Analyzing Data

Parabolic Distance vs. Time

A careful calculation of the area under the velocity versus time graph for a constantly accelerating object leads to the following distance equation:
d = 1/2at2 + vit + di

As you can see, this equation is a quadratic in t.  Therefore, with the distance on the y-axis and the time on the x-axis, a parabolic plot indicates an object with a constant acceleration.

The values from a quadratic regression of such data must be analyzed thoughtfully because the regression produces the coefficients for the standard quadratic form:

y = Ax2 + Bx + C

With 'y' representing 'd' and 'x' representing 't', it is clear that 'C' represents 'di' and 'B' represents 'vi'.

What might be less clear, however, is that the 'A' variable from the regression equation actually represents 1/2a.  As a result, the 'A' value from the regression must be doubled to find the acceleration.  Or, stated algebraically, we have:

A = 1/2a
a = 2A

This already confusing situation is made worse by the fact that the TI-82 uses lower-case variables for its quadratic regression.  Unless we use some sort of notation to differentiate the regression coefficients from the distance equation's variable, we are left with the expression "a = 2a", which doesn't make any sense (except when a = 0, but that is not the case here).

In any event, for our sample data set, we have:
 

Sample d vs. t Data
Time 
(s)		 Distance 
(m)
0.00 
0.25 
0.50 
0.75 
1.00 
1.25 
1.50 
1.75 
2.00 
2.25 
2.50 		  0.19 
 3.48 
 6.30 
 8.86 
 9.84 
11.39 
12.09 
11.82 
10.96 
 9.21 
7.20 
 

Which indicates that, for this data:
di = C
             di = 0.0181 m

vi = B
           vi = 15.3 m/s

1/2a = A
   a = 2A
                      = 2(-4.95 m/s2)
              a = -9.90 m/s2

Notice, though the calculator performs the regression, it is still up to you to determine the correct units and significant figures.
 
 

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