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T =T₁^2p= 1−(n1−n2)² (n1+n2)²

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2Jvω_c²=m g b+1 2Jhω²_c

#1$2

ωc= s

2m g(a−b) Jh−Jv

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84 '&6 9321&'0 1' 02"/4 :1' 1&# ;2$<1'= > b≈2,25 cm? 12 !2&6'&0 >a≈3,25 cm@

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ωc≈

r2×10×1×10⁻²

1×10⁻⁴ = 66 rad·s⁻¹

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B S23$A'82 = NMP * B ,'31' *;"'33' = TOU B ,'31' */I$*/8' = MVU

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B Y$1'##' '3 8$63' = NZP *[#

B 4'* # @' *;312' = V*3 NO#

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/5 &$##54)' )211'# 249543' '#3 924) 5 12I$*53$6'*'43 500×10⁶/3600 = 140 kW=

/5 657'&1 312&6@' '#3 7' <24 2191' 9' :1549'&1 &$#%&' 75 657'&1 1@'77' '#3 9' 225 kW=

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,2&1 75 1'*$#' '4 12&3' #&1 &4' 9&1@' 9' ∆t? 24 '&3 @657&'1 7'# 651$53$24# 9' 6$3'##' '3 98573$3&9' L M 651$53$24 9' 6$3'##' L 9' K E v= 2,5 m·s⁻¹ '4∆t?

M 5))@7@153$24 *2"'44' )211'# 249543' L a= _∆t^v M 9$#354)' 51)2&1&' L∆x=¹₂a∆t²= ¹₂v∆t

M 651$53$24 98573$3&9' L ∆z= ∆x×0,3 = ^0,3₂ v∆tA '43' *2"'44' 9' NKOB=

/5 651$53$24 98@4'1:$' 2&1 7'# #$C:'# A9' *5##' msiegeB '#3 A24 4' 1'49 5# '4 )2* 3' 75 651$53$24 98@4'1:$' 23'43$'77' %&$ '#3 :72<57'*'43 4&77'B L

∆Esieges= 119msiege

v² 2

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∆Eskieurs= 4×119/2×mskieur

v² 2 +0,3

2 g v∆t

+2**'

Pmax= ∆Esieges+ ∆Eskieurs

∆t ≈225 kW

#2$3

Pmax∆t= 119msiegesv²

2 + 2×119×mskieur

v² 2 +0,3

2 g v∆t

⇒

(Pmax−0,3×119mskieurg v) ∆t= 119 (msiege+ 2mskieur) v² 2 S6')msiege ≈200 kg? '3mskieur≈80 kg? 24 312&6' L

∆t= 119 (msiege+ 2mskieur) ^v₂²

Pmax−0,3×119mskieurg v = 119 (200 + 2×80) ^2,5₂²

225×10³−0,3×119×80×9,81×2,5 = 0,9 s A2& 0,4 s#$ 24 1'49msiege= 0B? )' %&$ #'*<7' 954# 32&# 7'# )5# $4#354354@=

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r < R⇒ −→rot~v= 2~Ω r > R⇒ −→rot~v=~0 ,5&#$'&:# 72$'# #286 2##$;5'# 9

−→rot~v

<:=)' 1&

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−→rot~v

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~v=r.Ω.~uθ 2&: r < a

~v=^Ω.a_r²~uθ 2&:r > a

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/ 01 12**'R 3' 45"21 6& 75$##'5& # 58$539 Oz#21 5:'9 '8 3' 7')8'&4 7$8'##' 51;&35$4' ~Ω =ω.~ez <ω >0=>

/ 01 2&445 4'164' δR≈1,75 m <35 6$#851)' '184' 3'# $'6# '8 35 8?8'=>

/ @5 '#518'&4 #&4 A'44' '#8 g= 9,81 m·s⁻²>

B24)' 6C'1845D1'*'18

E51# 3' 824'9 $3 " 5 35 F24)' 6C$1'48$' 6C'1845D1'*'18 %&$ 75&8m R ω²> 01 )!'4)!' 621)R ω²=g9 57')^δR_R = 10%>

EC2GR= 17,5 m> H$1#$ω=q

9,81

17,5 = 0,75 rad·s⁻¹9 #2$8 &1' I4$26' 6' 42858$21 6' 8 s>

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f~iC=−2m ~Ω∧~v=−2m ω ~ez∧vθ~eθ= 2m ω vθ~er

