A capacitor discharges through a resistor according to

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Uncertainty in measurement

Computational Physics, Ch Numbers a machine can hold, and the two errors that follow

Problem

V(t) = V_0 e^{-t/\tau}

, where

V_0 = 9.00\,\mathrm{V}

is the initial voltage and

\tau = 2.50\,\mathrm{ms}

is the (exactly known) time constant. The elapsed time is measured as

t = 7.50\,\mathrm{ms}

with a relative uncertainty of

0.30\%

. (a) Compute the condition number

\kappa = |t f'(t)/f(t)|

V

as a function of

t

, evaluated at

t = 7.50\,\mathrm{ms}

. (b) Use

\kappa

to bound the relative uncertainty in the computed voltage

V(t)

, and give

V(t)

with its absolute uncertainty. (c) Suppose instead

t

had no measurement uncertainty at all and were simply stored as an IEEE 754 binary64 value. Give the resulting relative error in

t

from storage alone, and say whether it changes the answer to part (ii).

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