
Note that images of the projectile and target pucks before launching cannot be used for your analysis, so when analyzing the video, move to the first active frame. In fact, it is wise to begin data analysis for images when the projectile puck physically clears the rubber band, as the rubber band can possibly rub on the surface of the projectile puck as it is leaving the launcher.
On your lab bench computer, activate Video Point Fundamentals by clicking on the icon on the desktop, then on Open Movie. A dialog box will appear. From your movie in the desktop, open your elastic collision movie.
You should now see a Movie Window, a Data Table, and a Coordinate System Window. Click on the size expander box in the upper right hand corner of the Movie Window to double its size.
You are now ready to digitize points. Take S1 to be the projectile puck. Position the cursor at the center of S1 and “click”. . You can see the point marked and the cursor moves to the next position. Continue to click the center of S1 until you have data for each frame and a message box informs you that you have completed the data series. Chose the option “Track another point”
Click OK
Center the cursor on puck S2 and “click”.
The new data points will have a new color as displayed on the right hand side . . Repeat measurements for puck S2 as you did for S1. Note, until the collision occurs, you would be clicking on the same point, S2 does not move. However, just after the collision occurs, you would be automatically tracing S2. Keep clicking, until you have completed the data series. Once a puck collides with the wire “fence” on the edge of the air table, disregard the remaining points.
5.
Clicking continue will bring you to the data page. Click file then export, then excel. Save this file to the desktop and open it in an excel spreadsheet. Copy and paste all relevant data (time, xposition, and y-position) into the Graphical Analysis program. DO NOT copy column headings into Graphical Analysis.
Graphical Analysis.
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You are now in a position to graph and analyze the tabulated data, but first the Graphical
Analysis program must be started.
Click on the Graphical Analysis.
Click on the upper left hand element of the table (currently blank).
Pull down the Edit Menu and select Paste. Your data table from Video Point should now be pasted into the table. If you have pasted your data correctly into Graphical Analysis your data should appear in five separate columns each with a heading of X, Y, Column, Column 1, and
Column 2. Change them to: Column 1, T; Column 2, X1; Column 3, Y1; Column 4, X2;
Column 5, Y2 (with associated units).You are now ready to analyze projectile motion.
Now go to the Data Menu and select New Calculated Column. Do this six times so that a total of
6 new Columns are created in your table.
The header of Column 6 should be labeled “drel” for the relative distance between the two pucks in a given frame. The functional form for drel should be recorded in the text line as: drel = ((X1-X2)^2+(Y1-Y2)^2)^0.5 drel will be measured in CCD pixels (rather than meters).
The header of Column 7 should be labeled “vrel” for the relative velocity between the two pucks in a given frame. The functional form for vrel should be recorded in the text line as: vrel = delta(drel) note that the velocity would be normally written in a form like v =
 x/
 t. Since the time between frames is always the same (1/30 th of a second), we can measure time in frame units (ie.
1,2,3,...), which is just the row number in the table, called the “index”. In any case the time between successive rows in the table is always
 t = delta (index) = 1. Hence we can write: vrel = delta(drel)/delta(index) = delta(drel) / 1 = delta(drel).
Measure the masses of two pucks and make sure that mass2 is the mass of puck S2 which was initially at rest. The header for Column 8 should be labeled “PX” for the x-component of total momentum, numerically is written:
PX = mass1*delta(X1) + mass2*delta(X2) where you see the mass labels, you should type in the proper numerical masses for your pucks instead of the names “mass1” and “mass2”. Remember, mass1 refers to puck S1 and mass 2 refers to puck S2.
Note that momentum is normally written P = mv = m
 x/
 t.
Since the
 t = delta(index) = 1, we can ignore the time interval in our calculations.
The header for Column 9 should be labeled “PY” for the y-component of total momentum. numerically it is written: PY = mass1*delta(Y1) + mass2*delta(Y2) again use numerical values for “mass1” and “mass2” instead of the names.
The header for Column 10 should be labeled “PCM” for the magnitude of the total CM momentum. numerically it should be written: PCM=(PX^2+PY^2)^0.5
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The header for Column 11 should be the direction of motion of the center of mass in the CCD camera coordinate system, and labeled “SLOPE”. numerically: SLOPE=PY/PX
Now save your work, by pulling down the File Menu and selecting “Save as”. Enter a convenient name (using the key word “elastic”) and save your analysis in the movie folder.
As this point, you have completed the preparatory work for analysis of the elastic collision.
6. Analysis.
In Graphical Analysis, your Elastic Scattering data should be in your data and should be labeled data set 1.
Momentum Conservation: In all collisions, momentum is conserved. This means the xcomponent of momentum (PX) is conserved, and separately the y-component of momentum (PY) is conserved. Additionally the magnitude of the total momentum (PCM) is conserved and the direction of the total momentum (SLOPE) is conserved.
Example: for PCM to be conserved means that PCM before collision, PCM(i), should be equal to PCM after the collision, PCM(f). Equivalently, we can take the difference in these two quantities and ask that the difference be consistent with zero:

