Equation making question

Question

This is the question:

An experiment produced the set of values shown below for separation (d) and energy (E). A graph was drawn of E as a function of d. Determine an equation relating E and d. Explain how you obtained it.

The values for d are:

2.2

4.4

6.1

11.1

The values for E are:

16.2

8.2

5.4

3.2

Answer 1

1) The results go in opposite directions, so the relation is negative linear or inverse.

2) It's pretty clear that if you compare the change in E / Change in d, for the 1st to 2nd point and the 2nd to 3rd point that you do not have a linear relationship

3) Since it is inverse in some way: E = a/dⁿ where n is not known, you should calculate 1/d and see how it compares to E. (You could also compare d_i/d_j to E_j/E_i for pts i and j to see if the ratio is = which would imply a simple inverse relation)

4) The inverse relation fits pretty well as the first to the second point halves 1/d and the E halves as well. Checking 3rd and 4th point, we find the 1/d ratio is .55 and the E ratio is .59.

If you plot lnd vs lnE you will get a slope of -1 which tells you that E = ad^-1