Conservation of Momentum  

Purpose

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.

Prior Knowledge

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.

Time and Evaluation of Activity

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.

MSDE CORE Learning Goals

  1. Goal 1: Skills and Processes
    1. Expectation 4: The student will demonstrate that data analysis is a vital aspect of the process of scientific inquiry and communication
      1. Indicator 6: The student will use spreadsheet, graphing, and database programs and probeware on computers and/or graphing calculators
    2. Expectation 5: The student will use appropriate methods for communicating in writing and orally the processses and results of scientific investigation
      1. Indicator 1: The student will demonstrate the ability to summarize scientific concepts, processes and data through written communications
      2. Indicator 5: The student will use computers and/or graphing calculators to produce tables, graphs and spreadsheet calculations
    3. Expectation 6: The student will use mathematical processes.
      1. Indicator 1: The student will use ratio and proportion in appropriate situations to solve problems
      2. Indicator 3: The student will manipulate quantities and/or numerical values in algebraic equations
      3. Indicator 4: The student will judge the reasonableness of an answer
  2. Goal 5: Concepts of Physics
    1. Expectation 1: The student will know and apply the laws of mechanics to explain the behavior of the physical world.

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 “better” 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.

Sample glider data

180.1	0.6723		0		190.2	0		0.6343
180.1	0.6309		0		190.2	0		0.5936
180.1	0.6209		0		190.2	0		0.5835

180.1	0.6884		-0.1814		290.7	0		0.488
180.1	0.7527		-0.1898		290.7	0		0.5365
180.1	0.7221		-0.1736		290.7	0		0.5167

280.7	0.3326		0		190.2	0		0.3567
280.7	0.377		0		190.2	0		0.4125
280.7	0.427		0		190.2	0		0.4712

180.1	0.8604		-0.6338		190.2	-0.672		0.7033
180.1	0.8411		-0.6709		190.2	-0.7477		0.6501
180.1	0.8546		-0.6585		190.2	-0.7074		0.6686

180.1	0.5447		-0.7376		290.7	-0.5125		0.2385
180.1	0.5176		-0.766		290.7	-0.5398		0.2041
180.1	0.5949		-0.8154		290.7	-0.5722		0.2557

190	0.7678		0.3501		190	0		0.3654
190	0.6958		0.3147		190	0		0.3312
190	0.7858		0.3594		190	0		0.375


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