Fizika 1i 1ea - fizipedia.bme.hu

Fizika 1i

Fizika Tsz3 h előadaacutes + 1 h gyakorlat

1előadaacutes

Mieacutert eacuteppen fizika

Fizikai kutataacutesok Alkalmazaacutesok

Szaacutemiacutetoacutegeacutepes haacuteloacutezat Internet (www )

Tranzisztor Feacutelvezető elektronika

Nemlin Egyenletek(aacuteramlaacutestan)

Szaacutemiacutetoacutegeacutep

GPS(atomoacutera rel elm) Helymeghataacuterozaacutes

40


CT (NMR)


Gyoacutegyaacuteszat raacutekdiagnosztika

Holograacutefia 3D keacutepalkotaacutes 3D TVbankkaacutertya stb

Anyagtudomaacuteny Uacutej anyagok DNS


Kaacuteosz elmeacutelet Modell


Mert eacuterdekes


Mert izgalmas a joumlvőKvantumszaacutemiacutetoacutegeacutep Nagy szaacutemolaacutesi sebesseacuteg

RSA koacuted feltoumlreacutese stb

Nanofizika Laacutethatatlan repuumllőgeacutepOumlntisztuloacute ruhaOumlngyoacutegyuloacute szaacutemiacutetoacutegeacutep

Robot kutyaYoutube robot dog boston dynamics httpsyoutubeM8YjvHYbZ9w

Mit kell tudni Matematik aacuteboacutel

Emleacutekeztet őI Vektorok

ar

br

ar

Vektorok oumlsszeadaacutesa

bavr +

b-r

cr

ar

ar

ar c

r

cbarrr ++

br

λ

abbarrrr +=+

br

br

br

br

br

Vektorgeometria

Vektor(ok) kivonaacutesa

ar

br ba =minus

vr

)b(-abarrrr +=minus

b-r

ar

br

)b(avr minus+

ar

br

barr minus a

r

br

abrr

minus

barr minus

Konponensek eacutes egyseacutegvektorok

x

y

ir

jr

rr

xr

yr

2

y

2

x rrrr +==r

Θ

x

y

r

rtan =Θ

Descartes koordinaacutetaacutek xr yramp

Polaacuter koordinaacutetaacutek r amp Θ

Θsdot= cosrrx

Θsdot= sinrry

jrirr yx

rrr +=

1ji ==rr

)rr(r yx=r

)r(r Θ=r

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd

cj)b(ai)b(aba yyxxrrrrr =minus+minus=minus

xc yc

j)cb(ai)cb(acba yyyxxx

rrrrr +++++++=+++

Skalaacuterszorzat ϕsdot=sdot cosbabarrrr

ar

br

ϕ

Def

1jjii =sdot=sdotrrrr

eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr

yyxx bababa +=sdotrr

ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+

Szuperpoziacutecioacute

Vektoriaacutelis szorzat

γsdot=times sinbabarrrr

kjirrr

=times ikjrrr

=timeseacutes

jikrrr

=timeseacutes de 0kkjjii =times=times=timesrrrrrr

Jobbkeacutez-szabaacutely

Peacutelda forgatoacutenyomateacutek

FrMrrr

times=

Vektoriaacutelis szorzat kiszaacutemiacutetaacutesa

bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr

( ) ( ) ( ) kbabajbabaibababa xyyxzxxzyzzyrrrrr

sdotminus+sdotminus+sdotminus=times


II Trigonometria

βα+βα=β+α sincoscossin)sin(

αα=α cossin2)2sin(

βαminusβα=β+α sinsincoscos)cos(+

αminusα=α 22 sincos)2cos(

βαminusβ+α=β+α

tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α

Joacute tudni helliphellip

MATEMATIKA BEVEZETŐ1 Differenciaacutelszaacutemiacutetaacutes

Mieacutert hasznos a differenciaacutelszaacutemiacutetaacutes

Aacutetlagsebesseacuteg Pillanatnyi sebesseacuteg

Sebesseacuteg = uacutetidő

Peacutelda

Uacutet-idő meacutereacutese diszkreacutet pontokban

s

t

Mekkora az aacutetlagsebesseacuteg a 3 eacutes a 4 s koumlzoumltt

t=1s

s=7m

Mekkora az aacutetlagsebesseacuteg a 3 eacutes az 5 s koumlzoumltt

s=16m

t=2s

Geometriai jelenteacutesα

A sebesseacuteg a viacutezszintessel bezaacutert szoumlg tangenseacuteta meredekseacuteget mutatja meg