T$ vθ > 09 )C'#8(R(6$4' #$ 35 7$8'##' 6' 3C5#84215&8' '#8 651# 3' *?*' #'1# %&' 35 7$8'##' 6& 75$##'5& <7$8'##' 6C'1845D1'*'18=9 3C5#84215&8' #'*J3' 5324# 3&# 32&469 )2**' #2&*$# R &1' '#518'&4 g+δg 57')

δg= 2.ω.vθ

T$v= 10 km·h⁻¹9 δg= 2×0,56×3,6¹⁰ = 3,1 m·s⁻² K $3 Q#'30% 3&# 32&46 U

T$ 5& )21845$4' vθ <09 )C'#8(R(6$4' #$ 35 7$8'##' 6' 3C5#84215&8' '#8 2 2#I' R 35 7$8'##' 6& 75$##'5& <7$8'##' 6C'1845D1'*'18=9 3C5#84215&8' #'*J3' 5324# *2$1# 32&469 )2**' #2&*$# R &1' '#518'&4 g−δg 57')

δg= 2.ω.|vθ|

+5# 3$*$8' K #$ |vθ|=R ω9 3C5#84215&8' '#8 '1 5 '#518'&49 )54 $3 '#8 L:' 651# &1 4IFI4'18$'3 ;53$3I'1>

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/ Q8##' *;68$5' 1' 6@'8& ? Meau= 18 g·mol⁻¹<<

/ +;2#3823' 1'# 78O 85B8$3# ? R= 8,31 J·K·mol⁻¹<

-5821'&5# D '#3$*'5 ?

/ R;6&*' 1& A'55' 1@'8& '2A$5;2 Veau= 100 mL9

/ &2 5CB5$7C583'&59 *;1C6$#8:6' );**' &2 '2)'$23' 1' A;6&*' '2A$5;2 Vf rigo = 50 cm×50 cm×1,0 m = 0,25 m³9

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K8 *8##' 1@'8& 182# 6' A'55' '#3 ?m=Veauρeau= 100 g<

K8 %&823$3C 1' *83$>5' '2 '8& 182# )' A'55' '#3 ?v= ^Veau_M^ρeau

eau = 5,6 mol<

K' )!827'*'23 1@C383 1@C%&83$;2 H2O(`)H2O(g) );21&$3 D &2 C383 1@C%&$6$:5' '3 68 %&823$3C 1' *83$>5' );23'2&' 182# 68 A8 '&5 #83&5823' '#3 ngU '2 8 5;H$*823 68 A8 '&5 #83&5823' D &2 78O 85B8$39 ;2 35;&A' 68

%&823$3C 1' *83$>5' 1@'8& 86;5# 5C#'23' 182# 68 !8#' 78O'&#' ? ng= ^Psat_{R T}^V = 0,087 mol<

V )!8%&' ;&A'53&5' 1& 5CB5$7C583'&59 68 %&823$3C 1' A8 '&5 1@'8& ns %&$33823 '#3ns= 10%ng = 0,0087 mol<

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'235' 6'# 1'&H !8#'#9 ;2 8 1;2) '235' )!8%&' ;&A'53&5' 1' ;53'9 ns = 0,0087 mol 1@'8& %&$ #@CA8 ;5'23 ? 68 ;53' '#3 ;&A'53' WX B;$# 85 F;&59 6' 2;*:5' 1' F;&5# N 2C)'##8$5' D 6@CA8 ;583$;2 );* 6>3' 1' 6@'8& 182# 6' A'55' '#3 1;2) ? N= ₁₀ⁿ_ns = 64F;&5#<

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λ 4 = c

4ν = 3×10⁸

4×2,45×10⁹ = 3 cm

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Πmax= Pmax

4π r²_min

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Pmax= 4π r²_minΠmax= 2π×0,04²×10≈0,1 W +;**'Pmax= 4π r²_min <Π>8I')

