The conservation of momentum is a fundamental law of physics.
Typical labs require that students take data for several trials and
several cases. This can mean that students are required to do dozens
of calculations. In addition, when setting up the equations for each
case, students sometimes fail to see that it is the same fundamental
relationship at work in all cases. This activity forces the point that
one equation can be used for all the various experimental cases. This
activity can meet several MSDE Core Learning Goals.

Before this activity, students have worked extensively with data analysis
using either graphing calculators or graphing software. They have
learned that physical laws and physics equations are based on experimental
evidence. They have used CBLs and calculators to gather data. However,
this is the first exposure, in this class, to spreadsheets, ULIs and air
tracks and to MATLAB.

It seemed from close observation that most of them were familiar with
spreadsheets. They required little direct instruction other than a few
basics about how this particular package worked. Their only MATLAB
instruction came directly from the handout, though some students had used
MATLAB in another class.

Some students were able to complete the spreadsheet in the space of 40 min
and to do the MATLAB activity in about 30 min. Other students took much
longer figuring out the relationships between the various quantities
needed just to set up the spreadsheet. Once this was determined though,
they breezed through the MATLAB portion.

The combination of working with the spreadsheet and MATLAB resulted in
many valuable student-teacher interactions about the relationship between
the variables. It appeared that there was a much greater understanding of
the underlying equations than is usually achieved with the calculator and
just having the students plug in the numbers. In addition, there was a
greater willingness to search out sources of mathematical error, such as
putting the wrong number in the wrong cell, because the spreadsheet was
doing all the work.

Most interesting to watch were the groups who found that their MATLAB
analysis produced different results than the spreadsheet. In each case,
it turned out that the spreadsheet was in error and the students could go
back and correct it.

#### Analysis of AirTrack Momentum Data Using Spreadsheets and MATLAB

Spreadsheet:
1. Using the spreadsheet suggested by the teacher, enter your data as follows:
column A should be the mass of glider 1,
column B should be the initial velocity of glider 1,
skip a column,
column D should be the final velocity of glider 1,
skip a column,
column F should be the mass of glider 2,
column G should be the initial velocity of glider 2,
skip a column,
column I should be the final velocity of glider 2.
2. Fill numbers for each run into the appropriate columns. Remember, a glider
as rest has a velocity of 0 and velocities may be positive or negative depending
on direction of motion. When all experimental values are filled in, save the
spreadsheet first in its native format and then once again as an ASCII text file.
Be sure to name this second file something different from the first.
3. Using the first spreadsheet saved, enter formulas for the momentum (initial
and final) in each of the skipped columns. Thus cell C2 becomes A2*B2 or
A2*B2/1000 (if your mass was in g). This formula can be copied down the
column. Repeat for the other skipped columns, using the appropriate
combinations of mass and velocity: column C should be the initial
momentum of glider 1, column E should be the final momentum of glider 1,
column H should be the initial momentum of glider 2, and column J should
be the final momentum of glider 2.
4. Now determine the total initial momentum and the total final momentum.
The total initial momentum should be in column K and represents the sum of
columns C and H. The total final momentum should be in column L and represents
the sum of columns E and J. Determine the average momentum for and the
percent difference between these two columns. If there is a sign change between
the two columns, remember to take the absolute values of the numbers.
5. In two more columns, determine the momentum change for glider one and the
momentum change for glider two. Compare these two columns and determine
a percent difference.
6. Record the results of the spreadsheet and save it.
MATLAB:
1. Open MATLAB. Open the text file saved earlier and edit out all words.
You can leave spaces and even empty lines in the program, but it should
consist of nothing but numbers - columns of masses and velocities. When
the file is edited, save it as glider.dat . MATLAB will ask about a .m
extension, but click on No .
2. From the command window, type the following:
load glider.dat loads in the data file
mass1 = glider(:,1); reads the first column of the glider
file and names that column mass1
mass2 = glider(:,4);
intvel1 = glider(:,2);
finvel1 = glider(:,3);
intvel2 = glider(:,5);
finvel2 = glider(:,6);
What did these commands do?
3. Now we calculate the total initial and final momentum. All the values are
entered as long lists (1 column matrices). We need to right our commands so
that MATLAB knows we want to multiply the first number in one column by the
first number in the next, the second by the second and so forth. We do not
want to do matrix multiplication. To achieve this, multiplication signs are
followed by a space and a period. So the commands below must be typed exactly
as they appear as far as the spaces around the multiplication and
division signs are concerned. Remember you only need the 1000 at the end if
your masses are in kg.
momint = (mass1 .* intvel1 + mass2 .* intvel2)/1000
This time we leave off the semicolon so that we can see the results.
momfin = (mass1 .* finvel1 + mass2 .* finvel2)/1000
Remember that you may need to use the absolute value function in one of these
equations. Compare these values to those generated by your spreadsheet. Comment
on any significant differences.
4. Now the percent error:
error = 100 * (momfin-momint) ./(.5*(momint +momfin))
Again, compare these results to those from the spreadsheet and comment on any
significant differences.
5. Now determine the change in momentum for glider one and the change in
momentum for glider two (You write the commands!) . Also, determine a
percent difference error.
Questions
1. Does your data support or refute the conservation of momentum? Are some
cases ÒbetterÓ then others? Suggest some sources of error and explanations
for differences between cases.
2. Does your data consistantly support a residual slope to the air track or
the existence of some friction? Explain
3. Finally, which method - the spreadsheet or MATLAB seemed to be a timesaver?
Give reasons for your choice.