Az uacutet eacutes idő koumlzoumltt ismert a fuumlggveacutenykapcsolat

peacutelda

x(t2)

x(t1)

t1 t2

A sebesseacuteg meacuteg mindig aacutetlagsebesseacuteg (a szelő meredekseacutege)a kifejezeacutes a differenciahaacutenyados

Ha t2 nagyon megkoumlzeliacuteti t1-et (t2 = t1 + Δt eacutes Δt rarr 0 )

a differenciahaacutenyados hataacutereacuterteacuteke a differenciaacutelhaacutenyados a derivaacutelt

amely megmutatjaa pillanatnyi sebesseacuteget (az eacuterintőmeredekseacutegeacutet) t1-ben

x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0

A differenciaacutelaacutes (derivaacutelaacutes) alkalmazaacutesa

Hataacuterozzuk meg az y=x2 fuumlggveacuteny grafikonjaacutenak meredekseacutegeacutet x=3 pontban

f(x)=x2

Keacutepezzuumlk a fuumlggveacuteny derivaacuteltfuumlggveacutenyeacutet vagy derivaacuteltjaacutet

f(x)=x2 frsquo(x)=2x

Helyettesiacutetsuumlk be az eacuterinteacutesi pont x koordinaacutetaacutejaacutet

frsquo(x=3)=2bull3=6

Az f(x)=x2 fuumlggveacuteny grafikonjaacutenak meredekseacutege az x=3 helyen 6

tgα=6

Derivaacutelaacutesi szabaacutelyok

( ) xx ee =prime

Oumlsszetett fuumlggveacuteny

Oumlsszetett fuumlggveacuteny derivaacutelaacutesa

(f(g(x)))rsquo=frsquo(g(x))bullgrsquo(x)

Peacutelda

(sin(3x2))rsquo=cos(3x2)bull6x

f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)

Maacutesodik derivaacuteltf(x)=5x3

frsquo(x)=53x2=15x2

frsquorsquo(x)=152x=30x

Peacutelda

Alkalmazaacutes (pl)

x(t)=5t3

F=ma F kiszaacutemiacutethatoacute

xX(t)

1 D mozgaacutes

0

Szeacutelsőeacuterteacutek meghataacuterozaacutesa

Peacutelda f(x)=2x3-21x2+60x+3

Hol van az f(x) fv szeacutelsőeacuterteacuteke

f(x) fuumlggveacuteny szeacutelsőeacuterteacuteke ott talaacutelhatoacute ahol frsquo(x)=0

frsquo(x)=6x2-42x+60

6x2-42x+60=0 x1=5 x2=2

Minimum vagy maximum

frsquo(x)=6x2-42x+60 frsquorsquo(x)=12x-42

frdquo(5)=18 frdquo(2)=-18Minimum Maximum

f(x)=2x3-21x2+60x+3

Szokaacutesos jeloumlleacutes azidő szerinti derivaacuteltra

xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2

)a(f)ax)(a(f)a(ff(x) 2 +minus

primeprime+minusprime+=

2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=

2 Integraacutelszaacutemiacutetaacutes

CEacuteL

Goumlrbe alatti teruumllet meghataacuterozaacutesa

x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt

Alsoacute-felső koumlzeliacutető oumlsszeg

Mineacutel finomabb a beosztaacutes az alsoacute eacutes a felső koumlzeliacutető oumlsszeg eacuterteacuteke annaacutel inkaacutebb megkoumlzeliacuteti egymaacutest

s(f) S(f)s(f) lt S(f)

IntegraacutelHa a beosztaacutes minden hataacuteron tuacutel finomodik akkor

s(f)=S(f)

=s(f)=S(f)

ba

Az integraacutel kiszaacutemiacutetaacutesaNewton-Leibniz teacutetel

Ha leacutetezik F(x) uacutegy hogy

F(x) az f(x)fuumlggveacuteny primitiacutev fuumlggveacutenye

Frsquo(x)=f(x)

A primitiacutev fuumlggveacuteny segiacutetseacutegeacutevel a hataacuterozott integraacutel kiszaacutemiacutethatoacute

int= dx)x(f)x(F

(Hataacuterozatlan integraacutel)

Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233

Integraacutelaacutesi szabaacutelyok ndash Primitiacutev fuumlggveacuteny

Primitiacutev fuumlggveacuteny meghataacuterozaacutesa

intsin5(x)cos(x)dxPeacuteldaintsin(3x+5)dx

intsin(x5)x4dx

H F

Parciaacutelis integraacutelaacutes

Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika

A kinematika alapjai A toumlmegpont helyeacutenek megadaacutesa az id ő fuumlggveacutenyeacuteben