<Π>=c < eem>=c

ε0E² 2 + B²

2µ0

=c

ε0E_max²

4 +B_max² 4µ0

=cε0E_max²

2 =cB²_max 2µ0

;2 35;&I' 75P)' 8& )!8* ?6')35$%&' 1' 5?F?5'2)' A Pmax= 4π r_min² cε0E_max²

2 = 4π×0,04²×3×10⁸×8,85×10⁻¹²×0,5×61²≈0,1 W

;2 35;&I' 75P)' 8& )!8* *872?3$%&' 1' 5?F?5'2)' A Pmax= 4π r²_mincB_max²

2µ0

= 4π×0,04²×3×10⁸×0,5× 2×10⁻⁷2

4π×10⁻⁷ ≈0,1 W 3"(&4'$

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m=4

3π R³µ≈4

3π×(0,1)³×10³≈4 kg

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I= P 4π d² 3I') 8

• I '0W·m⁻²Z

• P '0 [ Z

• d'0 *9

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/01'2*$341$53 6' 74 6$#143)' '312' 7' 8'29'2 '1 :4&1' ;'364< = d= 9 km>

?20%&'3)' 6& #53 @5&0 =

/5)&*'31 A = B 74 351' 74 7&# 924C' 5##$87' #&2 7D$3#12&*'31> ,5&2 7' )52 6'# E7 '#F '77' '#1 411'$31' 752#%&' 74 7539&'&2 6D536' 6' 74 351' G4$1 6'&H G5$# 74 7539&'&2 6& )52> I

+47)&7 6' 74 G20%&'3)' J 7D4$6' 6'# 65330'# = v=λ f⇔f =^v_λ = 50 Hz>

K3 351' dA = 1 m '1 dB = 9,0 km> IA = _{4π d}^P2

A '1 IB = _{4π d}^P2

B #5$1 _I^IA_B = ^d_d^2B2

A L7' 5$31 M 60#$93' :4&1

;'364<N>

O01!56' P =

E1103&41$53 6&' J 74 6$#143)' = ∆L=LA−LB = 10 log

IA

I0

−10 log

IB

I0

= 10 log

IA

IB

653)∆L= 10 log_d²

B

d²_A

= 20 log

dB

dA

= 20 log 9×10³

= 79 dB E$3#$LB=LN endaz = 100−79 = 21 dB>

O01!56' A =

E 421$2 6' 74 2'741$53 LA= 10 log

IA

I0

F 53 )47)&7'IA =IA= 10⁻²W·m⁻²>

K3 '3 606&$1IB =IB= 1,2×10⁻¹⁰ W·m⁻² '1 653)LB =LN endaz= 10 log

IB

I0

= 21 dB>

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2

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• >4 'L')8&' *9$48'4948 &4 <$194 6;74'0?$' /&0 '4 676&$0' 19 &$##94)' 9##/)$7'= >4 0' 0'46 1' *H*'

#"#8@*'3 '8 1;/4 &8$1$#' 1' +01'!2*% (% ,31#%!4"% $"#1+"./% '480' 6'&E $4#8948# 8 '8(t+δt)3 '4 1;9<#'4)' 6' '08'# JG$#)/#$87 47?1$?7'K O ¹₂(Dmδt)v² = δW = P δt 6/48 /4 676&$8 19 &$##94)' 9##/)$7' 5 1;&4' 6'#

:/0)'# 6' /&##7' P =¹₂Dmv²= ¹₂ρSv³=

• U1 :9&8 )/& 1'0 1'# 6'&E <$194# O 9G') F1 = F2 = ^mtot₂ ^g = ρSv² /4 '&8 $#/1'0 19 G$8'##' v = q_m

totg 2ρS = N9 &$##94)' 9##/)$7' 9&E 6'&E 8&"@0'# '#8 91/0# Ptot= 2P =ρSv³=ρS

mtotg 2ρS

3/2

=¹₂q

(mtotg)³ 2ρS =

• U1 '#8 8'* # 6;7G91&'0 4&*70$%&'*'48 19 &$##94)'= >4 )/4#$6@0' &4 /060' 6' ?0946'&0 8" $%&' 6' 19 *9##' 8/891' 6'mtot= 200 kgJ&4' '0#/44' B #/4 7%&$ '*'48K I 1' 6$9*@80' '#83 6;9 0@# 19 !/8/3 6' 1;/060' 6' ₂₀¹ 6' 19 89$11' 6;&4' '0#/44'3 #/$8 8" $%&'*'48 5 cm3 6/4)S'20 cm²= 2,0×10⁻³m²I 19 *9##' G/1&*$%&' 6' 1;'9& '#8 6' ρ = 1000 kg·m⁻³3 '8 1;/4 0'4609 g = 10 m·s⁻²= >4 /<8$'48 91/0# Ptot'22 kW #/$8 1;7%&$G91'48 6;&4 */8'&0 6' VW )!'G9&E J1 ch = 0,736 kWK3 )'19 909X8 0'198$G'*'48 )/!70'483 '8 )/* 98$<1' 9G') 1'# <0/)!&0'# 6'# 0/6&$8# )/**'0)$9&E=

