ejercicios resueltos de cinematica.todos los movimientos.cuarto eso y primero bachillerato

7/26/2019 Ejercicios Resueltos de Cinematica.todos Los Movimientos.cuarto Eso y Primero Bachillerato

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96 EJERCICIOS RESUELTOS DE CINEMTICA. TODOS LOS MOVIMIENTOS4 E.S.O. Y 1 BACHILLERATO

EJERCICIOS RESUELTOS DE CINEMTICA. 4 E.S.O. Y 1 DEBACHILLERATO

NOTA DEL PROFESOR:

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ESTUDIAREMOS"( *+$ 7a 7$'7( $" ,(f$%( $INTENTARENTENDER"( #$%a(""a#(. S %$ $!&$!#$ es!u#e('o.S !(ENTENDIS"( #$%a(""a#( ,( $" ,(f$%(8 a!(&a $" !5$(#$ $3$''( $! "a ,)a '"a%$8 si( 'ej&r e%#e/&r & !r&$&j&r &

#ro)esor8 ,$#"$ % (% ,+$#$ $%("/$ $" si-uie(!e ejercicio

E3$,"( $%+$"&( N 1U! )/" #$%'0$ +!a &a$'&(a %$''+"a8 #$ a#( 8 $! +!

&$,( #$ 1; %. D$&$!a %+ a,#$< /$"('#a#.

Resouci*(

Lo(-i!u'circu()ere(ci&0 . 2 . r3eAB

3eAB= &a# #$ "a "(!2&+# #$ "a''+!f$$!'a.

P(f$%(: A. >aa2(


2/137


A B 3eAB= ? . @ . ? = ? . 814 . ? == 156 %.

7 3 r 7= ? . = ? . = 18 %.

Ri'e/ 0 3eAB9 3! = 18 1; % = 156 %9s

:eoci'&' 0 7 3 r 7 9 3! = 1; 1; % =1 %9s

E3$''( $%+$"&( N ?U! '('7$ ,a&$ #$%#$ $" ,+!&( "(&'( #$ "a NIV. U!a 7(a -%&a#$ ""$2a a" ")$&( 11;.A"" 2a %$ #a "a /+$"&a8 $!'(!&-!#(%$ $! $" ")$&( 66 #(% 7(a%#$%,+% #$ 7a0$ ,a(.

aG Ca"'+"a $" #$%,"a


3/137


K = ; CG & = 1 7. 11;

AG BG DG 11; G

aG =ACAB = 66 = == >%.

Q $ =ADABG ADACG=11;G11;66G = 44 = 11 >%

0GV = & = ?7 = 1?5 >%9,

Ra,#$% 9 ,

No coi(ci'e( #ues!o +ue e es#&cio recorri'o (o es i-u& & %*'uo 'e


4/137


Q =ADAC = 11; 66 = 44 >%

V = Q & = 44 1 7 = 44 >%9,

E si-(o (e-&!iaa2(


5/137


E" &$,( *+$ $" (03$&( %$ $%&- (/$!#( $% $" %( $! $" $3$ O*+$$! $" $3$ OY8 "+$2( #$%,$3a$(% $" !#$G 0 =! "( ""$/a$(% a 0 4

F !:

! 0 G9= = 4 G? = 4 ?9G 0 4 F G9K

E" #$%,"aaa2(aa2(aa2(


6/137


Ca"'+"a:aG La ,(%')! ,aa & = ? %.0G La /$"('#a# ,aa & = ? %.

Resouci*(

&G = & &? ? G 3 S.I.G

aG?G= ? 4 ? G 3 ?G= ? 6 3

?G= ?? 6?G1?= ?5= % 'e sis!e%& 'e re)ere(ci&.

0GV = ##&

V = ?& 3 V?G= ? .? 3 V?G= 4 3

:@7= 1? 4?G1?= 11?= 451 % . s1E3$''( $%+$"&( N 6 F+$!&$ E!+!'a#(: D,&(. F #$" I.E.S. La A%+!')! #$ E"'7$. R$%("+')!: A.>aa2(


7/137


r = 4?G1?= 4 %

0G A" %$ "a &a$'&(a +!a (e& rec!&8 "a Ra,#$< "a V %(!2+a"$%.

V = 6 6 V = V = 4 V = 1

:% = 1?G1?= 1 % . s1

L+$2(:

Ri'e/= 1 . %1= :%

E" $%,a'( $'(#(:

Ri'e/ 0 s 9 ! % = Ra,#$< . & % = 1 . %1. 6 G %

s= 4 %

")2'a$!&$ '(!'#$ '(! $" #$%,"a


8/137


Resouci*(

aG C(,(!$!&$% 'a&$%a!a% #$" /$'&( ,(%')!:

= & & = = ? . 0 G9 = ?&

0G = & ?& 3 S.I.G

V = G ;G ;

r8@= . ; . ? . ; . 3 = 8r@= . . ? . . 3 = i 18 jV = ? 1; 3 ; :% = i j

:% = ? ??G1?= ?91?= 5= % . s1

'GLa $'+a')! #$ "a &a$'&(a:

0 G 9

'($%,(!#$ a "a $'+a')! #$ +!a $'&a $! +!a &a$'&(a$'&"!$a %$ '+,"$ "a '(!'#')! #$ *+$ $" $%,a'( $'(#($! "a &a$'&(a $% 2+a" a" )#+"( #$" /$'&( #$%,"aaa2(


9/137


Resouci*(La #$/a#a #$ "a /aa0"$ G'(( 0a%$ $,(!$!'a"G %$,$ /a"$ 1 ,( "( &a!&( "a ,(#$(% $"!a #$ "(% '-"'+"(%.

aG = ? = . ? ?1 1. 11 ; 0 ?G F 0G = . 1 . 1 ; 0 =G'G = 4 . 1 . 41 . . 1 ; 0 4G=F K G

E! F%'a $% -% '($!&$ "a% f+!'(!$%:

aG $ = &?

& 0G $ = & 4'G $ = &4 &

E! $%&$ 'a%( $% $" E%,a'( *+$! #$,$!#$ #$" &$,(. La% f+!'(!$% %(!"a% %a% "(% $%+"&a#(% %$-!:

&@ e0 ?! F $@ e0 =!c@ e0 4!=F K!

'@E3$''( $%+$"&( N 9 F+$!&$ E!+!'a#(: D,&( F #$" I.E.S. ALCARIA. R$%("+')!: A. >aa2(


10/137


E3$''( $%+$"&( N 1; F+$!&$ E!+!'a#(: D,&( F #$" I.E.S. ALCARIA. R$%("+')!: A. >aa2(aa2(


11/137


A'$"$a')! & 0 ': 9 '! a = ? 3 . %? a = ??G1?= ? . %?

0G &%0 :@F :8@9 F 8

V?G= ? . ? 3 V?G= 4 3

V( = ? . ; 3 V( =

a= 4 3 G ? a= ?3 &% = ??G1?= % . s

E3$''( $%+$"&(N 1? F+$!&$ E!+!'a#(: D,&( F #$" I.E.S. ALCARIA. R$%("+')!: A. >aa2(aa2(


12/137


E" /$'&( ,(%')! #$ +! )/" /$!$ #a#( ,( : = ?&? 4 3 S.I.G.Ca"'+"a: aG La /$"('#a# $#a $!&$ 6 %$2+!#(% 0G La /$"('#a#!%&a!&-!$a.

Resouci*(

aG&G = ?&? 4 3

:% 0 r?@F r=@9 ? F =

6G= ? . 6?. 4 3 6G= ? 4 3 G= ? . ?. 4 3 G= 1 4 3

V= ? 4 3 G 1 4 3 G :% 0 49= i 0 1 i

:% = 1?G1?= 1 . %1

aG: 0 'r 9 '! := 4! iE3$''( $%+$"&(N 1 F+$!&$ E!+!'a#(: D,&( F #$" I.E.S. ALCARIA. R$%("+')!:

A. >aa2(


13/137


: 0 'r 9 '!C(( $" /$'&( ,(%')! $% '(!('#(8 "a /$"('#a# /a"#-:

: 0 4! i

La a'$"$a')! %$-:

&= 4 i & = 4?G1?= 4 %. s

0G La a'$"$a')! &a!2$!'a":

&!0 ' 7 : 7 9 '! : = 4& G?1?= 4!

