FIZYKA I ASTRONOMIA

ARKUSZ PRÓBNEJ MATURY Z OPERONEM

FIZYKA I ASTRONOMIA

POZIOM ROZSZERZONY

Czas pracy: 150 minut ,QVWUXNFMDGOD]GDMÈFHJR

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Fizyka i astronomia Poziom rozszerzony

1

Listopad 2013

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MHVW]JRGQ\]SRGDQ\PVFKHPDWHPRUD]LQQHSRSUDZQHRGSRZLHG]LZQLPQLHSU]HZLG]LDQH

Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów 1. 1.1

SNWļ]DSLVDQLH,,]DVDG\G\QDPLNL1HZWRQDEH]XZ]JOÚGQLHQLDNUÈĝND

a Q Q

m m

m m

m m g

= −

+

= −

+

1 2 ⋅

1 2

1 2

1 2

SNWļREOLF]HQLHSU]\VSLHV]HQLDFLÚĝDUNöZ

a = ⋅ =

40

50 10 m 8 s  m

2 s2

10

1.2

SRSNWļ]DSLVDQLH,,]DVDG\G\QDPLNLGODNDĝGHJRFLDïD Q₁−N₁=m a₁⋅

N₂−Q₂=m a₂⋅ N R₁ −N R₂ = ⋅ FI

SNWļZ\HOLPLQRZDQLH]W\FKWU]HFKUöZQDñQDSLÚÊQLFLN1 i N₂RUD]XZ]JOÚGQLHQLH

]DOHĝQRĂFLSRPLÚG]\SU]\VSLHV]HQLHPNÈWRZ\PDOLQLRZ\P m g m a m a m g R m R a

1 1 2 2 3 R

1 2

− − − 2

( )

⋅ = ⋅

SNWļZ\]QDF]HQLHZ]RUXQDSU]\VSLHV]HQLHOLQLRZHFLÚĝDUNöZ m1 m g2 m3 m1 m2 a

1

− 2

( )

= + +









⋅

a m m g

m m m

=

(

−

)

+ +

1 2

3 1 2

1 2

SNWļREOLF]HQLHSU]\VSLHV]HQLDOLQLRZHJR a =

(

−

)

+ + ⋅ =

45 5

30 45 5 10 m 5  s

m

2 s2

m

₂

m

₁

m

₃

R

N

₁

N

₁

Q

₁

Q

₂

N

₂

N

₂

Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów 1.3

SNWļZ\]QDF]HQLHZ]RUXQDSUÚGNRĂÊNÈWRZÈZUXFKXSU]\VSLHV]RQ\P ε

 %ω

%t a R Xt X = ⋅a t

R

SNWļREOLF]HQLHSUÚGNRĂFLNÈWRZHMSRF]DVLHV

X = ⋅

= 5 10

0 05 1000

  , m

s s

m

rad s

2

2. 2.1

SNWļ]DVWRVRZDQLH]DVDG\]DFKRZDQLDHQHUJLL Ep1−Ep2=Ek

mg H h mv

(

−

)

= ⁰

2

2

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v0= 2g H

(

−h

)

SNWļREOLF]HQLHSUÚGNRĂFLv0

v0= 2 10⋅  m⋅500 =100 s

m m

2 s

12

2.2

3U]\NïDG\SRSUDZQ\FKRGSRZLHG]L 6SRVöE,

SRSNWļ]DSLVDQLHUöZQDñUXFKXZNLHUXQNXSR]LRP\PLZNLHUXQNXSLRQRZ\P x=v0x⋅t

y v t gt

= ⁰y⋅ −

2

2

SNWļZ\]QDF]HQLH]DVLÚJXORWXSRSU]H]Z\HOLPLQRZDQLHF]DVX]UöZQDñLSU]\UöZ- nanie y PLHMVFHXSDGNXFLDïDEÚG]LHPLDïRZVSöïU]ÚGQHx  z, y  0)

