A particle's velocity along a guide rail is (in m s

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Integration: error expansions, Gauss rules and adaptivity

Problem

A particle's velocity along a guide rail is

v(t)=t\,e^{-t}

(in m s

^{-1}

t

in s). Its displacement over

0\le t\le1

s is

s=\int_0^1 v(t)\,\mathrm{d}t

, with exact value

s=1-2/e=0.264241\,\mathrm{m}

(to six figures). ((a)) Compute the trapezium estimate

T_1

with one interval (

h=1

s) and

T_2

with two intervals (

h=0.5

s). ((b)) Form the Romberg extrapolate

R=T_2+\dfrac{T_2-T_1}{3}

. ((c)) Compare the errors of

T_2

and

R

against the exact value and state the factor by which extrapolation improved the estimate.

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