&! =4 % . s

Ta!&( "a a'$"$a')! '(( "a a'$"$a')! &a!2$!'a" so( co(s!&(!es ,( "( &a!&( i('e#e('ie(!es 'e !ie%#o.

E3$''( $%+$"&(N 14 F+$!&$ E!+!'a#(: D,&( F #$" I.E.S. ALCARIA. R$%("+')!: A. >aa2(


14/137


&= 4! i = j

& = 4& G? ?1?= 1?! K @19

Paa & =1 %

a = 16 9 G1?= ?1?= % . sE3$''( $%+$"&(N 1 F+$!&$ E!+!'a#(: D,&( F #$" I.E.S. ALCARIA. R$%("+')!: A. >aa2(


15/137


E3$''( $%+$"&(N 1 F+$!&$ E!+!'a#(: D,&( F #$" I.E.S. ALCARIA. R$%("+')!: A.>aa2(aa2(


16/137


Sa0$!#( *+$V&G = &? & G 3:

V = &? & G?1? V = &? & . %1

&! =! F % . s

a! = V ? R

A" %$ "a &a$'&(a +!a "!$a $'&a R = ,( "( *+$ a" %+%&&+

R $! "a $'+a')! a!&$( %a0$!#( *+$ &(#( !5$( #/##(,( $% 2+a" a '$(8 !(% *+$#aa:

a! = V ? = ;

A" ""$/a +!a &a$'&(a $'&"!$aNO EISTE &(8 !( $%&$c&%$io e( & 'irecci*( 'e %o


17/137

m


: 0 i ! j C(!('#a "a $'+a')! #$ "a /$"('#a# ,(#$(% '(!('$ V?G V;G:

:@= ? . ? . 3 :@= 4 3

V8@= ? . ; . 3 :8@=

E" /a"( #$ & :

&%= 4 3 G ? ; &%= ? 3

&% = ??

G1?

= % . s

E'+a')! #$ "a a'$"$a')! !%&a!&-!$a:

& 0 ': 9 '!

&= # #& ?& 3 G &= ? 3 & = ??G1?= % . s

E3$''( $%+$"&( N 19 F+$!&$ E!+!'a#(: D,&(. F #$" I.E.S. La A%+!')! #$ E"'7$. R$%("+')!: A.>aa2(


18/137


La $'+a')! #$ "a /$"('#a# %$-:

:= # #& ?&? 4 3G := 4! i

0GE'+a')! #$ "a a'$"$a')!:

a = #V #&Sa0$!#( *+$ : 0 4! i8 $" /a"( #$ "a a'$"$a')! %$-:

a = 4 a = 4?G1?= 4 . %?La a'$"$a')! $% '(!%&a!&$ $ !#$,$!#$!&$ #$" &$,(. Paa & =

?% "a a'$"$a')! %$- #$ 4 % . s

.

'GLa $'+a')! '($%,(!#$!&$ (0$#$'$ a "a $,$%)!:

7 &! 7=' 7 : 7 9 '!

Sa0$!#( *+$ : 0 4! i8 ,(#$(% '(!('$ $" )#+"( #$ "a /$"('#a#

V = 4&G?1?=4&8 "+$2(:

7 &! 7=4 % . sE3$''( $%+$"&( N ?; F+$!&$ E!+!'a#(: D,&(. F #$" I.E.S. La A%+!')! #$ E"'7$. R$%("+')!: A.>aa2(aa2(


19/137


L+$2( ,aa & = 1:

V1G= ?&?G & 3 :1@= 3 V = ? ?G1?= 41?= 8 . %1

0GE'+a')! #$ "a a'$"$a')!:

& 0 ': 9 '!

S$!#(V:

V = ?&?G & 3

& =4! i = jE! "( $f$$!&$ a" )#+"(:

& = 4&G? ?1?= 1?! K @19E3$''( ,(,+$%&( N ?1 F+$!&$ E!+!'a#(: D,&(. F #$" I.E.S. La A%+!')! #$ E"'7$GLa ,(%')! #$ +! )/" /$!$ #a#a ,(: = ?& = ?&? 18 $! $" S.I..Ca"'+"a:

aG La $'+a')! #$ "a &a$'&(a.0G La /$"('#a# !%&a!&-!$a.'G La a'$"$a')! a "(% 1; %.

E3$''( $%+$"&( N ??La% $'+a'(!$% ,aa&'a% #$" (/$!&( #$ +!a ,a&'+"a /$!$!#a#a% ,(: = ?& = ?&? 1. Ha""a:

aG La $'+a')! #$ "a &a$'&(a.0G E" /$'&( #$%,"a


20/137


Resouci*(

aG E'+a')! #$ "a &a$'&(a:

= ?& & = G ?

= ?&? 1 = ? G ?? 1 : = ? 6 9 G ? 1

= ?? 9? 1 =G?9 F =G 69

0G E" /$'&( ,(%')! /$!$ #a#( ,( "a $'+a')!:

&G = ?& G ?&? 1 G 3

V$'&( #$%,"aaa2(


21/137


VG= ? 4 . 3 :=@= i 1 j

:=@ = ?? 1??G1?= 141?= 151? % . s1

$G & 0 ' : 9 '!

V = ? 4& 3

&= 4 j & = 4?G1?= 4 % . s

fG 7 &!7 0 ' 7 : 7 9 '!

V 0 i 4! j : = ?? 4&G?1? : = 4 16&?G1?

a& = # V #& a& = Z 4 16&?G1? 1. ?&G

a& = Z 4 16&?G1?. ?& a& = Z . ?& 4 16&?G1?

&! = 16& 4 16&?G1?= 16 . 1 4 16 G1? = 1? 9 4546 % . s

==56 % . s

a = a& ? a! ?G1?

4 = 8G? a!?G1? 16 = 1?84 a!? a!?= 8?6

&( = 8?6G1?= 15 % . s

2G 7&( 7 0 7 : 79 R 1G

: 0 i 4! j & = 1 V = ? 4 3

P(f$%(: A. >aa2(


22/137


: = ?? 4?G1? : = 4546 % . s1

D$ 1G:

R= 7 : 79 &( R= 484G? 18 = 1989 18 = 11 %.

E3$''( ,(,+$%&( N ?U! '+$,( %$ +$/$ $! $" $%,a'( %$25! "a $'+a')! #$ %+ ,(%')!:

&G = & ?G &?3 ?&D$&$!a:

aG V$"('#a# !%&a!&-!$a $! f+!')! #$" &$,(.0G Ca"'+"a %+ /$"('#a# !%&a!&-!$a ,aa & = ? %.'G Ca"'+"a "a /$"('#a# $#a $! "(% #(% ,$(% %$2+!#(%.#G Ca"'+"a "a a'$"$a')! !%&a!&-!$a ,aa & = 4 %.$G S+,(!$!#( +!a &a$'&(a ''+"a #$&$!a $" Ra#( #$

'+/a&+a #$ "a &a$'&(a.

E3$''( $%+$"&( N ?4La ,(%')! #$ +!a ,a&'+"a )/" /$!$ #a#a ,( %+% '(,(!$!&$%'a&$%a!a%:

= ?&? = & &? < = & ?D$&$!a:

E" )#+"( #$ "a /$"('#a# a'$"$a')! ,aa & = ? %.

E" a#( #$ '+/a&+a #$ "a &a$'&(a %$2+#a ,( $" )/".

Resouci*(

L( ,$( *+$ 7a$(% $% $%&a0"$'$ $" /$'&( ,(%')! #$ "a,a&'+"a:

&G = ?&

?

G &

&

?

G3 & ? G

P(f$%(: A. >aa2(


23/137

2 1/2

3 1/2

1/2


Sa0$(% *+$ $" /$'&( /$"('#a# $%:

: 0 'r 9 '!= # #& ?&? G & &? G3 & ? G =

= 4& 9&? ?&G3

V = 4&G? 9&? ?&G?