(PCM) = PCM(f) - PCM(i) where the error in this difference is:

[

(PCM)] = sqrt { (

[PCM(f)]) 2 + (

[PCM(i)]) 2 }
How can you know what is “before” collision and “after” collision in your data table? Look at the column “vrel”, which is the relative velocity of the two pucks. It should be negative before collision (the relative distance between the pucks is decreasing with time), and positive after the collision (the distance between pucks is increasing with time). It should then be obvious that rows in the table where vrel is negative correspond to data for physical quantities before the collision. Note there may be a row or two between the “before (initial)” and “after (final)” conditions where vrel may take on an intermediate or “transition” value. Data in these rows should be avoided. We now describe a useful analysis procedure.
 From the Data Menu, select “new data set”. A data set 2 will be created.

Click on data set 1, and select the first five columns - and also select all rows corresponding to the data before collision. [Do not include a row or rows where vrel is in a transition value.] Then go to the Edit Menu and select Copy.

Now click on data set 2, and click on the upper left hand data element in this new table.
Go to the Edit Menu and select Paste. The data subset, corresponding to the data before collision only, will now be included in this table.

Go to the file menu and select Save. This is just a precautionary measure. From the File
Menu, select page setup. Select landscape (sideways) printing mode. Then Print your data tables for data set 1 and data set 2.
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
On the graph (Data set 2) :
Click and hold on the label on the horizontal axis. Select “t” or “Index”.
Click and hold on the label on the vertical axis. Select “more” and check all (vrel, PX,
PY, PCM, and SLOPE) All they should be appear on the graph.
 To find the mean values and standard deviations for each data set, use the “Statistics” command on the “Analysis” menu, and check each box in the dialog box that appears.
Statistics boxes containing the mean and standard deviation should appear on the graph for each of your runs. Print the graph.

Fill in a new table of values of physical quantities before collision: elastic scattering mean value standard deviation (

) n vrel (i)
PX(i)
PY(i)
PCM(i)
Slope(i) error in mean

= (

)/

(n)
 Now from the Data Menu, select “new data set”. A data set 3 will be created.

Click on data set 1, and select the first five columns - and also select all rows corresponding to the data after collision. Again avoid a possible row or two just at collision time, where vrel may be in transition. Go to the Edit Menu and select Copy.

Now click on data set 3, and click on the upper left hand data element in this new table.
Go to the Edit Menu and select Paste. The data subset, corresponding to the data after collision only, will now be included in this table.

Go to the file menu and select Save. Again, this is just a precautionary measure. Print a copy of your data set 3.

Return to data set 3 and build a table for physical quantities after collision. Note that the number of measurements (n) in this case may be different than for that above. elastic scattering mean value standard deviation (

) n vrel (i)
PX(i)
PY(i)
PCM(i)
Slope(i) error in mean

= (

)/

(n)
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Energy Conservation : In a perfectly elastic collision, both momentum and kinetic energy are conserved. In such a collision, it can be shown that the relative speed of approach before the collision and the relative speed of separation after the collision have the same magnitude. Hence vrel(i) should be equal in magnitude to vrel(f).

The magnitude of the ratio (e) of the final and initial speeds: e = |vrel(f)/vrel(i)| is called the elasticity of the collision.

For a perfectly elastic collision e = 1.0, while for a completely inelastic collision e = 0.0.

The error in the elasticity is given as:

 e
 e


   vrel (
( f ) f )


2



   vrel ( i )
( i )


2
In your physics analysis outside of class, you will be asked to evaluate just how “elastic” your elastic collision is.

You have now completed all work required in the laboratory.
You should have printed a copy of the 3 data sets for elastic collision along with two graphs corresponding to data set 2 and 3. You should have filled in the 2 analysis tables. You should have recorded the masses of the pucks used.
6. Physics Analysis (outside the laboratory).
1.
Is

(PX) consistent with zero within errors?
2.
Is

(PY) consistent with zero within errors?
3.
Is

(PCM) consistent with zero within errors?
4.
4. Is

(SLOPE) consistent with zero within errors?
5.
Find the elasticity (e) and its error. Is the collision consistent with being completely elastic within errors?
6.
If your answers to any of questions 1-5 are “no”, can you think of reasons why there might be an inconsistency?
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