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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 258 39 "Vulk\341n kit\366r\351s (
ballisztikus probl\351ma)\n" }{TEXT 259 9 "Probl\351ma:" }{TEXT 260 
51 " Milyen messzire rep\374l a vulk\341nb\363l kil\366k\365d\365 szik
la?" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Param\351terek, kezdeti f
elt\351tel" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "restart;\nwit
h(plots):\ng:=9.81;\n#  S\373r\373s\351gek\nrholev:=1.2;rhoko:=2500;\n
# Szikla \341tm\351r\365je\nd:=0.5;\n# Kil\366v\351si sz\366g\nphi:=Pi
/4;\n# Vulk\341n magass\341ga\nh:=500;\n# Kezdeti sebess\351g\nv0:=400
;v0x:=v0*cos(phi);v0y:=v0*sin(phi);\n# T\351rfogat\nV:=4*(d/2)^3*Pi/3;
\n# T\366meg\nm:=rhoko*V;\n" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}
{PAGEBK }{PARA 11 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT 256 17 "M1. Ferde haj\355t\341s" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 143 "##############################\ninit:=x(0)=0,y(0)=h,
D(x)(0)=v0x,D(y)(0)=v0y:\nsys1:=m*diff(x(t),t$2)=0,m*diff(y(t),t$2)=-m
*g:\nfcns:=\{x(t),y(t)\}:\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 328 "sol1:=d
solve(\{sys1,init\},fcns,numeric,method=rkf45,\n             output=li
stprocedure):\nx1:=subs(sol1,x(t)):y1:=subs(sol1,y(t)):\nmaxt1:=fsolve
(y1,17..100);maxx1:=x1(maxt1);\nsol1:=dsolve(\{sys1,init\},fcns,numeri
c,method=rkf45):\nm1:=odeplot(sol1,[x(t),y(t)],0..maxt1,labels=[x,y],
\n            numpoints=50,color=blue):\ndisplay(m1);" }}{PAGEBK }
{PARA 13 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 
22 "M2. M1+k\366zegellen\341ll\341s" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 593 "##############################\nCd:=0.5:\nalpha:=rho
lev*(d/2)^2*Pi/2*Cd;\ninit:=x(0)=0,y(0)=h,D(x)(0)=v0x,D(y)(0)=v0y:\nsy
s2:=m*diff(x(t),t$2)=-alpha*sqrt(diff(x(t),t)^2+diff(y(t),t)^2)*diff(x
(t),t),m*diff(y(t),t$2)=-m*g-alpha*sqrt(diff(x(t),t)^2+diff(y(t),t)^2)
*diff(y(t),t):\nfcns:=\{x(t),y(t)\}:\nsol2:=dsolve(\{sys2,init\},fcns,
numeric,method=rkf45,output=listprocedure);\nx2:=subs(sol2,x(t)):y2:=s
ubs(sol2,y(t)):\nmaxt2:=fsolve(y2,0..100);maxx2:=x2(maxt2);\nsol2:=dso
lve(\{sys2,init\},fcns,numeric,method=rkf45):\nm2:=odeplot(sol2,[x(t),
y(t)],0..maxt2,labels=[x,y],numpoints=50,color=green):\ndisplay(m2);" 
}}{PAGEBK }{PARA 13 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT 261 59 "M3. M2 kiterjeszt\351se hangn\341l \ngyorsabban kirep
\374l\365 szikl\341kra" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 739 "#
#############################\n# Sebess\351g f\374gg\365 Cd \ninit:=x(
0)=0,y(0)=h,D(x)(0)=v0x,D(y)(0)=v0y:\nT:=273+25;gam:=1.4:R:=286.98: \n
c:=sqrt(gam*R*T);                 #Hangsebess\351g\nCdfun:=x -> 0.75+0
.25*tanh(x-c):\na:=seb -> rholev*(d/2)^2*Pi/2*Cdfun(seb):\nv:=t -> sqr
t(diff(x(t),t)^2+diff(y(t),t)^2):\nsys3:=m*diff(x(t),t$2)=-a(v(t))*v(t
)*diff(x(t),t),\n      m*diff(y(t),t$2)=-m*g-a(v(t))*v(t)*diff(y(t),t)
:\nfcns:=\{x(t),y(t)\}:\nsol3:=dsolve(\{sys3,init\},fcns,numeric,metho
d=rkf45,\n             output=listprocedure);\nx3:=subs(sol3,x(t)):y3:
=subs(sol3,y(t)):\nmaxt3:=fsolve(y3,0..100);maxx3:=x3(maxt3):\nsol3:=d
solve(\{sys3,init\},fcns,numeric,method=rkf45):\nm3:=odeplot(sol3,[x(t
),y(t)],0..maxt3,labels=[x,y],\nnumpoints=50,color=red):\ndisplay(m2,m
3);\n" }}{PAGEBK }{PARA 13 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 30 "Az M1, M2, M3 \366sszehasonl\355t\341sa" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(m1,m2,m3);" }}{PAGEBK }
{PARA 13 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 
5 "Euler" }{TEXT -1 8 " kontra " }{TEXT 263 22 "4-ed rend\373 Runge-Ku
tta" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Sz\341mol\341s" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1123 "##############################\nC
d:=0.5:\na:=seb -> rholev*(d/2)^2*Pi/2*Cd:\nv:=t -> sqrt(diff(x(t),t)^
2+diff(y(t),t)^2):\ninit:=x(0)=0,y(0)=h,D(x)(0)=v0x,D(y)(0)=v0y:\nnums
tep:=8;\ndeu:=array(1..numstep):drk4:=array(1..numstep):\n#\ns:=1:\nfo
r n from 1 to numstep do\nsys4:=m*diff(x(t),t$2)=-a(v(t))*v(t)*diff(x(
t),t),\n      m*diff(y(t),t$2)=-m*g-a(v(t))*v(t)*diff(y(t),t):\nfcns:=
\{x(t),y(t)\}:\nsolrk4:=dsolve(\{sys4,init\},fcns,numeric,method=class
ical[rk4],\n               output=listprocedure,stepsize=s):\nsolEu:=d
solve(\{sys4,init\},fcns,numeric,method=classical[foreuler],\n        \+
      output=listprocedure,stepsize=s):\nxeu:=subs(solEu,x(t)):yeu:=su
bs(solEu,y(t)):\nteu:=fsolve(yeu,maxt2-4..maxt2+4);xeu0:=xeu(teu);\nxr
k4:=subs(solrk4,x(t)):yrk4:=subs(solrk4,y(t)):\ntrk4:=fsolve(yrk4,maxt
2-4..maxt2+4);xrk40:=xrk4(trk4);\ndeu[n]:=[log10(s),xeu0]:drk4[n]:=[lo
g10(s),xrk40]:\nprint(s,xeu0,xrk40,maxx2);\ns:=s/2;\nod:\nm4:=plot([[[
deu[j][1],maxx2] $j=1..numstep],\n     [deu[j] $j=1..numstep],[drk4[j]
 $j=1..numstep]],\n     symbolsize=40,thickness=4,\n     style=[point,
point,line],color=[green,magenta,blue],\n     labels=[\"ln(h)\",\"t
\341vols\341g\"]):" }}{PAGEBK }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "E
redm\351ny" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#A z\366ld neg
ysz\366g az rkf45 eredmenyt mutatja" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "display(m4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{PARA 3 "
" 0 "" {TEXT -1 0 "" }}}{MARK "6 1 1 0 0" 31 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }