A small ball of mass falls from rest under gravity

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Letting the computer do advanced algebra: operators, series and differentiated programs

Problem

A small ball of mass

m

falls from rest under gravity

g

, subject to a linear air-resistance force

-bv

opposing the motion (a valid model at moderate speed, before drag becomes quadratic), so that

m\dot v = mg - bv

with

v(0)=0

. Here

m,b,g

are positive real constants. ((a)) A computer algebra system returns the solution

v(t) = \dfrac{mg}{b}\left(1-e^{-bt/m}\right)

. Verify by substitution that this satisfies both the differential equation and the initial condition. ((b)) For

m = 0.050\ \mathrm{kg}

b = 0.020\ \mathrm{kg\,s^{-1}}

and

g = 9.8\ \mathrm{m\,s^{-2}}

, evaluate the terminal velocity

mg/b

and the speed at

t = 5.0\ \mathrm{s}

, each to three significant figures.