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!"#$ %&##$%"'&(

!"#$%& &' ()*+#,%"',)!

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8+%)#0',)!

K$530 6' %&301$17 6' *:&4'*'01 3& 0$4'3& 6& $5:1'

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;2306'&2# 2'531$4'# E 53 321$' ):**&0' 3& L&$6' E t'1 Et+dt< /0 0:1'δm=Dmdt53 *3##' 6'# 6'&B 321$'#

*7#:#): $%&'# 6$#1$0)1'# E 5A'0127' '1 E 53 #:21$'<

D' #"#1J*' 3$0#$ 67=0$ '#1 #:&*$# E #:0 :$6# P~ =m0~g9m0):* 2'0301 53 *3##' 6& $5:1' *3$# 3&##$ )'55' 6' #:0 7%&$ '*'019 6& 1&"3& ?53 321$' 6'##$07' #&2 53 =;&2'>?H@@ '1 6' 53 321$' 6& L&$6' ):01'0&' 6'630#< ^! /0 :&223$1 R12' 1'017 6A:&H5$'2 630# &0 2'*$'2 1'* # 5A3)1$:0 T~ =−T−u→z6' 53 H3#' 6& 1&"3& #&2 53 321$'

!3&1' 6' )' 6'20$'2 %&$ 83$1 321$' 6& #"#1J*'< M$ :0 53 2'06 0730*:$0# '0 ):* 1'9 53 27#&51301' 6'# 8:2)'#

#A7)2$1F~ =−m0g−→uz−T−u→z<

/0 3 5$%&' 5' 1!7:2J*' 6' 53 %&301$17 6' *:&4'*'01 #:&# 53 8:2*' S d~p=F dt~ 34') S

~

p(t) =p~0+δm(−~u) =p~0−Dm~udt

~

p(t+dt) =p~0+δm(~u) =p~0+Dm~udt F:0)d~p= 2Dm~udt<

/0 2'12:&4' &0' 8:2)' 6' :&##7' 3035:;&' E )'55' 6A&0' 2'##$:0 6' 236$31$:0 5:2# 6A&0' 27L'B$:09 #:$1 Π =~ −2Dm~u9 '1 53 ):06$1$:0 6A7%&$5$H2' 6& $5:1' #A7)2$1 S F~ +Π =~ ~0 ⇔ m0g+T = 2Dmu (1)<

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B C097&24$2C 9' 7$4# 4& $976' ===

D$903 4' %&036$65 4' *7&;'*'36 B 9@'3#'*:9'

/@0&62' #79&6$73 )73#$#6' B >0$2' &3 :$903 4' %&036$65 4' *7&;'*'36 #&2 9' #"#6<*' >'2*5 2' 25#'365 #&2 90 E1&2'FG0H= /' #"#6<*' '#6 )73#6$6&5 B t 02 9' I&$4' #$6&5 '362' 9@'3625' 0;036 90 7* ' G7J #0 ;$6'##' '#6

%&0#$*'36 3&99'H '6 90 #726$' 4' 90 6&"<2'= K3 " 0L7&6' 90 7* '8 9' 6&"0&8 9' $976' '6 #73 5%&$ '*'36= .362' t '6 t+dt8 9' I&$4' #@'#6 45 90)5 4' udt '3 #726$' '6 73 3519$1' 90 ;$6'##' B 9@'3625'= K3 376' 4@&3 $34$)' 0 9'#

12034'&2# 2'906$;'# B 90 026$' )7**&3' 0& I&$4' B t'6 Bt+dt= K3 376'δm=Dmdt90 *0##' 4'# 4'&M 026$'#

