A quadratic drag law is tested for a sphere falling through air: for some constant

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Conditions of Validity, Ch Inner products, orthogonality and the adjoint

Problem

A quadratic drag law is tested for a sphere falling through air:

F = cv^2

for some constant

c

. Five measurements of speed

v_i

and drag force

F_i

are collected into the vectors

\mathbf{a} = (v_1^2, v_2^2, \ldots, v_5^2)

and

\mathbf{b} = (F_1, F_2, \ldots, F_5)

\mathbb{R}^5

, using the standard real dot product

\mathbf{u}\cdot\mathbf{v} = \sum_i u_i v_i

, which is the inner product on real vectors used throughout. Data:

v_i=2,\ 4,\ 6,\ 8,\ 10\ \mathrm{m\,s^{-1}}

and

F_i=1.0,\ 4.5,\ 8.0,\ 15.0,\ 24.5\ \mathrm N

. ((a)) Starting from the projection theorem for the best approximation of

\mathbf{b}

by a vector in the one-dimensional subspace

\mathrm{span}\{\mathbf{a}\}

, derive the normal equation fixing the least-squares coefficient

c

\mathbf{b}\approx c\mathbf{a}

, and show it reduces to a ratio of inner products. ((b)) Using the Cauchy-Schwarz inequality, derive a bound on

|c|

in terms of

\|\mathbf{a}\|

and

\|\mathbf{b}\|

alone, and state the condition under which the bound is attained exactly. ((c)) Evaluate

c

and the bound found in part (b) for the tabulated data (four significant figures), verify the inequality holds, and comment on what the closeness of the two numbers tells about the quality of the quadratic-drag fit.

Browse the problem library