Toumlmegpont helyzete )(trr

Elmozdulaacutes )()( 12 trtrrrrr minus=∆

Megtett uacutet rΔs

i

isum=r

Kinematika rarr toumlmegpont helyzete rarr pl tenisz challange

Apophis kisbolygoacute

Legegyszer űbb modell 1 D - mozgaacutes

0 x(t) x

Definiacutecioacutek

Aacutetlagsebesseacuteg

oumlssz

oumlsszaacutetl t

sv =

Pillanatnyi sebesseacutegdt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0

xsd [m] pontosabban keacutesőbb t [s]

Meacuterteacutekegyseacuteg ms

Elmozdulaacutes sumint ==minusi

ii

t

t

12 Δtvv(t)dt)x(t)x(t

2

1

Poziacutecioacute += 0x)t(x

v = 72 kmh = 20 ms

elmozdulaacutes

Legegyszer űbb mozgaacutes egyenesvonaluacute egyenletes mozgaacutes

v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS

Egy egyszer ű feladat

A B

s

Aacutetlagsebesseacuteg (laacutettuk)

Average velocity Average speed

12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =

Egy paradoxon Achilleus eacutes a tekn ősbeacuteka

Achilleus nem eacuteri utol a teknősbeacutekaacutet mert mire odaeacuter ahol a teknősbeacutekavolt eredetileg addig a teknős előbbre jutott eacutes iacutegy tovaacutebb hellip

x(t)

tt

Megoldaacutes Achilleus nem eacuteri utol a teknősbeacutekaacutet amiacuteg nem eacuteri utol a teknősbeacutekaacutet

Hol a hiba

Hosszuacutesaacuteg eacutes id őegys eacutegA maacutesodperc A maacutesodpercet eredetileg az aacutetlagosNap-nap segiacutetseacutegeacutevel lehetett meghataacuterozni annak 186400-ad reacutesze

Atomoacutera nagy pontossaacuteg1ms eacutev vagy jobb

A meacuteter 1 meacuteter a Foumlld keruumlleteacutenek (a Paacuterizson aacutetmenő deacutelkoumlrnek)140000000-od reacutesze rarr ősmeacuteter

A maacutesodperc az alapaacutellapotuacuteceacutezium-133 atom keacutet hiperfinomenergiaszintje koumlzoumltti aacutetmenetnek megfelelő sugaacuterzaacutes 9192631770 perioacutedusaacutenak időtartama

1 meacuteter Kr86 narancssaacuterga spektrumvonalaacutenak 165076373 - szorosa

Pontosabb definiacutecioacute jegyzet

v ne const rArr v = v(t)

Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva

aacutetl==Def aacutetlagos gyorsulaacutes

2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t

aaacutetl = tgαv(t)

Def pillanatnyi gyorsulaacutes

dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum

Mozgaacutes aacutelland oacute gyorsulaacutessal

a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21

Elmozdulaacutes eacutes poziacutecioacutev

t

2

21

attvso

+sdot=Elmozdulaacutes

( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===

Feladatmegoldaacuteshoz hasznos formulaacutek

Szabadeseacutes

g = 981 ms2 asymp 10 ms2

Mintapeacuteldaacutek hellip

2 ea 2D eacutes 3D mozgaacutes + koordinaacutetarendszerek

2D eacutes 3D mozgaacutes

Aacutetlagsebesseacuteg (vektor) t

rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur

eacuterintő iraacutenyuacute egyseacutegvektor

dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==

ϕϕϕ ϕϕϕ ererererervarr

ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==

( ) ( ) ϕϕϕϕ errerrar

rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd

Polaacuterkoordinaacutetaacutek r ϕ

ϕ

(siacutekbeli)

00

)()( rdvtrt

rrr += int ττA toumlmegpont helyzete

A toumlmegpont aacuteltal megtett uacutet ττ dvst

int=0

)(

A toumlmegpont gyorsulaacutesa ( )ttt

uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar

v csoumlkken 0 at lang

0 at rangv noumlvekszik

Egy speciaacutelis eset consta =r

2

21

tatvr(t)roo

sdot+sdot+= rrrrtav(t)v

osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=

Viacutezszintes mozgaacutes

Fuumlggőleges mozgaacutes

Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=

fuumlggőleges mozgaacutes

2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ

viacutezszintes elmozdulaacutes

A nagy Bertha eacutes tsi

vo = 1700 ms

θ = 55deg

s =

h =

Egy joacute strateacutegia a h oacutegolyoacutecsataacutehoz avagy hogyan lehet a laacutenyokat (fiuacutekat) hoacutegolyoacuteval eltalaacutelni