V = 16&? 1&4 4&? 6 & 9G

V = 1&4

6&

?;&?

9 G:@= 4 . ? 6 4G 3 = 4;3

7 : 7 = ? 4;? ?G1?= 16G1?= 485K % . s1

P( (&a ,a&$ %a0$(% *+$ $" /$'&( a'$"$a')! $%:

& 0 ': 9 '!= ##& 4& 9&? ?&G3 = 4 1& ?G3

a?G= 4 6 ? G 3 &@= 4 3

7 & 7= 4? ?G1?= 16 1444 G1?= =5 % . s

Paa '(!('$ $" a#( #$ '+/a&+a +&"


24/137


a&= # V #& 1&4 6& ?;&? 9 G1?

a&= Z 1&4 6& ?;&? 9 G Z 1. ?4& 1;&? 4;&G

a&= Z 1&4 6& ?;&? 9 G1?. ?4& 1;&? 4;&G

a&= ?4& 1;&? 4;& ? . 1&4 6& ?;&? 9 G1?

a&= ?9? 4? ;G ? . 1?96 ? ; 9 G1?

&!= 1;4 ? . 4;89 = 1;4 18 ==65K4 % . s

S !(% /a(% a "a $'+a')! ?G:

| a | = ( at2+ an

2)1/2

8? = 894? a!?G1?

E"$/a!#( "(% #(% $0(% #$ "a $'+a')! a" '+a#a#(:

1498?4 = 149844 a!? a!= 1498?4 149844 G1?

&(0 454 % . sN(% /a(% a "a $'+a')! 1G:

R = V? a! R= 4;89G? 484 = =65K1 %

E3$''( $%+$"&( N ?E" /$'&( ,(%')! #$ +! )/" /$!$ #a#( ,( "a $,$%)!:

&G = ?& &?G D$&$!a:

T,( #$ (/$!&( #$'')! #$" )/".

A'$"$a')! &(&a" #$" %( ,aa & = 1 %.

P(f$%(: A. >aa2(


25/137


Resouci*(

S$25! "a $'+a')! #$" /$'&( ,(%')!:

&G = ?&&?G

%("( $%&$ '(,(!$!&$ #$" %( $! "a #$'')! #$" $3$ OK.E" &,( #$ (/$!&( "( $(% #$&$!a!#( $! "a '+$%&)! 0G.

R$'(#$(% *+$: & 0 ':9 '! 1G

P( (&a ,a&$:

: 0 'r 9 '! V = # #& ?& &? G = 6&? 1;&G

E" /$'&( /$"('#a# #$,$!#$ #$" &$,( ,( "( &a!&( %$ &a&a#$ +!Mo


26/137


Resouci*(

&G = & 3 &?

: 0 'r 9'r= # #& & 3 &?G

: 0 i ! V = ? ?&G?1? 7 : 7 0 4!@19

7 :1@7= ? 4 G1?= 5= % . s1

R$'(#$(% *+$: & 0 ': 9'!= # #& ?& G1?= ?

7&1@7= ?G1?= 1541 % . s

Sa0$(% *+$: & 0 &! &( 7 & 7 0 &! &(@19 1G

P( (&a ,a&$: &!0 ' 7 : 7 9'!= # #& ? 4&?G1?

a&= Z ? 4&?

G11?

. & = & ? . ? 4&?

G1?

&!= . 1 ? . 8 = 1;86 = 8564 % . s

S !(% /a(% a "a $'+a')! 1G:

1841 = ;84? a!?G1?

$"$/a!#( "(% #(% $0(% a" '+a#a#(:

189 = ;84 a!? a!?= 189 ;84 &(= 1844G1?= 15 % . s

E3$''( $%+$"&( N ?E" a#( #$ '+/a&+a #$ "a &a$'&(a ''+"a #$%'&a ,( +! )/"&$!$ +! /a"( #$ ; ' "a $'+a')! #$ %+ (/$!&( /$!$ #a#a8 $!f+!')! #$" &$,(8 ,( "a $,$%)!:

$ = &? 6& 4

P(f$%(: A. >aa2(


27/137


D$&$!a "a% '(,(!$!&$% !&!%$'a% #$ "a a'$"$a')! a% '(( $"/a"( #$ "a a'$"$a')! ,aa & = %.

Resouci*(

R = ; ' . 1 1;; ' = ;8;

$ = &? 6& 4

Pa&$(% #$" '(!'$,&( #$ /$"('#a#:

: 0 'e 9 '! V = # #& &? & 4G V = 6&

7 : 7 0 =?! @19

La a'$"$a')! $% 2+a" a: & 0 ': 9'!= # #& 6& G = 6

a = 6?G1? 7 & 7 0 ? % . s

P( (&a ,a&$: &!0 ' 7:7 9'! = # #& 6&? ? G1?

a&= Z . 6&? ? G 11?. ?& = ? & ? 6 . ? ? G1?

&!= ? . ? . ;841 = 6; 6;8? = 5K1 % . s

R$'(#$(%: 7 & 7 0 &! &(@19

6 = 891G? a!?G1?E"$/a!#( "(% #(% $0(% #$ "a $'+a')! a" '+a#a#(:

6 = 489 a!? &(= 6 489 G1?= 1584 % . s

E3$''( $%+$"&( N ?I!&$,$&a $" (/$!&( *+$ ""$/a +! )/" '+a 2-f'a $%:

P(f$%(: A. >aa2(


28/137

m

20


&; &1 &? & %Resouci*(

INTERVALO TIPO DE MOVIMIEBTO

&( &1 M.R.U. V= C(!%& G

&1 &? No ,& %o


29/137

ot1 t

2 t

3 t

4 t

5 t

6


O&a ,(%0"#a#:

V

& &

V [;

E" )/" %$ $%&aa #$%,"a


30/137


S$2+!#( &a(:V = ; #$%'a!%aG

& = 1; !+&(% ] 176; !+&(% = 8P16 ,

e0 8$%&- #$%'a!%a!#(G

Tercer !r&%oV =

& = 4 !+&(% ] 176; !+&(% = ;\ 7.

C(( $2$%a a" ,+!&( #$ ,aa8 #$0$ $'($ "(% ??\ !'a"$%8

,( &a!&(8 %+ /$"('#a# #$ $2$%( $%:

: 0 e9!= ??\;\ 7 = =8 %9,

La% 2-f'a% %$-! ,( &a!&(:

P(f$%(: A. >aa2(


31/137


P(0"$a $%+$"&( N 1S(0$ +!a $'&a %$ #$%,"aaa2(


32/137


Resouci*(

La 2-f'a $,$%$!&a $" (/$!&( #$ +! '+$,( *+$ &$!$ "+2a

$! &$% &a(%:

Ta( I: M.R.U. a *+$ $! 1; % $'($ ; 8 ,( &a!&(:

:10e9!= ; 1; % = %9s

Ta( II:$" '+$,( ,$a!$'$ ,aa#( #+a!&$ ; % a ; $&(% #$"(2$!. V?= ; ,$a!$'$ ,aa#(G

Ta( III: M.R.U. E" '+$,( $2$%a a" (2$! $! %::=0 e9! = ; % = 18 %9s,$( $" %$!( #$ "a /$"('#a# $% $"

'(!&a( a" #$" &a( I8 a *+$ $2$%a a" (2$!G

P(0"$a ,(,+$%&(U! '('7$ %$ #$%,"aaa2(


33/137

e (m)

9000

7500

6000

4500

3000

1500

100 200 300 400 500 600


D$0$(% '(!('$ "a% ,(%'(!$% *+$ ('+,a $" )/" $! f+!')! #$"&$,(. Paa $""( +&"aa2(


34/137

e(m) 12

9

6

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35/137


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Pa%a$(% &(#a% "a% +!#a#$% a" S%&$a I!&$!a'(!a":

4 7 . 1;;; 1 . 1 7 6;; % = 1 %9s= 1 % . s1

? 7 . 1;;; 1 . 1 7 6;; % = 8 %9s= 8 % . s1

. 1;;; 1 = ;;;

