04 Phys I Stiliaris - eclass.uoa.gr · ΔΙΑΝΥΣΜΑΕΠΙΤΑΧΥΝΣΗΣ v dt duˆ dt dv...

&

NN ,201,2016620120177

Stathis STILIARIS, UoA 2016-2017 1

NN ,201,2016620120177

ALONSOALONSOFINNFINN

GIANCOLIGIANCOLI HALLIDAYHALLIDAYRESNICKRESNICKWALKERWALKERYOUNGYOUNG

FREEDMANFREEDMAN

5.1,5.2,5.3,5.1,5.2,5.3,5.45.4

2.1,2.2,2.3,2.1,2.2,2.3,2.4,2.5,2.6,2.4,2.5,2.6,2.7,2.8,2.92.7,2.8,2.9

2.1,2.2,2.3,2.4,2.5,2.1,2.2,2.3,2.4,2.5,2.6,2.7,2.8,2.92.6,2.7,2.8,2.9

2.1,2.2,2.3,2.4,2.1,2.2,2.3,2.4,2.5,2.62.5,2.6

5.5,5.6,5.8,5.5,5.6,5.8,5.115.11

3.5,3.63.5,3.6 4.1,4.2,4.3,4.44.1,4.2,4.3,4.4 3.1,3.23.1,3.2

5.9,5.105.9,5.10 5.2,5.55.2,5.5 4.74.7 3.43.4

5.75.7 3.7,3.83.7,3.8 4.5,4.64.5,4.6 3.33.3


::

:: ((x)x) ((t)t)

tsvav

=

O x

A B

x

::ToTo vvavav tt 00

txlimvlimv

0tav0t

== dt

dxv =

+===t

t0

t

t0

t

t

x

x 0000

dtvxxdtvxxdtvdx


::

:: ((v)v) ((t)t)

tvaav

=

::ToTo aaavav tt 00

tvlimalima

0tav0t

== dt

dva =

+===t

t0

t

t0

t

t

v

v 0000

dtavvdtavvdtadv

O x

A B

x

v


::

::

2

2

dtxda

dtdx

dtda

dtdva =

==

dxadvvdtdxdtadvvdtadv =

==

=

=x

x

20

2x

x

v

v00

0dxa

2vvdxavdv


a=a=

)tt(avvdtavv 00t

t0

0

+=+=

[ ] ++=++=+=t

t0

t

t00

t

t000

t

t0

0000

dt)tt(adtvxdt)tt(avxxdtvxx

20000 )tt(a2

1)tt(vxx ++=


200

0

at21tvxx

tavvconsta

++=

+==

v=v(t)v=v(t) x=x(t)x=x(t)

a=cta=ct


200

0

at21tvxx

tavvttanconsa

++=

+==

20

20

20000

2

200

00

0

vva)xx(2

)vv()vv(v2a)xx(2a

)vv(a21

avvvxx

avvt

=

+=

+

+=

=

a)xx(2vv 020

2 += ::

)xx(a2vv)xx(a2vvdxavdvadxvdv 0

20

20

20

2x

x

v

v 00

+==

==


200

0

at21tvxx

tavvttanconsa

++=

+==

t2vvxx

t)vv(21tvxx

ttvv

21tvxx

tvva

00

000

2000

0

+=

++=

++=

=

t2vvxx 00

+=

,, v(t)v(t)::

t2vvxxvdtdxvdtdx 00

t

0

x

x0

+===


(( g=g=9.8ms9.8ms22).).

200

0

gt21tvyy

tgvvga

+=

==

gr

0y0 =

0vr

( )g2v

g2vg

21

gvvHg/vtyHy

gvt0v

20

200

00max

0

=====

==

g2vH20=


1rr

2rr

kzjyixrrrr zyx ++=++=rrrr

( ) ( )

kzjyixr

k)zz(j)yy(i)xx(r

kzjyixkzjyixrrr

121212

11122212

++=

++=

++++==

r

r

rrr

ktzj

tyi

tx

tkzjyix

trvav

+

+

=

++=

=r

r

kdtdzj

dtdyi

dtdx)kzjyix(

dtd

dtrdv

trlimv

0t++=++==

=

rr

rr


kdtdzj

dtdyi

dtdx)kzjyix(

dtd

dtrdv ++=++==r

r

2z

2y

2x

zyx

vvvv

kvjvivv

++=

++=r

tv

tvva 12av

=

=rrr

r

kdtdvj

dtdv

idtdv)kvjviv(

dtd

dtvda

tvlima zyxzyx0t ++=++==

=

rr

rr

2z

2y

2x

zyx

aaaa

kajaiaa

++=

++=r


ss ..

