In this lab you will again work as individuals, but discussions are OK. The assignment is due at the beginning of lecture on Monday, 30 Mar. In this lab you'll explore the properties of curve-fitting algorithms. You'll need to submit your codes, plots and answers to the questions, below.

1. For this problem, you'll need the data set lab6_1.mat. These represent velocity (m/s) data from an acoustic ranging device as a function of time (s). Use the normal equations method to determine the best fit line for the model equation v = v_o + at.
   • Plot the raw data and a best-fit line on the same axes.
   • State the values determined for v_o and a.
   • Determine the uncertainties in those parameters.
   • Describe a physical experiment that might produce these data.
     Since a ≈ -g/2, this could be a puck slid upward (some initial velocity v_o > 0) on a ramp poised at a 30° angle with the horizontal.


2. For this problem, you'll need the data set lab6_2.mat. These data represent the signal voltage (V) across a circuit element as a function of time (s). Use the normal equations method to determine the best fit curve for the model equation s = Asin(t) + Bsin(2t) + Csin(3t).
   • Plot the raw data and a best-fit curve on the same axes.
   • State the values determined for A, B and C.
   • Determine the uncertainties in those parameters.
   • Look at your result for B. What change could you make to your model equation based on this result.
     Since B ≈ 0 and δB ≈ B, we might assume that there are only two sinusoidal components to the signal and model the problem as s(t) = Asin(t) + Csin(2t).

See plots below and code for details.

3. For this problem, you'll need the data set lab6_3.mat and my code expFit.m. In a lab you have a sample of ¹¹⁶Xe gas. You have a device to determine the amount of that particular isotope present and let it run for a few seconds prior to t = 0 (all t data are in s; N are in # of ¹¹⁶Xe atoms). ¹¹⁶Xe gas naturally decays and you're trying to determine the half-life.
   • What is the half-life of ¹¹⁶Xe according to your data?
     λ ≈ 54.0 s, which is close to the accepted value of 56 s (only ∼4% small). This result was obtained with a realistic estimate of K ≈ 10²⁴ as our seed.
   • Did your lab technician properly set up your equipment? Use your fitted parameters to justify your answer.
     Since K ∼ 10²⁴ ∼ N_o, you can assume that this was set up with a bias in the system. Your main result for λ is ∼ 4% off the accepted value of 56 s; not a bad result, but certainly affected by the bias in the zero reading of your laboratory setup.

See plots below and code for details. Note that in the bottom panel we assume that the technician did set up the apparatus correctly (i.e., K = 0). Under this assumption, we find a very similar looking fit, but a much improved value for λ ≈ 56.1 s (only ∼0.2% large). If we delve into the statistics and look at the coefficient of determination (or "goodness" of fit), we find that it is R² &asymp 0.998 for both cases! This is an extremely good fit for both assumptions. It is difficult here to determine whether or not the technician is competent or not. [In this case the data were generated (see details in the code) with K = 4×10²¹, so a bias was in the original data. Still the assumption of whether or not it was there is essentially lost in the noise – I used the proper value for λ when generating the data.]