The speed of a body falling through a resistive fluid is modelled as

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory

Computational Physics, Ch Finding the best parameters: minimisation and fitting

Problem

v(t) = v_\infty \tanh(kt)

, where

v_\infty

is the terminal speed and

k

is a rate constant. Three speeds are measured, each with the same uncertainty:

$t$ (s)	1	2	3
$v$ (m s $^{-1}$ )	8.0	13.5	16.0

Starting from the guess

v_\infty = 15.0\,\mathrm{m\,s^{-1}}

k = 0.50\,\mathrm{s^{-1}}

, carry out one Gauss-Newton step. You may use

\partial v/\partial k = v_\infty t\,\mathrm{sech}^2(kt)

with

\mathrm{sech}^2(x) = 1-\tanh^2(x)

. (a) Evaluate the residuals and the two partial derivatives

\partial v/\partial v_\infty

and

\partial v/\partial k

at the starting guess, for the three times. (b) Assemble the

2\times2

normal equations. (c) Solve for the update and report

v_\infty

and

k

after the step, to three significant figures.

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