0

0 2

0 0

2

= ⋅









− ⋅











v z 

v

g z

y v

x x

z v v

g

v g

x y

= ⋅ ⋅

= ⋅ ⋅ ⋅

2 _{0 0} 2 ₀² sinB cosB

SNWļREOLF]HQLH]DVLÚJXORWX

z =

⋅ ⋅ ⋅

=

2 10000 2

2 2 2 10

1000  

  m s

m s

m

2 2

2

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w w w. o p e r o n . p l 3

Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów 6SRVöE,,

SNWļ]DSLVDQLHUöZQDñUXFKXZNLHUXQNXSR]LRP\PLZNLHUXQNXSLRQRZ\P x=v0x⋅t

y v t gt

= 0y⋅ −

2

2

SNWļZ\]QDF]HQLH]DVLÚJXORWXSRSU]H]Z\HOLPLQRZDQLHF]DVX]UöZQDñL]DSLVDQLH

UöZQDQLDWRUX

y v x

v

g x

y v

x x

= ⋅









− ⋅













0

0 0

2

2

SNWļSU]\UöZQDQLH y PLHMVFHXSDGNXFLDïDEÚG]LHPLDïRZVSöïU]ÚGQHx  z, y LZ\]QDF]HQLHZ]RUXQD]DVLÚJ

0

0 2

0 0

2

= ⋅









− ⋅











v z 

v

g z

y v

x x

z v v

g

v g

x y

= ⋅ ⋅

= ⋅ ⋅ ⋅

2 _{0 0} 2 ₀² sinB cosB

SNWļREOLF]HQLH]DVLÚJXORWX

z =

⋅ ⋅ ⋅

=

2 10000 2

2 2 2 10

1000  m

s m s

m

2 2

2

2.3

3U]\NïDG\SRSUDZQ\FKRGSRZLHG]L 6SRVöE,

SRSNWļZ\]QDF]HQLHVNïDGRZ\FKSR]LRPHMLSLRQRZHMSUÚGNRĂFLSRF]ÈWNRZ\FK v0x=v0⋅ cos B

v0_y=v0⋅ sinB

SNWļ]DVWRVRZDQLH]DVDG\]DFKRZDQLDHQHUJLL mv^y mghmax

0 2

2 

SNWļZ\]QDF]HQLHZ]RUXQDPDNV\PDOQÈZ\VRNRĂÊ

h v v

max

= 02y= ⋅

( )

0

2 2

2g 2g

sinB

SNWļREOLF]HQLHPDNV\PDOQHMZ\VRNRĂFL

h_max=

⋅











⋅

=

10000 2

2 2 10

250

2

 m s

m s

m

2 2

2

6SRVöE,,

SRSNWļZ\]QDF]HQLHVNïDGRZ\FKSR]LRPHMLSLRQRZHMSUÚGNRĂFLSRF]ÈWNRZ\FK v0x=v0⋅ cos B

v0y=v0⋅ sinB

SNWļSU]\UöZQDQLHZLHONRĂFLx 1z

2 i yhmax

SNWļZ\]QDF]HQLHZ]RUXQDPDNV\PDOQÈZ\VRNRĂÊ

h v

z v

g z

max y v

x x

= ⋅



















− ⋅









0

0 0

1 2

2 1 2













2

h z g

max= ⋅ v

− ⋅

⋅ z sin

cos cos

B

B B

2 4

2

0

2 2

SNWļREOLF]HQLHPDNV\PDOQHMZ\VRNRĂFL

hmax=

⋅

⋅

−

⋅

⋅ ⋅











1000 2

2 2 2

2

1000000 10

4 10000 2

2

m m2 m

s m

s

2

2

2 

=

2 250 m

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Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów 3. 3.1