( )απα minus= sinsin

Θ)(g

vs o 2sin

2

=

Tudjuk (alg)

( ) ( )βπ 2sin2sin)2sin( =minus= ΘΘ

Θminus=2πβ

Egy uacutejabb p eacutelda avagy mieacutert eacutepiacutetetteacutek a vaacuterakat hegytetőre

1 megoldaacutes ax ay

2 megoldaacutes y(x)

Ciklois goumlrbe

x(t) =

y(t) =

Koordinaacuteta rendszerek

Descartes-feacutele koordinaacuteta rendszer

)( zyxkzjyixr =++=rrrr

Henger koordinaacuteta rendszer

)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp

rampr ++== ϕρ ϕρρ

kzeeavrrampamp

rrrampr

ampampr ++=== ϕρ ()()

Siacutekbeli polaacuter

Goumlmbi koordinaacutetarendszer

x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer

θϕ θϕθ ererervrr

rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr

Segiacutetseacuteg kjΘiΘer

rrrr)cos(sinsincossin Θ++= ϕϕ

jierrr ϕϕϕ cossin +minus=

kΘjΘiΘeΘ

rrrrsinsincoscoscos minus+= ϕϕ

Kinematika rarr dinamika

Kepler toumlrveacutenyek

Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)



Szaacutemiacutetoacutegeacutepes haacuteloacutezat Internet (www )

Tranzisztor Feacutelvezető elektronika

Nemlin Egyenletek(aacuteramlaacutestan)

Szaacutemiacutetoacutegeacutep

GPS(atomoacutera rel elm) Helymeghataacuterozaacutes

40


CT (NMR)








Mert eacuterdekes








ar

br

ar


bavr +

b-r

cr

ar

ar

ar c

r

cbarrr ++

br

λ

abbarrrr +=+

br

br

br

br

br

Vektorgeometria


ar

br ba =minus

vr

)b(-abarrrr +=minus

b-r

ar

br

)b(avr minus+

ar

br

barr minus a

r

br

abrr

minus

barr minus


x

y

ir

jr

rr

xr

yr

2

y

2

x rrrr +==r

Θ

x

y

r

rtan =Θ



Θsdot= cosrrx

Θsdot= sinrry

jrirr yx

rrr +=

1ji ==rr

)rr(r yx=r

)r(r Θ=r

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd


xc yc


rrrrr +++++++=+++


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


CT (NMR)








Mert eacuterdekes








ar

br

ar


bavr +

b-r

cr

ar

ar

ar c

r

cbarrr ++

br

λ

abbarrrr +=+

br

br

br

br

br

Vektorgeometria


ar

br ba =minus

vr

)b(-abarrrr +=minus

b-r

ar

br

)b(avr minus+

ar

br

barr minus a

r

br

abrr

minus

barr minus


x

y

ir

jr

rr

xr

yr

2

y

2

x rrrr +==r

Θ

x

y

r

rtan =Θ



Θsdot= cosrrx

Θsdot= sinrry

jrirr yx

rrr +=

1ji ==rr

)rr(r yx=r

)r(r Θ=r

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd


xc yc


rrrrr +++++++=+++


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)




Mert eacuterdekes








ar

br

ar


bavr +

b-r

cr

ar

ar

ar c

r

cbarrr ++

br

λ

abbarrrr +=+

br

br

br

br

br

Vektorgeometria


ar

br ba =minus

vr

)b(-abarrrr +=minus

b-r

ar

br

)b(avr minus+

ar

br

barr minus a

r

br

abrr

minus

barr minus


x

y

ir

jr

rr

xr

yr

2

y

2

x rrrr +==r

Θ

x

y

r

rtan =Θ



Θsdot= cosrrx

Θsdot= sinrry

jrirr yx

rrr +=

1ji ==rr

)rr(r yx=r

)r(r Θ=r

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd


xc yc


rrrrr +++++++=+++


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)