E%&a0"$


36/137

e(m)

8000

6000

4000

2000

0 200 400 600 t(s)

tB= 533,33 s

A V = 20 m/s V =- 15 m/s B


; = ;;; 1G . &B 1 &B= ;;; &B= ;;; 1 %G

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Va(% a $%("/$"( !+'a$!&$ /$$(% % "a a,(a')!$a"


37/137


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E" &$,( *+$ &a#a B $! ""$2a a" ,+!&( #$ '+'$ $AG8 %$-:

eB0 e8B :B. !B

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C(( $" )/" A ""$/a -% /$"('#a# *+$ $" B $" ,+!&( #$ $!'+$!&($%&a- -% '$'a #$" ,+!&( #$ ,aa #$ B. E" '(*+% #$ "a$,$$!'a $%:

P. E!'+$!&( A

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38/137


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! 0 e 9 :

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G= 1;;; 8? = 41??5?? %G= 41668 . 1 1;;; = 415? >%

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39/137


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P$( ,a%a$(% "a% +!#a#$% a" S.I:

VA= ; 7 . 1;;; 1 . 1 7 6;; % = 19844 % = 19844 .%1

VB= ; 7 . 1;;; 1 . 1 7 6;; % = 18 % = 18 .%1

1;; . 1;;; = 1;;;;;

E%&a0"$


40/137


18 . 1;;;;; $BPG = 19844 $BP 1;;;;;G

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S$ $! $!'(!&a-! a 99?;8 #$ B ( 0$! a:

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41/137


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42/137

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43/137

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44/137

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46/137


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L"$/a#( $" /a"( #$ea ?G:

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47/137


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48/137


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e= 16866 % . % = =5== %

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49/137


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16686 $?= Z . . $?? ?86

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$?= ?86 981 49?4986 G1?

$?= ?86 49811

$?= 1?489

$?= ?86 49811 [ ; NO VALIDO FISICAMENTE

L+$2( $" '('7$ &a#a0a $! $'($ $?:

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E%&$ &$,( $a 2+a" a" &$,( '(!%+#( ,( "a (&( $! $'($ $1 $?G8 ""$2a(% a "a '(!'"+%)! *+$ "a (&( &a#a- 65 s$! a"'a!


50/137


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51/137


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e 0 : . != ; % ] ?; % = 1888 %

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52/137


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La ,(%')! %$a:

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53/137


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54/137


Vf= 1;; 7 = ?8 . %1

& = 4 % &= ?8 ;G .%1G 4 % =?5K % . s

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Da&(% a" S.I.:

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e 0 . & . !

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P(f$%(: A. >aa2(


55/137

e(m)

120

100

80

60

40

20

0

0 3,6 4,5 6,4 t(s)


P(0"$a $%+$"&( N 44

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espacio (m) tiempo (s)

0 0

5,83 3,6

18,23 4,5

39,37 5,4

71,44 6,3

116,64 7,2

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56/137


Paa $ = 8 & = 86 %:

8 = Z . a . 86 %G? 8 = Z . a . 1?896 %?11866 = a . 1?896 %? a = 11866 1?896 %?

&10 85K %9s0 85K % . s

Paa $ = 18? & = 48

18? = Z . a . 48G? a?= 6846 ?;8? = 15 % . s

Paa $ = 98 & = 84 %:

98 = Z . a .84G? a= 84 ?9816 = 56 % . s

Paa $ = 1844 & = 68 %:

1844 = Z . a . 68G? a4= 14?8 9869 = =5? % . s

Paa $ = 116864 & = 8? %:

116864 = Z . a . 8?G? a= ?8? 184 = 45 % . s

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57/137


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58/137


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59/137


:=@0 15 %.s1F 65 %.s10 %.s1

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& = 1?8 ?G .%1 ?8 .%? ! 0 45 s

#G :)0 :o . & . e ; = 1?8 .%1G ? . ?8 .%?G . $

; = 168? ?. %? .%?. $ .%?. $ = 168? ?.%?

$ = 168? ?.%? .%? e 0 =15

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60/137


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$ = ?; .%?. %? e 0 %

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61/137

V(m%s-1)

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62/137


aG D$%'0$ #'7( (/$!&( '(! #$&a""$.0G Ca"'+"a "a a'$"$a')! $! 'a#a &a('G Ca"'+"a "a #%&a!'a &(&a" *+$ $'($.

P(0"$a $%+$"&( N ;U! P(%'7$ /a3a a +!a /$"('#a# #$ 166 78 $" '(!#+'&( a#/$&$*+$8 $! $#( #$ "a 'a$&$a8 7a +! !`( 3+2a!#( a "a% 'a!'a%.S+,(!$!#( *+$ !'a "a f$!a#a '+a!#( %$ $!'+$!&a a 9; #$" !`(8 *+$ "(% f$!(% $!&$2a! +!a a'$"$a')! +!f($ #$ 1? ]%?: S$%a"/a $" '7*+""(

Resouci*(

E" S%&$a #$ $f$$!'a *+$#aa #$ "a %2+$!&$ f(a:

C('7$ N`(9;

K = ;

V( = 166 7 . 1;;; 1 . 1 7 6;;; % = 46811 .% 1

C+a!#( $" '(!#+'&( %$ # '+$!&a #$ "a $%&$!'a #$" !`( a,"'a- "(%f$!(%8 *+$ "$ ,(,('(!a! +!a a'$"$a')! #$ 1 %.s8 $! $"$!+!'a#( !( a,a$'$ $" %2!( $!(%8 ,$( +!a f$!a#a %$,$,"'a +!a #%!+')! #$ "a /$"('#a# ,( "( &a!&( "a a'$"$a')!%$- !$2a&/a.A" a,"'a "(% f$!(%8 7a%&a ,aa%$ Vf= ;G8 $" '('7$ 7a $'(#( +!$%,a'( #$:

:)0 :8 . & . e; = 46811 .%1G? ? . 1? .%?G . $ ; = ?1?681 ?.%? ?4 .%?. $

P(f$%(: A. >aa2(


63/137


?4 .%?. $ = ?1?681 ?.%? e 0 1?51= %.s9 4 %.s0 5? %

C(( $" !`( %$ $!'(!&a0a a 9; NO SER ATROELLADO OREL COCHE.

P(0"$a $%+$"&( N 1U!a (&(''"$&a %$ +$/$ %$25! "a $'+a')!: =?; 1;& ;8&?

aG Ra


64/137


:)0 :o . & . e

; = 1; .%1G? ? . 1 .%?G . $ ; = 1;; ?.%? ? .%?. $

? .%?. $ = 1;; ?.%? e 0 188 %.s9 %.s0 8 %

E! "( $f$$!&$ a" &$,( *+$ &a#a $! #$&$!$%$8 "a $'+a')!:

:)0 :8 & . !

1 = 1; .%1 1 .%?G . & ; = 1; .%1 1 .%?. &

? .%?. & = = 1; .%1 & = 1; .%1 1 .%?

! 0 18 sE3$''( $%+$"&( N ? F+$!&$ E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(


65/137


P( (&a ,a&$ %a0$(% *+$:

a = Vf V( & ! 0 :) F :o 9 &G

L"$/a(% G a ?G:

;86 = ?8 . ; ?8Ga Z . a . ; ?8 G? a?

;86 = 181 a 8 a ;86 a = 181 8

;86 a = 89 a = 89 ;86 &= ?45? % . s

E! "( $f$$!&$ a" &$,(:

a = Vf V( & ! 0 :) F :o 9 & != ; ?8G 64?86G = 8584= s

E3$''( $%+$"&( N

U! '(!$3( '($ 7a'a %+ a#+2+$a a ? 7. C+a!#( $%&- a ?;; #$ $""a +! ,$(8 %&+a#( a 4; #$&-% #$" '(!$3(8 %a"$ $! %+,$%$'+')! $'($!#( 9; '(! +!a a'$"$a')! #$ %?'(!&!+a!#( #$%,+% a /$"('#a# '(!%&a!&$.

aG A"'a!


