A ball is thrown vertically so that its height obeys

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

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

Problem

y(t) = v_0 t - \tfrac12 g t^2

, with

v_0 = 20.0\,\mathrm{m\,s^{-1}}

and

g = 9.80\,\mathrm{m\,s^{-2}}

(air resistance ignored). Someone estimates the instantaneous velocity at

t = 1.00\,\mathrm{s}

using the forward-difference formula

[y(t+h)-y(t)]/h

in place of the true derivative

y'(t)

. (a) State the binary64 layout (sign, exponent and stored significand bits) and use it to derive

\epsilon_{\rm mach} = 2^{-52}

. (b) Expand

y(t+h)

as a Taylor series about

t

and derive the leading truncation-error term of the forward difference, as a function of

h

and

y''(t)

. (c) Say why, for this particular

y(t)

, that leading term is in fact the entire truncation error, with nothing left over. (d) Evaluate the truncation error at

h = 0.100\,\mathrm{s}

and at

h = 0.0500\,\mathrm{s}

, and give the ratio of the two errors.

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