SRSNWļQDU\VRZDQLHNDĝGHJRZHNWRUDLQGXNFML

PDJQHW\F]QHM3U]\NïDGRZRZSXQNFLH3

16

3.2

SNWļQDU\VRZDQLHZHNWRUDVLï\G]LDïDMÈFHMQD

SU]HZRGQLNSRFKRG]ÈFHMRGSU]HZRGQLND

SNWļZ\]QDF]HQLHZ]RUXQDZDUWRĂÊVLï\

F I I L a

I L

41 a

0 1 4 0 1

2

2 2

3 2

= 4

⋅

= ⋅ ⋅ ⋅

⋅ µ

π

µ π

SNWļREOLF]HQLHVLï\

F₄₁

7

3

3 2 4 10 81 1

4 0 03 1 15 10

=

⋅ ⋅ ⋅ ⋅

⋅ ≈ ⋅

−

−

Q Q

N

A A m

m N

2 2

, ,

3.3

SNWļZ\]QDF]HQLHRGOHJïRĂFLSXQNWXPRGNDĝGHJRSU]HZRGQLND r 1a

2 2

SRSNWļZ\]QDF]HQLHZ]RUöZQDZDUWRĂÊLQGXNFMLPDJQHW\F]QHMSRFKRG]ÈFHMRG

NDĝGHJRSU]HZRGQLND

B I

r

I

1 a

0 1 0 1

2 2

=µ = 2 ⋅ π

µ π

B I

a

I

a B

2

0 2

0 1

1

2 2

2 1

2 2

1

= ⋅ 2

=

⋅

= µ

π

µ π B32B1

B43B1

SNWļZ\]QDF]HQLHZ]RUXQDZDUWRĂÊZHNWRUDZ\SDGNRZHJRLQGXNFMLPDJQHW\F]QHM B² B4 B1 B B

2

3 2

=

(

−

)

+

(

−

)

2

SNWļREOLF]HQLHZDUWRĂÊZHNWRUDZ\SDGNRZHJRLQGXNFMLPDJQHW\F]QHM

B B B B B B I

=

(

−

)

+ − a









 = ⋅ = ⋅ ⋅

3 2 1

2 2 5 2 5 2

1 1 2

2

1 1

2

1

0 1

, , µ

π

SNWļREOLF]HQLHZDUWRĂFLLQGXNFMLZSXQNFLHP

B = ⋅

⋅ ⋅ ⋅

⋅ ≈ ⋅

−

2 5 −

2 4 10 9

2 0 03 2 12 10

7

, 4

, ,

Q Q

N

A A

m² T

B

₁

B

₂

B

₃

B

₄

B

P

I

₁

I

₂

I

₃

I

₄

B

₄₁

F

₄₁

I

₁

I

₂

I

₃

I

₄

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Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów

 

SNWļ]D]QDF]HQLHQDU\VXQNXNLHUXQNXW\ONRMHGQHJRSUÈGXI:JDïÚ]L]DZLHUDMÈFHM

NRQGHQVDWRUSUÈGQLHSï\QLH

R

₂

R

₁

E

r

B A

I I

c I

– +

12



SNWļVNRU]\VWDQLH],,SUDZD.LUFKKRIID F = +Ir IR₂

SNWļREOLF]HQLHQDWÚĝHQLDSUÈGX I=r R

+ F

2

I = +

= 200

2 18 A 10A



SNWļREOLF]HQLHVSDGNXQDSLÚFLDQDRSRU]HR1

U10

3RQLHZDĝZWHMJDïÚ]LSUÈGQLHSï\QLHQLHPDVSDGNXQDSLÚFLDQDRSRUQLNXR1.

SNWļREOLF]HQLHVSDGNXQDSLÚFLDQDRSRU]HR2

U₂= ⋅I R₂

U2=10A⋅188=180V

SNWļREOLF]HQLHUöĝQLF\SRWHQFMDïöZQDNRQGHQVDWRU]H UCU2180 V

5öĝQLFDSRWHQFMDïöZSRPLÚG]\ZÚ]ïDPL A i B MHVWWDNDVDPD



SNWļREOLF]HQLHïDGXQNXQDNRQGHQVDWRU]H Q=C U⋅ C

Q =0 5 10, ⋅ ⁻⁶F⋅180V=9 10⋅ ⁻⁵C

SNWļREOLF]HQLHOLF]E\HOHNWURQöZRGSRZLDGDMÈFHM]JURPDG]RQHPXïDGXQNRZLHOHN- WU\F]QHPX

N Q

e N = ⋅

⋅

= ⋅

−

−

9 10

1 6 10 5 625 10

5 19

C 14

C

, ,

SNWļREOLF]HQLHHQHUJLL]JURPDG]RQHMZNRQGHQVDWRU]H WCUC²

2

W = ⋅ ⋅

( )