ar

br

ar


bavr +

b-r

cr

ar

ar

ar c

r

cbarrr ++

br

λ

abbarrrr +=+

br

br

br

br

br

Vektorgeometria


ar

br ba =minus

vr

)b(-abarrrr +=minus

b-r

ar

br

)b(avr minus+

ar

br

barr minus a

r

br

abrr

minus

barr minus


x

y

ir

jr

rr

xr

yr

2

y

2

x rrrr +==r

Θ

x

y

r

rtan =Θ



Θsdot= cosrrx

Θsdot= sinrry

jrirr yx

rrr +=

1ji ==rr

)rr(r yx=r

)r(r Θ=r

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd


xc yc


rrrrr +++++++=+++


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


ar

br ba =minus

vr

)b(-abarrrr +=minus

b-r

ar

br

)b(avr minus+

ar

br

barr minus a

r

br

abrr

minus

barr minus


x

y

ir

jr

rr

xr

yr

2

y

2

x rrrr +==r

Θ

x

y

r

rtan =Θ



Θsdot= cosrrx

Θsdot= sinrry

jrirr yx

rrr +=

1ji ==rr

)rr(r yx=r

)r(r Θ=r

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd


xc yc


rrrrr +++++++=+++


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


x

y

ir

jr

rr

xr

yr

2

y

2

x rrrr +==r

Θ

x

y

r

rtan =Θ



Θsdot= cosrrx

Θsdot= sinrry

jrirr yx

rrr +=

1ji ==rr

)rr(r yx=r

)r(r Θ=r

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd


xc yc


rrrrr +++++++=+++


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Elemi vektoralgebra

jaiaa yx

rrr += jbibb yx

rrr+= ba =+

rr

dj)b(ai)b(aba yyxx

rrrrr =+++=+

xd yd


xc yc


rrrrr +++++++=+++


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


ar

br

ϕ

Def


eacutes 0ji =sdotrr

jaiaa yx

rrr += jbibb yx

rrr+= ba =sdot

rr


ba

bacos rr

rr

sdotsdot=ϕ

Peacutelda munka

sFWrr

sdot=

zzba+




kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)



kjirrr

=times ikjrrr

=timeseacutes

jikrrr




FrMrrr

times=


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


bbb

aaa

kji

ba

zyx

zyx ==times

rrr

rr




II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

II Trigonometria






tgtg1

tgtg)(tg

+

1cossin 22 =α+α

HF )2(tg =α

)3cos( =α

2

cos =

α






Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)





Peacutelda


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


s

t


t=1s

s=7m


s=16m

t=2s




peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


peacutelda

x(t2)

x(t1)

t1 t2





x

t

xX(t)

1 D mozgaacutes

0

dt

dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)



f(x)=x2






tgα=6


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


( ) xx ee =prime




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)




Peacutelda


f(g(x))

f=sin(x) g=3x2

f(g(x))=sin(3x2)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


frsquo(x)=53x2=15x2


Peacutelda


x(t)=5t3


xX(t)

1 D mozgaacutes

0






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)






6x2-42x+60=0 x1=5 x2=2




f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

f(x)=2x3-21x2+60x+3


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


xdt

dxv amp==

xdt

xdv

dt

dva

2

2ampampamp ====

3D-benr

dt

rdv ampr

rr ==

rdt

rdv

dt

vda

2

2ampampr

r

amprr

r====

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Taylor-sor

)ax(2



2

x1)xcos(

2+minus=

3

xx)xsin(

3+minus=

2

xx1e

2x +++=


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


CEacuteL


x1 x2 x1 x2

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Peacutelda

F F

s t

W=Fs I=Ftv

ts=vt





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)





s(f)=S(f)

=s(f)=S(f)

ba




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)




Frsquo(x)=f(x)


int= dx)x(f)x(F


Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Peacuteldaf(x)=x2

f(x)=x2

F(x)=x33

Ellenőrzeacutes

(F(x))rsquo=f(x)

=(2)33-(1)33=73=233




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)




intsin(x5)x4dx

H F


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


Peacutelda

Peacutelda2 H F

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Peacuteldaacutek

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Kinematika





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)





i

isum=r




0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


0 x(t) x



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)



oumlssz

oumlsszaacutetl t

sv =


dx

t

txttxtv

t=

∆minus∆+=

rarr∆

)()()( lim

0




ii

t

t


2

1


v = 72 kmh = 20 ms

elmozdulaacutes


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


v = const

tx-x(t)

v o=

tvxx(t) o sdot+=

ts

v = tvs sdot=

x

t

xo

x(t)

v(t)

t

v

t

tvs sdot=

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

GPS


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


A B

s



12

12 )()(tt

txtx

idő

elmozdulaacutes

minusminus=

oumlssz

oumlsszaacutetl t

sv =

oumlssz

oumlsszaacutetl t

sv =



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)



x(t)

tt


Hol a hiba








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)








