Data Analysis
After all of the graphs and charts were done, the group determined that the classification of this system must be a power law relation. The graph constructed showed a power law. A power law is resembled on a graph in this form:
Since it was determined that it was a power law relationship, the formula for the equation as a result was:
y = kxn
After this was determined, the resulting steps below were done using the principles learned in the math classes throughout the year. So to find the values of constants, k and n, the log of both sides of the equation was taken.
logy = logklogxn
logy = logk + logxn
logy = nlogx + logk
From this equation above, it was seen that this equation was very similar to the equation that was on Figure 1.2:
y = mx + b
logy = nlogx + logk
It was very similar as the letters are only different of the variables and log was added to every variable.
b and logk = y-intercept m and n = slope
Using Figure 1.3 done on graphical analysis below, the constants k and n were calculated:
Figure 1.3
To solve for k:
logk = y-intercept
y-intercept = 1.821
k = 101.821
k = 66.22
To solve for n:
Using Figure 1.3 on the previous page, two points on the line were chosen to calculate the slope. These two points were chosen as they were very accurate because they were on the line of best fit and were easy to read.
x2 − x1
= 2.0 – 1.45
0.1 – (-0.2)
= 0.55
0.3
n = 1.833
After the constants were found, they were substituted into the power law equation and the formula for the experiment was:
y = kxn
y = 62.22x1.833