66/137


P$( C(!$3( Ma#2+$a?;; Caa'&$%&'a% #$" (/$!&( #$" ,$(:

a,$(= %?V( = ; M.R.U.A$ = 9;

E" ,$( '(!%$2+- +!a /$"('#a#$ f!a" #$:

:)

#erro0 :o#erro . & . e V(,$(= ;

Vf?,$(= ? . . 9; :)#erro= ? . . 9; G1?= =8 %9s

Ta#a- +! &$,( $! $'($ "(% 9; :e 0 :o . ! . & . ! e 0 . & . !

9; = Z . . &? != 9;?8G1?= ? s

Caa'&$%&'a% #$" (/$!&( #$" '(!$3(:V = '(!%&. = ?; %M.R.UE! $%&(% 6 % $" '(!$3( M.R.UG $'($- +! e:

e 0 : . ! e= ?; % . 6 % = 18 %A ,a& #$ $%&$ ($!&( co(ejo #erro si-ue( %o


67/137


1;

E" '(!$3( &a#a- $! a"'a!


68/137


Ta#a- +! &$,( $! $'($ "(% 9; :e 0 :o . ! . & . ! e 0 . & . !

9; = Z . . &? != 9;?8G1?= ? s

Caa'&$%&'a% #$" (/$!&( #$" '(!$3(:

V = '(!%&. = ?; % M.R.U

E! $%&(% 6 % $" '(!$3( M.R.UG $'($- +! e:

e 0 : . ! e= ?; % . 6 % = 18 %

La !+$/a %&+a')! $%:

A #&r!ir 'e es!e %o%e(!o co(ejo #erro se %ueaa2(


69/137

&(m/s)

20

10

5

t(s)


E3$''( $%+$"&( N 4

E" (/$!&( #$ +! )/" /$!$ #a#( ,( "a 2-f'a:

; 1; ?; ? 4;

D$&$!a ,aa 'a#a !&$/a"( #$ &$,(:E,"'a "a 2-f'a #$&$!a $" &,( #$ (/$!&(.A'$"$a')!.

E%,a'( $'(#(.E%,a'( &(&a" $'(#(

Resouci*(

INTERVALO %G

; 1;E" )/" ""$/a +!a /$"('#a# !'a" *+$ /a a+$!&a!#( #$ f(a

U!f($8 "( *+$ !(% !#'a *+$ $! $%&$ !&$/a"( $%&$ +!M.R.U.A.

INTERCALO %G

1; ?;

La /$"('#a# ,$a!$'$ '(!%&a!&$8 1; %. E%&( !(% !#'a *+$ $"(/$!&( $! $%&$ !&$/a"( $%M.R.U.

P(f$%(: A. >aa2(


70/137


INTERCALO %G

?; ?

La /$"('#a# /+$"/$ a a+$!&a #$ f(a +!f($ 7a%&aa"'a!


71/137


V( = 1; % Vf = ?; % & = %

& 0 :) F :o 9 ! &= ?; 1; = 15 % . s

INTERCALO %G

? 4;

Vaa')! #$ /$"('#a#8 $%&$!'a #$ a'$"$a')!.

V( = ?; % Vf = ; & = 1? %

& 0 :) F :o 9 ! &= ; ?; 1? = 15?? % . s

INTERVALO %G

; 1;

e 0 :o . ! . & . !

e= . 1; Z . ;8 . 1;; = ; ? = 6 %

INTERVALO %G

1; ?;

e 0 : . ! e= 1; . 1; = 188 %

INTERVALO %G

1; ?;

e 0 :o . ! . & . !

e= 1; . 1; Z . 18? . 1;; = 1;; 6?8 = 1?5 %

INTERVALO %G

P(f$%(: A. >aa2(


72/137


? 4;

e 0 :o . ! . & . !

e= ?; . 1? Z . 1866G . 144 = ?4; 1198? = 1854 %

eT0 e1 e e= e40 1;; 16?8 1?;84=465K %

E3$''( $%+$"&( N F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(


73/137


= 1;;; 1;;;;;; 6;;;;G1? ?

= 1;;; ;; ?

G1= 1;;; ;; ? = K88 % NO ES :LIDA XISICAMENTE ,+$%&(*+$ "a %$,aa')! $!&$ "(% )/"$% $% #$ ;;

G= 1;;; ;; ? = 188 %

E" $!'+$!&( %$ ,(#+'$ a 188 % 'e A.

E3$''( $%+$"&( N 6F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(


74/137


!%o!oc.0 !c&%i.0 !

M(/$!&( #$ (&('.: M.R.U.A

e0 . & ! $= Z . ? . &? e=!1G

M(/$!&( #$" 'a.:M.R.U

e 0 :c&%i . ! e 0 1 . !1G

U!$!#( "a% $'+a'(!$% 1G ?G f(a(% +! %%&$a:

$ = &?$ = 1 . & ! 0 e 9 1 e= 1G? e= %

S$ $!'(!&a-! a?? #$" %$-f((.

V((&('. = ; Vf= a = ? %?

R$'(#$(% *+$:

Vf?= V(? ? . a . $ Vf?= ? . a . $ Vf?= ? . ? . ?? =9;;

:X= 9;;G1?==8 % . s1

P(0"$a $%+$"&( N S$ "a!aa2(


75/137


2 = 98 .%?

Vf = ; C(( "a /$"('#a# #%!+$ "aVf = ; a'$"$a')! $% !$2a&/a ,( &a!&(2 = 981 .%?

2 = 981 .%?

V( = ? .%1

E% +! "a!aa2(


76/137


= 68 18 = =1PK %

E" &$,( $,"$a#( $! 0a3a %$ ,+$#$ (0&$!$ $%&+#a!#( $"(/$!&( #$ 'a#a "0$ /;= ;8 a = 2 = 9\ %?G. La% $'+a'(!$% %(!"a% #$" M.R.U.A.:

e 0


77/137

V0= 0 ( se e 'a ae )

* = 30 m

+ = 9,81 m%s-2


S$ &a&a #$ +! (/$!&( M.R.U.A ,( "( &a!&( ,aa 'a"'+"a $"

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


78/137


L( ,$( *+$ 'a"'+"a$(% %$- "( *+$7 = ; V;= ? .%1 &a#a $" '+$,( $! ,aa%$ VfA= ;G.P(#a ('+ *+$ $! ? % $" '+$,( a"'a!'$ "a -a a"&+a $%& 0a3a!#(:

7 = ; = ? .%1 981 .%?G . &

981 .%?. & = ? .%1S+$"( ! 0 %.s19 K51 %.s0 5 s.

A" 'a0( #$ ? % $" '+$,( %2+$ %+0$!#(8 "+$2( %+ /$"('#a# %$-:

:)0 :o - . ! :)= ? .%1 981 .%?G . ? % =

= ? %1 1986? .%1= 5= .s1.

0G E" &$,( ,aa a"'a!


79/137

$


a= 189 2 = 981 .%?

7 = 189 = 689

7 =

D$ "(% % %$ 7a! '(!%+#( ?8 %. E" '+$,( $,$


80/137


'G S$ 'a"'+") $! $" a,a&a#( aG. A "(% ?8 % #$ !'a#( $" (/$!&(

P(0"$a $%+$"&( N 6;La!


81/137


b 8 %.s17 = ; Vf =

""$2a a" %+$"( '(! "a %a /$"('#a# '(! "a *+$ ,a&) %$,$*+$ !( a'&5$ f+$


82/137


e 0 : . ! e= 18 % . ;8? % = 151= %

La '(!'"+%)! $% *+$ $" $'$$!&( (o c&e e(ci%& 'e 'e & c&$e/& 'e& #erso(&.

0GLa /$"('#a# *+$ #$0$ ""$/a "a ,$%(!a #$,$!#$- #$" $%,a'( a

$'($ #$" &$,( $,"$a#(.

