8 CHAPTER1. THECLASSICALSTATE
thistrajectoryexactly,butwewouldliketodevelopamethodwhichcanbeapplied
toaparticlemovinginanypotentialfieldV(x). SoletusbeginwithNewton’slaw
F=ma,whichisactuallyasecond-orderdifferentialequation
md^2 x
dt^2=−
dV
dx(1.1)
Itisusefultoreexpressthissecond-orderequationasapairoffirst-orderequations
dx
dt=
p
m
dp
dt= −
dV
dx(1.2)
wheremisthemassandpisthemomentumofthebaseball. Wewanttofind the
solutionoftheseequationssuchthatx(t 0 )=Xinandx(t 0 +∆t)=Xf,whereXin
andXf are,respectively,the(initial)heightofyourhandwhenthebaseballleaves
it,andthe(final)heightofyourhandwhenyoucatchtheball.^1
Withtheadventofthecomputer,itisofteneasiertosolveequationsofmotion
numerically,ratherthanstruggletofindananalyticsolutionwhichmayormaynot
exist(particularlywhentheequationsarenon-linear). Althoughtheobject ofthis
sectionisnotreallytodevelopnumericalmethodsforsolvingproblemsinbaseball,
wewill,forthemoment,proceedasthoughitwere. Tomaketheproblemsuitable
foracomputer,dividethetimeinterval∆tintoN smallertimeintervalsofduration
!=∆t/N,anddenote,forn= 0 , 1 ,...,N,
tn≡t 0 +n!,
xn=x(tn), pn=p(tn),x 0 =Xin, xN=Xf(1.3)
Anapproximationtoacontinuoustrajectoryx(t)isgivenbythesetofpoints{xn}
connected bystraightlines, asshowninFig. [1.1]. Wecan likewise approximate
derivativesbyfinitedifferences,i.e.
(
dx
dt
)t=tn→
x(tn+1)−x(tn)
!=
xn+1−xn
!
(
dp
dt)t=tn→
p(tn+1)−p(tn)
!=
pn+1−pn
!
(
d^2 x
dt^2)t=tn→
1
!
(
dx
dt)t=tn−
(
dx
dt)t=tn− 1
→
1
!
{
(xn+1−xn)
!−
(xn−xn− 1 )
!}
(1.4)(^1) Wewillallowthesepositionstobedifferent,ingeneral,sinceyoumightmoveyourhandto
anotherpositionwhiletheballisinflight.