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                                  A Difference Look at Newton's Apple

The cell below contains data from one experiment that we did dropping a pillow.  You can use your own data by uploading it from your graphing calculator using the appropriate TI-Graph Link program and cable and then cutting and pasting from the Graph Link window to this window.   You may need to do some editing -- putting commas between entries and deleting unusable data at the beginning and end of the experiment.   Evaluate the next cell now.

;[s]
4:0,1;69,0;491,2;517,0;520,-1;
3:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
Clear[Pillow, n]
Pillow = {5.98496, 5.97415, 5.95615, 5.92014,
          5.88773, 5.82651, 5.75449, 5.66086,
          5.54203, 5.43039, 5.29715, 5.18912,
          5.07029, 4.94065, 4.79661, 4.63456,
          4.47611, 4.28886, 4.08000, 3.86753,
          3.66227, 3.43541, 3.21214, 2.98887,
          2.76201, 2.51353, 2.27226, 2.01659,
          1.75371, 1.48723, 2.27947, 1.76812,
          1.45842};
          
n := 29
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The cell below illustrates how Mathematica  can plot this data.  Evaluate it now.
;[s]
5:0,0;31,1;42,0;65,2;80,0;82,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,2,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
ListPlot[Table[{k, Pillow[[k]]}, {k, 1, n}],
         PlotRange -> {0, 7}]
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The cell below looks at the first and second order Newton divided differences.  This particular data was collected at intervals of 0.02 seconds.
:[font = input; preserveAspect; ]
q := 0.02
first[k_]  := (Pillow[[k + 1]] - Pillow[[k]])/q
second[k_] := (first[k + 1] - first[k])/q
MatrixForm[Table[{k, first[k], second[k]},{k, 1, n-2}]] 
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Notice the velocity data isn't too bad but the acceleration data (the second order divided differences) is pretty poor.  This is not surprizing in view of the discussion in this module.

As we discussed in the module, one way to try to avoid these kinds of errors is by using measurements made at longer time intervals.  The cell below does this by using every fourth data point.  Evaluate the next cell now. 
;[s]
3:0,0;381,1;407,0;410,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;
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pick[k_]   := Pillow[[4 * k - 3]]
d := 0.08
first[k_]  := (pick[k + 1] - pick[k])/d
second[k_] := (first[k + 1] - first[k])/d
MatrixForm[Table[{k, pick[k], first[k], second[k]},      
           {k, 1, 6}]]
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This is a definite improvement but nothing to write home about.
^*)