E" $%,a'( $% #$ ? $" &$,( $% ;8? %:

: 0 e 9 ! := ? ;8? % = 54= %9s

E3$''( $%+$"&( N 6?D$%#$ +!a a


83/137


; = 6? 1986? 7 ,= 6? 1986? = =5 %

Ta#a +! &$,( $! a"'a!


84/137


P$#a 1: = Z . 2 . &? 1G P$#a ?: 98?9 = V1,?. & Z . 2G . &?

?G

D$ 1G #$%,$3a(% W&X "( ""$/a(% a ?G:

& = ? 2 G1? 98?9 = 8;9 . ?2G1? Z . 2 ?2G1??

98?9 = 8;9 . ?2G1? 98?9 = 8;9 . ?2G1?

E"$/a!#( "(% #(% $0(% a" '+a#a#(:

68 = ?89 . ?2 68 2 = 18 68 = 18

= 6818 : G= 15?? %

Ta#a-! $! $!'(!&a%$:& = ? 2 G1? & = ? . 1866 981 G

1?

!T= ?84 &1 &?= ?84 ;8 ;8 = =56 s

La ,$#a ? %+0) $! ;8 % +!a a"&+a #$:

,= 98?9 1866 =65?= %

S$ $!'(!&a-! #$" %+$"( a +!a a"&+a #$:

H= 4486 86 =51K %

0G

La ,$#a ! 18 $! $" ,+!&( #$ $!'+$!&( a"'a!


85/137


La ,$#a ? a"'a!


86/137


7,$#a1 = 7 = Z . 2 . & ;8?G?

P$#a ! ?

7,$#a?= ; 7 =V(,?. & Z . 2 . &?

U!(% "a% #(% $'+a'(!$%:

7,$#a1 = 7 = Z . 2 . & ;8?G? 1G S$ f(a +! %%&$a #$

#$ $'+a'(!$%. L"$/a(%,7,$#a? = ; 7 =V(,?. & Z . 2 . &

?

?G #$ "a 1G "( ""$/a(% a "a ?G:

; Z . 2 . & ;8? G?= 1 . & Z . 2 . &?

6; 2 . & ;8? G?= ; .& 2 . &?

6; 981 . &? ;8;4 ;84 & G = ; & 981 &?6; 981 &? ;89 89? & = ; & 981 &?

1986? &? 89? & ; & ;89 6; = ;

1K5? ! ==5K ! F K5?1 0 8

& = 89? 11;86 46819G1? 98?4

& = 89? 684 98?4

!1= 89? 684 98?4 = 158 s

&?[ ; NO TIENE SENTIDO XZSICO

S$ $!'(!&a-!:

,= Z . 981 8 18;G?= 56 %.

P(f$%(: A. >aa2(


87/137


; 8? =45 % 'e sueoT(a!#( $" %+$"( '(( %%&$a #$$f$$!'aG.E3$''( $%+$"&( N 64 F+$!&$ #$" E!+!'a#(: D,&( #$ F #$" IES E" E%'(a". R$%("+')!:

A.>aa2(


88/137


S(!#(

7 = V%(!#(. &?

C(( "(% #(% ,$(% $0(% #$ "a% $'+a'(!$% a!&$($% %(!

2+a"$%8 "(% #(% %$2+!#(% &a0! "( %$-!:

Z . 2 . &1?= V%(!#(. &? ?GS &$!$(% $! '+$!&a "a $'+a')! 1G ,(#$(% $%'0 *+$:

!0 !1L"$/a(% &?a "a $'+a')! ?G !(% *+$#a:

Z . 2 . &1?= V%(!#(. ? &1G

S+%&&+$!#( #a&(%:

Z . 981 . &1?= 4; . ? &1G

489 &1?= 6; 4; &1 45K !1 =48 !1F ?8 0 8

!1= 4; 116;; 1?G1? 981

!1= 4; 98;9 981

!1= 15K4 s

P(f$%(: A. >aa2(


89/137

1

1

1

2

2


La %$2+!#a %("+')! #$ "a $'+a')! !( &$!$ %$!( f%'( ,+$%&( *+$!(% ,(,('(!a +! /a"( !$2a&/( ,aa $" &$,(.

La ,(f+!##a# #$" ,(


90/137


E" $%*+$a a!&$( $% +!a $,"'a')! +#a #$ "( *+$ $%&- ('+$!#($! !+$%&a $,$$!'a. A7(a ,a%a$(% a" '-"'+"( #$ /aa0"$%$%&a0"$'#a% $! $" '(*+% a!&$(.

C+$,( N 1

V('1 = 1; %

aG

M-a a"&+a a"'a!


91/137


V$"('#a# a"'a!aa2(


92/137


S #$%,$3a(% #$ G $" &$,( ! "( ""$/a(% a "a $'+a')! 4G8 !(%*+$#a:

! 0 , 9 45K@19

489 7 = 981 . 7489G1? Z . 981 . 7489G1??

489 7 = 981 . 7489G1? 489 . 7489G

489 7 = 981 . 7489G1? 7

$#+'$!#( &!(% %$$3a!&$%:

489 = 981 . 7489G1?

$"$/a!#( a0(% $0(% #$ "a $'+a')! a" '+a#a#(:

489G?= 981G?. 7489 ?48;1 = 19864 7 ,= ?48;119864 = 15 %

La a"&+a *+$ %+0$ $" '+$,( N? a" $!'+$!&( #$" N 1 $%:

,c= 489 18?? = =5? %

L(% '+$,(% %$ $!'+$!&a!:,ci(ici&@ , F 45K@= ;819 86 = 8 So$re e sueo +ue es (ues!ro sis!e%& 'e re)ere(ci&@

E! "( $f$$!&$ a "a /$"('#a# '(! "a *+$ ""$2a! "(% '+$,(% a" ,+!&( #$

$!'+$!&(:

C+$,( N 1

T$,( #$ 0a3a#a:

! 0 ,945K@19 != 18??489G1?= 854K s

:e(cue(!roc1

0 :oc1

- . ! :e(cue(!roc1

= ; 981 . ;849 = 45 %.s1

P(f$%(: A. >aa2(


93/137

a)

Vo+oo

5 m/s=

* = 200 m


C+$,( N ?

T$,( #$ %+0#a = 854K s

V$"('#a# #$ $!'+$!&(:

:e(cue(!roc= Vf'? 2G . & = 981 981 . ;849 = % . s1

E3$''( $%+$"&( N 66U! 2"(0( a%'$!#$ '(! +!a /$"('#a# '(!%&a!&$ #$ %. C+a!#( %$

$!'+$!&a a ?;; #$ a"&+a8 %$ #$3a 'a$ +! "a%&$. D$%,$'a!#((aa2(


94/137


V("a%&$ = %

Ca"'+"$(% "a a"&+a a"'a!


95/137

naao

.o


E" &$,( &(&a" $! ""$2a $" "a%&$ a" %+$"(:

!T0 !su$i'&&s!re !$&j&'&&s!re& sueo 0= ;8 684 =?5K s

0G

La /$"('#a# '(! "a *+$ ""$2a a" %+$"(:

:)&s!re0 :o&s!re - . !

Vf"a%&$= ; 981 . 684 =?5K6 % . s1

P(0"$a $%+$"&( N 6U! !a#a#( *+$$ a&a/$%a +! ( #$ 1; #$ a!'7+a '(! +!a/$"('#a# #$ %. La '($!&$ #$" a2+a ""$/a +!a /$"('#a# #$ ?8%. D$&$!a:La /$"('#a# '(! "a *+$ a&a/$%a- $" )E" -!2+"( #$%'&( ,( $" !a#a#( $! %+ #$%,"a


96/137

n


La /$"('#a# $%+"&a!&$:

7 : 70 7


97/137

Vo = 100 m/s

* = 1000 m



98/137


V$7'+"( ,aa#( V(= 1;; %7 = 1;;;

A" *+$#a "0$ "a 0(0a &a#a- $! 'a$ "(% 1;;; #$ a"&+a:

, 0 :o . ! . - . ! V( = ;

, 0 . - . ! 1;;; = 489 &? != 1;;;489G1?=145 s

C(! $%&$ &$,( "a 0(0a $'($- +!a #%&a!'a 2+a" a" a"'a!'$:

G 0 :oG . ! G= 1;; % . 148? % =14 %

E" a/)! #$0$- #$3a $! "0$&a# "a 0(0a 14 %a!&$% #$ ""$2a a"(03$&/(.