Gyorsulaacutes

12

12

-tt

))-v(tv(t

Δt

Δva


2s

m

t1 t2

∆v

∆t

α

v(t2)

v(t1)

t



dt

dv

Δt

tvttvta

t=minus∆+=

rarr∆

)()()( lim

0

0

i

ii vta)t(v +∆=sum

0

i

ii xtv)t(x +∆=sum


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


a = const

t

v(t)-va o=

tavv(t)o

sdot+=

v(t)

t

a gt 0 v(t)

t

vo

a lt 0

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

int intint ++prime

=+=

primett

xtvddaxdvtx0

00

0

0

0

)()()( ττττττ

t

vo

∆v=at

tvo sdot

2at21


t

2

21

attvso


( ) 2

21

attvxtxoo

+sdot+=

Laacutettuk

Poziacutecioacute

v1

t

v2

a = const

a

vvt

vvs

22

2

1

2

221 minus=+=

v

t

a = constv0 = 0

a

vvtats

2221 2

2 ===

t

vv

t

a = const v1 = v0

v2 = 0

t

v

v0

a

vtvtas

2221 2

002 ===


Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Szabadeseacutes






rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)





rv

aacutetl ∆∆=r

r

idő

elmozdulaacutesv

aacutetl=

r


oumlssz

oumlsszaacutetl t

sv =


rdtv

rr =)(

Mivel t

udrrdrr =

tur


dt

udru

dt

drtv t

t

rrr +=)(

tvr

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

x

y

rr

rer

ϕer

rerrrr =

vr

rrererr amprr

ampampr +=

ϕϕeer

rampamp

r =r

eerampamp

r ϕϕ minus=

)(ter

r

)(teϕ

r

)( dtter

+r

redr

)( dtte +ϕ

r

ϕedr

rvr

tvr

ϕϕererrvr

ramp

rampamp

rr +==


ampramp

rampamp

rampampamp

ramp

rampampamp

rr ++++==


rampampampamp

rampampamp

r ++minus= 22

rvr

tvr

ϕd


ϕ

(siacutekbeli)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

00

)()( rdvtrt



int=0

)(


uvuvuvdt

d

dt

vda amp

rramp

rr

r +===

R

v

dt

du

R

vdt

u

dut

t

t =rArr= r

nR

vuva t

rramp

r2

+=

(egyszerűen)

ta

cpa

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

cpar (t)v

r

tar

R

ar

tcp aaarrr +=

cpar

tar

ahol t

vlima

0tt ∆

∆=rarr∆

22

tcpaaa +=

v ne const

cpar

(t)vr

tar

R

ar




2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


2

21

tatvr(t)roo


osdot+= rrr

2

21

tatvxx(t)xxoo

+sdot+=

tav(t)vxoxxsdot+=

2

21

tatvyy(t)yyoo

+sdot+=

tav(t)v yoyy sdot+=



Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Hajiacutetaacutes

ovr

oxvr

oyvr

s

Θvvooxcos=

Θvv ooy sin=

g

vt oy2=

Θ)(g

vs o 2sin

2

=


2

21

tatvyy(t)yoyo

++=

0== fo yy

2

21

0 tatvyoy

+=

g

Θv

a

vt o

y

oy sin22=minus=

Θtvtvx(t)ooxcos==

g

ΘvΘvΘtvtv o

ooox

sin2coscoss sdot===

Θ



vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


vo = 1700 ms

θ = 55deg

s =

h =



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)



Θ)(g

vs o 2sin

2

=

Tudjuk (alg)


Θminus=2πβ




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)




Ciklois goumlrbe

x(t) =

y(t) =





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)





)( zrr ϕ=r

x y

z

rr

ϕz

ρ

kzjyixvrramp

ramp

ramp

rampr ++==

kzjyixavrrampamp

rampamp

rampamp

rampr

ampampr ++===

kzerrrr += ρρ

kzeevrramp

ramp

ramp


kzeeavrrampamp

rrrampr




x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)


x y

rr

ϕ

θ

z

)( θϕrr =r

rerrrr =

rer

ϕer

θer


rampramp

ramp

rampr ++== sin

FHavr === rampr

ampampr




kΘjΘiΘeΘ




Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)



Nap

1

Nap

2

Nap

2a3

3

2

consta

T =

A1

A2

A1 = A2

(Tycho de Brahe)

Fizika 1i 1ea - fizipedia.bme.hu

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