A Chapter 1 drag-limited fall has mass and quadratic drag coefficient

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Research software practice, verification and validation

Computational Physics, Ch A workspace that can be rerun next year

Problem

A Chapter 1 drag-limited fall has mass m=3 kgm=3\,\mathrm{kg} and quadratic drag coefficient b=0.08 kg m−1b=0.08\,\mathrm{kg\,m^{-1}}, so the terminal speed is v∞=mg/bv_\infty=\sqrt{mg/b} with g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}. The script integrates this fall for N=2000N=2000 steps, and Chapter 2 established that each stored floating-point value satisfies fl(x)=x(1+δ)\mathrm{fl}(x)=x(1+\delta) with ∣δ∣≤12ϵmach|\delta|\le\tfrac12\epsilon_{\rm mach}, ϵmach=2.22×10−16\epsilon_{\rm mach}=2.22\times10^{-16}, so that after NN steps the rounding-error floor on a quantity of typical size v∞v_\infty is of order Nϵmachv∞N\epsilon_{\rm mach}v_\infty. (a) Compute v∞v_\infty to four significant figures. (b) Estimate the rounding-error floor on the final speed after N=2000N=2000 steps, using the formula above. (c) A known-answer test asserts that the final speed lies within tol=10−3v∞\mathrm{tol}=10^{-3}v_\infty of v∞v_\infty. Compute this tolerance and the ratio tol/floor\mathrm{tol}/\text{floor}, and say whether the tolerance sits safely above the rounding floor. (d) Following the convention that a computed number is quoted only to the digits its supporting analysis warrants, state to how many decimal places v∞v_\infty should be quoted given the tolerance in part 3.

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