0G

E%&$ a,a&a#( $% + a02+( ,+$%&( *+$ !( %a0$(% $! *+ ,(%')!

%$ $!'+$!&a! $" a/)! '(! $%,$'&( a" )/".

S+,(!2a(% )/" $! (/$!&( '(! +!a /$"('#a# #$ ? 7 = ?;%.

? 7 . 1;;; 1 . 6;; %1 7 = 8 %9s

C(! $%&a /$"('#a# $" )/" $'($- +! $%,a'(:

:%*


99/137


1;;; = 489 a"'a!'$V)/"G? 1;;; = 489 . a"'a!'$??;?

Ac&(ce= 4;; . 1;;; 489G1?=561 %

E" a/)! #$0$- %("&a $" ,($'&" $! $" ($!&( #$ ,a%a ,( "a


100/137


G 0 :%*


101/137


18 1? =5= %

E%&a "(!2&+#8 8 8 ,"'a +!a a"&+a #$ '7(*+$ *+$ $% "( *+$ !(%,#$ $" ,(0"$a. La ,$##a #$ #$%,"a


102/137

esoin

Vo

* = 80 m

Aane = 60 m


,c,o+ue0 ,TF ,c,o+ue01 8? = 65 %

E3$''( $%+$"&( N ; F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(aa2(


103/137

Atit e a&in

Vo = 500 m/s

Vo = 0

* =


A *+ a"&&+# /("a0a $" a/)!.

+ #%&a!'a $'() "a 0(0a 7(


104/137

Vo = 500 m/s


R$'(#a *+$ $" &( 7(aa2(


105/137


:0 :oG :) V = ;;G? 84G?1?= ?;;;; 619811G1?

:=8?51 % . s1E3$''( ,(,+$%&( F+$!&$ #$" E!+!'a#(: D,&(. #$ F #$" IES La A%+!')! #$ E"'7$GD$%#$ +! ,+!&( %&+a#( a 1;; . %(0$ $" %+$"( %$ #%,aa7(aa2(


106/137


:oG= 118;? ;89; G1?= =5 % . s1

E3$''( $%+$"&( N U! !`( #a +! ,+!&a, a + 0a")! *+$ $%&- a ?; ' #$" %+$"(8 '(! +!-!2+"( #$ 6; %(0$ "a 7(aa2(


107/137


?8 = V(?. %$!?6; 2 Z . 2 . V(?. %$!?c 2?

?8 = V(?. %$!?c 2 Z . V(?. %$!?c 2

?8 = Z . V(?. ;84 981 ?8 = ;8;4 . V(? ?8 = ;8;4 V(?

:o= ?8 ;8;4 G1?=54 % . s1

E3$''( $%+$"&( N 4S$ "a!


108/137


c = ;( V(V(

ESTADO II Vf?7 = 1; 7-.

ESTADO I

R$'(#$(%:

V( = V( . '(% cV( = V( . %$! c

E" &$,( !$'$%a( ,aa #$%a(""a $" A"'a!'$ M-( $% M.R.U.G:

Ac&(ce M;Gi%o 0 :oG . ! Ac&(ce M;Gi%o 0 :o . cos [ . !G

E" &$,( ,aa #$%a(""a $" a"'a!'$ -( $% $" #(0"$ *+$ $"'($%,(!#$!&$ $! a"'a!aa2(


109/137


:)0 :)1 - . ! :)10 8 :)0 - . ! ! 0 :)9 -

E%&$ !+$/( &$,( $% $" +&"


110/137


A!ur& %;Gi%& &c&(/&'& 0 ,es!&'o I ,es!&'o II = ?;4;8? 1; =

= 1K85 %E3$''( $%+$"&( N F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(


111/137


La /$"('#a#8 $! $" $3$ OY8 $! $" ,+!&( -% a"&( #$ "a &a$'&(a /$!$#a#a ,( "a $'+a')!:

:) 0 :o -@ . ! :) 0 8 ; = V( 2 . &

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


112/137


!0 ?88 . se( [9- !G0 != . ?88 se( [9-

S /("/$(% a "a $'+a')! #$" a"'a!'$! -(:

Ac&(ce %;Gi%o 0 :oG . !G

A"'a!'$ -( = V( . '(% c . ? . 6;; %$! c2

A"'a!'$ -( = V( . 6;; . ? %$! c '(% c2 = V( . 6;; . %$! ?c2

1;;; = 6;; . 6;; . %$! ?c2 %$! ?c = 1;;; . 2 6;;;;

se( [= 16;6;;;; = 854K

?c =?98 [=145?

E3$''( $%+$"&( N 6 F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(aa2(


113/137

Vo Vo

m%

Aane imo


7

c V(

E" A"'a!'$ -(8 '(( '(!%&a $! $" #0+3(8 $% $'(#( ,( $" )/"$! $" $3$ O'(!M.R.U:

Ac&(ce M;Gi%o 0 G%;G.0 :oG . !G 1G

S$25! $" &-!2+"( #$ "a f2+a:

:oG 0 :o . cos [

La $'+a')! #$" a"'a!'$ -( *+$#aa #$ "a f(a:

G%;G.0 :o . cos [ . !G ?G

E" /a"( #$!G"( ,(#$(% '(!('$ ,a&$!#( #$" /a"( #$" &$,(!$'$%a( ,aa (0&$!$ "a a"&+a -a -.G:

%;G.0 :o . ! . -@ . ! %;G.0 :o . !F . - . !

Paa +!A"'a!'$ M-( "aA"&+a M-a #$0$ %$ 2+a" acero:

R$'(#$(% *+$ #$" &-!2+"( #$ "a f2+a:

:o 0 :o . se( [

P(#$(% $%'0:

P(f$%(: A. >aa2(


114/137


; = V( . %$! c . & Z . 2 . &? ; = & V( . %$! c Z . 2 . &G

V( . %$! c Z . 2 . &= ; ? . V( . %$! c = 2 . &

!0 :o se( [9-

Paa $%&$ /a"( #$ &$" a"'a!'$ %$- -(:!0 !G

G%;G.= V( . '(% c . ? . V( . %$! c 2 =

= V(?. ? %$! c . '(% c2

G%;G.0 :ose( [9-

S$25! "a 5"&a $'+a')! "a-.#$,$!#$ #$"%$! ?c. E" /a"( -( #$"%$!( #$ '+a"*+$ -!2+"( $% "aUNIDAD.L+$2(:

se( [ 0 1 ?c = 9; [= 9;? =4

E" a"'a!'$ %$- -( ,aa +! -!2+"( #$4.

E3$''( $%+$"&( N F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(aa2(


115/137

Vo Vo

m%500 m=

Vo0

m%


S$25! $" &-!2+"( ;V(V(:

V( = V( . '(% c V( = V( . %$! c

R$'(#$(% *+$ $! $" $3$ OY $" (/$!&( #$" ,($'&" $% M.R.U.A:

:)0 :o . -@ . %;G. :) 0 8 ; = V(? ? . 2 . -.

V( = ? . 2 . -.G1? :o= ? . 981 . ;; G1?= KK584 %9s.

E" &$,( *+$ &a#a $! a"'a!


116/137


V$'&(a"$!&$ %$ '+,"$:

:o 0 :oG :o 7 :o7 0 7 :oG7 7 :o7

7 :o 7= 198;?G? 998;4 G?1? V( = 6?1189 9;89?G1?

:o 0 145 % . s1

0G

c

S$ '+,"$:

:o 0 :o . se( [ se( [= V( V( = 998;4 ?148? =854?

[ 0 65 'GE" &$,( #$ /+$"( '(!'#$ '(! $" &$,( $! $'($ $" -.:

!aa2(


117/137

Vo Vo

= 30


V(

:)0 :o . -@ . %;G

E! $" ,+!&( #$ -a a"&+a Vf = ;. La $'+a')! a!&$( *+$#a #$ "af(a:

; = V(? ? . 2 . -. V( = ? . 981 . -.G1?