= = ⋅

−

0 5 10 180 −

2 0 0081 8 1 10

6 2

, 3

,   ,

F V

J J

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Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów



SNWļZ\]QDF]HQLHLORĂFLHQHUJLLSRWU]HEQHMGR]DJRWRZDQLDZRG\

Q=mw⋅cw⋅ %T

SNWļZ\]QDF]HQLHZ]RUXQDF]DVSRWU]HEQ\GR]DJRWRZDQLDZRG\

P Q

t t Q

P

m c T

U I

w w

= = ⋅ ⋅

⋅

%

2

1 pkt – obliczenie czasu

t =

⋅

⋅

⋅

⋅

= =

1 4190 80

180 10 186 2 3 1

kg J

kg K K

V A s min

 

, ,

 

SNWļREOLF]HQLHVWDïHMVLDWNL

d = = ⋅ ⁻ =

1

200mm 5 10 ⁶ 5 m µm^α

10



SNWļVNRU]\VWDQLH]UHODFMLWU\JRQRPHWU\F]QHM

tg

( )

B =x L

x= ⋅L

( )

= ⋅L = ⋅L

−

( )

tgB B

B

B B sin

cos

sin sin

1 ²

SNWļVNRU]\VWDQLH]UöZQDQLDQDVLDWNÚG\IUDNF\MQÈ n⋅λ=d⋅sinα

SNWļZ\]QDF]HQLHSRïRĝHQLDSUÈĝNDQDHNUDQLH

x L n

d n

d

n L

d n

= ⋅

⋅

− ⋅











= ⋅ ⋅

−

(

⋅

)

M M

M 1 M

2 2 2

SNWļZ\]QDF]HQLHZ]RUXQDUöĝQLFÚRGOHJïRĂFLSRPLÚG]\SUÈĝNDPL

%x=x2−x1

%x L n

d n

n

d n

= ⋅

−

(

⋅

)

⁻ −

(

⋅

)

















M M M

2 2

2 2

1 2

1 2

SNWļREOLF]HQLHUöĝQLF\RGOHJïRĂFL

%x =3 5 10⋅ ⁻₇ ⋅0 5 2 − 1

, m , m







=

L

a

x

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Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów



SNWļZ\]QDF]HQLHZ]RUXQDUöĝQLFÚRGOHJïRĂFLPLÚG]\SUÈĝNDPLGODUöĝQ\FKGïXJR- ĂFLIDOL

%x=x₂−x₁

%x n L

d n d n

= ⋅

−

(

⋅

)

−

−

(

⋅

)

















M M

M M

2

2 2

2

1

2 1

2

SNWļREOLF]HQLHUöĝQLF\RGOHJïRĂFL

%x = ⋅ ⋅

(

⋅

)

⁻

(

^{⋅ ⋅}

)

⁻

⋅

⋅

−

− −

−

3 0 5 6 5 10

5 10 3 6 5 10

3 5 10 5 1

7

6 2 7 2

7

, ,  

,  

m m ,

m m

m 0

0^{−6 2} 3 3 5 10^{−7 2}

( )

⁻

(

^{⋅ ⋅}

)















=

  ,  

m m

0 104, m10 4, cm



SNWļREOLF]HQLHVWDïHMVLDWNL d n

= ⋅ λ sinα d = ⋅ ⋅

= ⋅ =

−

5 4 10 −

0 5 4 5 10 4

7

  6

, m ,

m µm^α

SNWļREOLF]HQLHOLF]E\U\V

N = ≈

1

4mm 250 µm

α

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