A meteorological sensor pod of mass is released from rest at height above a valley floor

PHYSICS · Subjective · 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 meteorological sensor pod of mass m=0.40 kgm = 0.40\,\mathrm{kg} is released from rest at height y0=20 my_0 = 20\,\mathrm{m} above a valley floor. Take downward as negative and g=9.81 m s−2g = 9.81\,\mathrm{m\,s^{-2}}. Above a fog bank that begins at height yfog=15 my_{\rm fog} = 15\,\mathrm{m}, the drag coefficient in mv˙=−mg+bv2m\dot v = -mg + bv^2 is bclear=0.030 kg m−1b_{\rm clear} = 0.030\,\mathrm{kg\,m^{-1}}; at or below that height the denser fog gives bfog=0.090 kg m−1b_{\rm fog} = 0.090\,\mathrm{kg\,m^{-1}}, and the switch is treated as sharp, with no transition region. Use the update rule with the order vn+1=vn+a(yn,vn) Δtv_{n+1} = v_n + a(y_n, v_n)\,\Delta t followed by yn+1=yn+vn Δty_{n+1} = y_n + v_n\,\Delta t, with a(y,v)=−g+(b(y)/m)v2a(y,v) = -g + \big(b(y)/m\big)v^2 and b(y)b(y) evaluated at the height yny_n that starts the step, and take a step Δt=0.25 s\Delta t = 0.25\,\mathrm{s}. (a) Beyond y0, v0, m, dt, t_end\texttt{y0, v0, m, dt, t\_end}, state the extra named arguments (with their SI units) that a function fall_two_layer\texttt{fall\_two\_layer} would need to model this descent, with gg fixed inside the function as in fall\texttt{fall}, and say in one sentence why each must be a named argument rather than a number fixed inside the function body. (b) Carry out the update rule by hand, keeping four decimal places, until you reach the first step number nn at which the state (yn,vn)(y_n, v_n) used to compute the acceleration has yn≤15 my_n \le 15\,\mathrm{m}. Give that nn together with yny_n and vnv_n. (c) Give the velocity produced by that step, that is vn+1v_{n+1}, the first velocity computed using bfogb_{\rm fog}.

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