=

=

=

tslim

srlimv

ts

srlim

trlimv

0t0s

0t0tr

r

rrr

ss rr,, ,, (()) ..

Tvr

vdtds

tslim

udrrd

srlim

0t

T0s

==

==

rr

vudtdsuv TT ==

r


:: Tar

C t v a. a .

: .

Tar

x

y

Nar a

rCC

vr

:: Nar


vdtud

dtdvuv)u(

dtda

dtvda TTT +===

rr

r

Tu

Nu

'uT

rr 'uT

Tu Tud

x

y

d

, uT :

vu

ddvu

dsdvu

dtds

dsdu

dtdu

dtudduud NNNNNTNT ======

duud1uduud NNNTT ===

uN, duT uT, .:


vdtud

dtdvuv)u(

dtda

dtvda TTT +===

rr

r

NT

2

NTNT aavu

dtdvuv

vu

dtdvua

rrr+=+=+=

aaTT:: .. (( vv)),, aaTT ..

aaNN:: .. ,, ..


RR

vr

:: RR,, ..

:

dtdR)R(

dtd

dtdsv ===r

dtd

=

dd//dtdt

,, ..

::rad/s=srad/s=s11


RR

vr

rr . :

vrr

RR

rr

rvrrr

=

dtdRR)asinr(sinrv ====

rrr


RR

vr

:

==dd//dt=constdt=const

::

== 22// =2=2ff

f:f:

f=1/Tf=1/T

:

====t

00 tdtddtddt

d

0

t0 +=


vr

ar Ta

r

Nar

NT

2

NT aavu

dtdvua rrr +=+=

:

uT uN , .

:

RR v v=v= RR dv/dt dv/dt=Rddv/dt=Rd/dt/dt

+=+= N22

T

2

NT uRRu

dtdR

vu

dtdvuar N

2T uRudt

dRa rrr +=


vr

ar Ta

r

Nar

aT aN, .

aT v

aN

N2

T uRudtdRa +=r

,, dd//dtdt 22RR,, ..

N2

N2

T uRuRudtdRa rrrr =+=


vr

ar

xar

yar

xvr

yvr

xx

yy

.

j)cosv(dtdi)sinv(

dtdj

dtdv

idtdv

dtvda yx

rrrrrr +=+==

v :

jsinvicosvjdtdsinvi

dtdcosva

rrrrr

==

RsincosRajsinRicosRa 222222 =+== rrrr


() gg.

v0=vi 0=i

jsinvicosvjvivv 00000y0x0 +=+=r

rr vv ,, aa ((aa==gg).).

!!

..


( )tcosvxxtvxx 000x00 +==

( ) 20002y00 gt21tsinvyygt

21tvyy +==

gtsinvv 00y =

a)yy(2vv

0

2y0

2y = ( ) ( )02002y yyg2sinvv =


, .

( )tcosvxx 000 +=

( ) 2000 gt21tsinvyy +=

00

0

cosvxxt

=

( )2

00

0

00

0000 cosv

xxg21

cosvxxsinvyy

+=

( )( )200

2

0 cosv2gxxtany

=

0yx 00 ==


RR ..

t)cosv(xxR 000 ==

,, yyyy00=0=0::

200 gt2

1t)sinv(0 =

::

= 0020 cossingv2R 0

20 2singvR =

00=45=45Stathis STILIARIS, UoA 2016-2017 26

04 Phys I Stiliaris - eclass.uoa.gr · ΔΙΑΝΥΣΜΑΕΠΙΤΑΧΥΝΣΗΣ v dt duˆ dt dv...

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