A small raindrop is modelled with a linear drag law instead of the quadratic law used…

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 small raindrop is modelled with a linear drag law instead of the quadratic law used for the hailstone:

m\dot v = -mg - kv

, where downward is negative,

k

(in

\mathrm{kg\,s^{-1}}

) is the linear drag constant, and the term

-kv

always opposes the motion whatever the sign of

v

. Air resistance is the only force besides weight; the raindrop has mass

m = 4.0\times10^{-6}\,\mathrm{kg}

and

k = 2.0\times10^{-6}\,\mathrm{kg\,s^{-1}}

, and is released from rest at height

y_0 = 100\,\mathrm{m}

. (a) Write the acceleration

a(v) = -g - (k/m)v

and the update rule that advances the state

(y,v)

by one step

\Delta t

, taking the position update to use the velocity already held before the step (the old velocity). (b) Hand-compute

v_1

and

y_1

after one step of

\Delta t = 1.0\,\mathrm{s}

, using

g = 9.81\,\mathrm{m\,s^{-2}}

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