*5#7#)7 $%&'# 4$#6$3)6'# B 9@'3625' '6 B 90 #726$'=

/' #"#6<*' 0$3#$ 45E3$ '#6 #7&*$# B #73 7$4#P~ =m~g8m)7* 2'3036 90 *0##' 4& $976'8 4' #73 5%&$ '*'368 4& 6&"0& '6 4' 90 026$' 4& I&$4' 4' 90 E1&2'FG:H= N9 >0&6 " 0L7&6'2 9' 7$4# 4& 2'#6' 4' 90 )79733' 4' I&$4' O

−ρSHg−→uz= ^{! "}/0 25#&96036' 4'# >72)'# #@5)2$6 473) F~ =−mg−u→z−ρSgH−→uz= K3 0 9$%&' 9' 6!572<*' 4' 90 %&036$65 4' *7&;'*'36 #7&# 90 >72*' O d~p=F dt~ 0;') O

p(t)~ =p~0+δm×(~0) =p~0

~

p(t+dt) =p~0+δm(~u) =p~0−Dmu−u→zdt P73)d~p=−Dmu−→uzdt=

K3 7:6$'36 473) 9@5109$65 O Dmu=mg+ρgSH GQH

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4+*-*(,77,!& :T = (m−m0)g+Dmu; '', -.74+,!( ,),-&$<,7,!& 5! &,+7, 4+.<,!#!& (5 4.$(% (, '# 4#+&$, =#%%, (5 &5>#5 (m−m0)g3 7#$% %5+&.5& ', "#-&,5+Dmu95$ ,%& '.$! (8?&+, !*@'$@,#=', 45$%95, -8,%& -, &,+7, 95$3 (8#4+A% /013 %,+# +,%4.!%#=', (, '8*95$'$=+, (5 4$'.&, ,& (, &.5&, ', -.'.!!, (8,#5 (, B#5&,5+H;

,&$##03)' 4' 90 7* '

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12034'&2# 2'906$;'# B 90 026$' )7**&3' 0& I&$4' B t'6 Bt+dt= K3 376'δm=Dmdt90 *0##' 4'# 4'&M 026$'#

*5#7#)7 $%&'# 4$#6$3)6'# B 9@'3625' '6 B 90 #726$'=

K3 0 9$%&' #&2 )' I&$4' 9' 6!572<*' 4' 9@53'21$' *5)03$%&' '362' t '6 t+dt= K3 #& 7#' 9' I&$4' 02>0$6 '6 $3)7* 2'##$:9'= /'# 620;0&M 4'# >72)'# 4' 2'##$73 B P0 B 9@'3625' )7**' B 90 #726$' #@033&9'36= K3 0 473) dEm=P dt7JP '#6 90 &$##03)' 4' 90 7* '= .3 3519$1'036 9@53'21$' )$356$%&' B 9@'3625' 4& 6&"0&8 73 0 O











Em(t) =E0+1

2δm ×(0)²+δm g×0 =E0

Em(t+dt) =E0+1

2δm u²+δm gH =E0+ (1

2Dmu²+DmgH)dt

S& 0##01'8 7&2 0##&2'2 90 )73#'2;06$73 4& 45:$6 ;79&*$%&' '362' 9@'3625' '6 90 #726$' 4' 90 7* '8 90 #')6$73 4' 9@5)7&9'*'36 B 9@'3625' 47$6 T62' :$'3 9&# 12034' %&@'3 #726$'= K3 2'627&;' :$'3 9' 20$#733'*'36 %&09$606$>

4& 259$*$30$2' 403# 9@'M 2'##$73 4' P O P =¹₂Dmu²+DmgH (3)

/@53'21$' δE =P dt '#6 &6$9$#5' 7&2 0*'3'2 90 *0##' δm B 9@096$6&4' H 0;') 90 ;$6'##'u8 473) 0;') &3' 53'21$' 76'36$'99' δm gH '6 )$356$%&' 1

2δm u²= !"#$%&'!" (

/0 2'906$73 GQH '6 9@'M 2'##$73Dm=ρSu4& 45:$6 '2*'6 4@'M 2$*'2 90 ;$6'##' u35)'##0$2' B 9@5%&$9$:2' 4&

