A pebble of mass falls from rest through air with drag coefficient

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Mechanics & Dynamical Systems · Kinematics and Newtonian dynamics of particles

Computational Physics, Ch A falling body that knows about air: the first simulation

Problem

A pebble of mass m=0.050 kgm = 0.050\,\mathrm{kg} falls from rest through air with drag coefficient b=0.0010 kg m−1b = 0.0010\,\mathrm{kg\,m^{-1}}, vv measured negative downward, so the drag term (b/m)v2(b/m)v^2 is always positive and must oppose the fall (reduce the downward acceleration). Someone writes the following loop, using a coarse step Δt=1.0 s\Delta t = 1.0\,\mathrm{s} chosen only to make the effect obvious by hand: (line 1) t, v = 0.0, 0.0 (line 2) for n in range(5): (line 3) a = -g - (b/m)*v**2 (line 4) v = v + a*dt (line 5) t = t + dt. Running it and tabulating vv against tt gives v=0, −9.81, −21.54, −40.64, −83.48 m s−1v = 0,\ -9.81,\ -21.54,\ -40.64,\ -83.48\,\mathrm{m\,s^{-1}} at t=0,1,2,3,4 st = 0, 1, 2, 3, 4\,\mathrm{s}, the magnitude roughly doubling at each step with no sign of levelling off. The terminal speed computed from mg/b\sqrt{mg/b} is 22.1 m s−122.1\,\mathrm{m\,s^{-1}}. ((a)) Identify the line responsible for this runaway growth and say what it should read instead. ((b)) Explain, in one sentence and in terms of the force balance, why that line produces a speed that never settles.

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