$976' O u= rmg

ρS +gH=

N9 3' 2'#6' %&@B 2' 726'2 )'66' 'M 2'##$73 403# )'99' 4' 90 &$##03)' 4' 90 7* ' 47335' 02 GFH8 #7$6 O P =Dm

u² 2 +gH

=ρSu u²

2 +gH

K3 7:6$'36 473) O P =ρS rmg

ρS +gH

mg

2ρS +³₂gH

+73)'23036 9'# ;09'&2# 3&*52$%&'# 35)'##0$2'#8 73 2'34 m= 100U1 0;') 9@5%&$ '*'368g= 10*=#⁻²= ,7&2 90 !0&6'&2 H8 90 !767 '2*'6 4@5;09&'2 '3;$273 H = 5*= /0 !767120 !$' *7362' %&' 9' 4$0*<62' 4& 6&"0&

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4 = 78)*²=

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*@#',7,!& 95, ', 4.$(% (5 E5$(, (, '# 4#+&$, B.+$F.!&#', (5 &5>#5 /$77,+@*,1 ,%& 95#!& G '5$ -.74,!%* 4#+ '# 4.5%%*, (8H+-B$7A(,;

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-,% %544.%$&$.!%;

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;$9'##'u= 13*@#⁻¹< #5$9 '2;$752 CD >*E!< ;14'&7 %&$ '&9 #'*=4'7 71$#5221=4' '9 )5* 19$=4' 1;') 4' F'9 ;$#$=4'

#&7 41 !595B71 !$'@ G' 36=$9 *1##$%&' '#9 1457# 1##'H )52#6%&'29 &$#%&I$4 '#9 3' 4I5737' 3' Dm= 100>BE#@

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!"# $%%&# '%"( %!)* +

! "#$%%&'$!( "&) *$ +,!-.(.,! -#/0&.*.1)2 -,!!/2 %$) 3456 ,! %2&( )27$)0&2) 0&2 *#,! %2&( 8,&2) .!-/%2!9 -$772!( "&) !"# $%&%'()&!*Dm2(u%,&) $""&)2) *#/0&.*.1)2 -& %.*,(2: ;! <$ -,!+ <,.) "#.* 2"( %,"".1*26 %,&)

&!2 =$&(2&)H ('%.0&2 -,!!/26 -2 +=,.".) $& 7.2&> +2 +,&%*2 3Dm6u5 %,&) 7.!.7."2) *$ %&.""$!+2 !/+2""$.)2:

?$!" *$ "&.(26 ,! @>2 H = 57:

.2 &9$4$#129 312# 41 )523$9$52 3I6%&$4$=7' KRM 4I'S 7'##$52 Dm=ρSu 3& 36=$9< 52 5=9$'29 T mg=Dmu−Dm

gH

u ⇔ Dm= mg

u−gH u

@

A2 $29753&$9 )'99' 'S 7'##$52 3'Dm312# 4I'S 7'##$52 KUM 3' 41 &$##12)' 3' 41 5* '< #5$9 T P = 1

2Dm(u²+ 2gH) =mg

2 ×u²+ 2gH u−gH

u

= mgu

2 ×u²+ 2gH u²−gH @

.2 $29753&$#129 41 ;$9'##' )171)967$#9$%&'v0=√gH< 3529 41 ;14'&7 2&*67$%&' '#9 3' v0= 7.1*E#< '9 41 ;$9'##' 763&$9'x= u

v0< 52 5=9$'29 T

P =P0f(x) 1;') P0= mgv0

2 '9 f(x) = x²+ 2 x²−1 x

G1 &$##12)' )171)967$#9$%&' P0 1 &2' ;14'&7 3'P0= 3.5>?@ +I'#9 '2 1##129 4I5737' 3' B7123'&7 %&' 4I52 5=9$'2371$9 #$ 52 #' )529'291$9 3I&2 71$#522'*'29 17 1214"#' 3$*'2#$522'44' ^!@

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!" #$ %&'$(&)*+ %)& ,-$+&' ,-..' .)//'m,'00' (1 %*0-+' ! ,'00' (' 02')1 ,-$+'$1' ()$/ +-1+' 0) %)&+*' 3'&+*,)0' (1 +14)15 ,' 61* ,-$(1*&)*+ 7 1$' 3)0'1& %01/ 80'38' 9(-$, %01/ &)*/-$$):0';

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