Sa0$(% *+$: V( = V( . %$! c ,( "( *+$ "a $'+a')! a!&$( *+$#a #$"a f(a:

V( . %$! c = 1986? -.G1?

E"$/a!#( a0(% $#(% #$ "a $'+a')! a" '+a#a#( !(% *+$#a:

V(?. %$!?c = 1986? -.

V( = ; % c = ;(

?;; . ;8? = 1986? -. 6? = 1986? -. -.= 6? 1986?

%;G.0 =15 %E" &$,( !$'$%a( ,aa a"'a!


118/137


0GE" &$,( #$ /+$"( '(!'#$ '(! $" &$,( !$'$%a( ,aa $'($ $"A"'a!'$ M-(8 -.. E" -.%$ $'($ $! $" $"$ OK '(! M.R.U.

G%;G0 :oG . !G

E" &$,( #$ /+$"( $% $" #(0"$ *+$ $" &$,( !$'$%a( ,aa a"'a!


119/137


-.= V( . & -.= V( '(% c . &

R$'(#$(% *+$:

&= ? . & !0 !G9

"( ""$/a(% a "a $,$%)! #$ -.:

%;G.0 :o se( [ . !F . - . !

%;G.0 :o se( [ . !G9 F . - . !G9@

Sa0$(% *+$:G%;G.0 :o . cos [ . !G

#$%,$3$(% &:

!G0 G%;G.9:o . cos [

-.= V( . %$! c . -.V( . '(% cG? Z . 2 . -.V( . '(% cG??

-.= &a2 c . -.? Z . 2 ?-.V(?. '(%?cG4

-.= ;8? -. 489 ?-. V(?. ;81

-.= ;8? -. ?8? ?-. V(?

D$ f(a 2$!$a" ,(#$(% $%&a0"$'$ *+$ "a $'+a')! #$ "a &a$'&(a$%:

0 85 G F 65 G9:o

P(f$%(: A. >aa2(


120/137

A

B

A


P(0"$a $%+$"&( N 9D$f!$ a#-! '(( +!#a# #$ $##a #$ -!2+"(%.C+-!&(% a#a!$% 7a $! +! -!2+"( #$ 1;; C+-!&(%2a#(% '(!&$!$ +! -!2+"( #$ @? a#a!$% C+-!&(%a#a!$% %(! ;'+-!&(% 2a#(% %$a2$%a"$% %(! 1 a#-!

Resouci*(

R&'i&( es e


121/137


Da&(%: A= ? B= \ /A= 6 % /B = AS$ &a&a #$ +! M.C.U8 ,( &a!&(8 / = d .

VA= dA . A6 = dA . ?dA= a#%.

C(( A B %$ $!'+$!&a! $! "a %a ,"a&af(a 2a&(a8 7a! #$2a "(% #(% '(! "a %a /$"('#a# a!2+"a8 ,$( #%&!&a /$"('#a#"!$a" ,( $%&a a #f$$!&$% #%&a!'a% #$" '$!&( ,( &a!&(8 $'($

''+!f$$!'a% #f$$!&$% a" %( &(.

dA= a#% dB= a#%

D$ $%&$ (#(:

VB= dB]B VB= ]\ VB= 1;\ %

P(0"$a $%+$"&( N 1U!a +$#a 2a a aaa2(


122/137


1 /+$"&a?;;; @ a# . =1=88


123/137


La /$"('#a# a!2+"a "a 'a"'+"a$(% #$ "a f(a:

V = d . R d = V R = 11811 %G ;8; = 8; a#%

Paa '(!('$ $" ,$(#( +&"aa2(


124/137


1 7 . 6;; % 1 7 = 6;; %.

1 /+$"&a = ? @ a#

\ 0 2 9 T 0 2 r&' 9 =?88 s 0 5 . 184r&' 9 s

A2+3a %$2+!#$(:D$%'0$ +!a /+$"&a '(,"$&a $! +! !+&(.

/+$"&a ? @ a# 1 !

d = 1 . . =8586 2 r&' 9 s! 1 /+$"&a 6; %

R$'(#$(% *+$:V = d . R 1G

' . 1 1;; ' = ;8;

V("/$!#( a "a $'+a')! 1G:

: 0 8586 2 r&'9s . 858= % 0 51 . 18=%.s1

C+$%&)! $%+$"&a N R$%,(!#$ 0$/$$!&$ a "a% %2+$!&$% '+$%&(!$%:aG D(% +$#a%8 +!a 2a!#$ (&a ,$*+$`a8 2a! '(! "a %a /$"('#a#

a!2+"a. '+-" #$ $""a% #a -% /+$"&a% $! $" %( &$,(0G '+-" #$ "a% +$#a% #$" 'a%( a!&$( &$!$ a( /$"('#a# "!$a"

aGRAe RB dA= dB &A= &B

d = $%,a'( a!2+"a & $%,a'( a!2+"a c G = d . &

E" c $% $" %( ,aa "a% #(% +$#a% dA= dB8 &A= &BG '(( $"

!5$( #$ /+$"&a% #$,$!#$ #$ c:

P(f$%(: A. >aa2(


125/137


1


126/137


P(0"$a $%+$"&( N D$&$!a "a /$"('#a# a!2+"a #$ (&a')! #$ "a T$a a"$#$#( #$ %+$3$ "a /$"('#a# "!$a" #$ +! ,+!&( %&+a#( %(0$ $" $'+a#(8 %a0$!#(*+$ %+ ,$$&( $% #$ 4;.;;; .

Resouci*(

Da&(%: La T$a #$%'0$ +!a /+$"&a $! %+ (&a')! #$ ?4 7.

?4 7 6;; % 17 = 64;; %

4;;;; . 1;;; 1 = 4 . 1;

La /$"('#a# a!2+"a #$ (&a')! $%:

\ 0 2 9 T 0 2 r&' 9 ?488 s 0 5= . 18r&'9s

E" ,$$&( '(!'#$ '(! "a "(!2&+# #$ "a &a$'&(a. La &a$'&(a $%+!a ''+!f$$!'a %+ "(!2&+# /a"$:

L = ? @ R R 0 L 9 2 0 4 .186%9 2 r&' 0 92 . 186%

Y '(( : V = d . R : 0 5= . 18r&'9s . 92 . 186% 0 856= %.s1

NOTA: E! $%&$ $3$''( ,$!%( *+$ $" #a&( #$ ,$$&( 2+a" a 4;;;; !( $% '($'&(.

P(0"$a ,(,+$%&(S %a0$(% *+$ "a #%&a!'a $#a S("T$a $% #$ 1;.;;;.;;; 8 %+,(!$(% *+$ %$ &a&a #$ +! (/$!&( ''+"a +!f($8 'a"'+"a "a%/$"('#a#$% a!2+"a "!$a" #$ !+$%&( ,"a!$&a. E,$%a "a /$"('#a##$ &a!%"a')! #$ "a T$a $! 7G.

P(0"$a ,(,+$%&(U! &(//( 2a #a!#( +!a /+$"&a 'a#a 11 %. R$a"aa2(


127/137


'G 'a"'+"a "a /$"('#a# '(! *+$ %$ #$%,"a


128/137


E3$''( $%+$"&( N 9 F+$!&$ E!+!'a#(: D,&(. F #$" IES E" E%'(a"GS(0$ +! ,+!&( #$ "a ,$f$a #$ +!a ,"a&af(a ''+"a 2a&(a #$ ;' #$ a#( %$ $!'+$!&a +! ,$*+$`( (03$&( *+$ 2a %("#aa$!&$ '(!"a ,"a&af(a. E" (03$&( ,(%$$ +!a a'$"$a')! '(!%&a!&$ #2#a 7a'a$" '$!&( #$ ? %?.

aG Ca"'+"a "a /$"('#a# a "a *+$ 2a "a ,"a&af(a.0G S %$ &a%"a#a $" (03$&( $! #$'')! a#a" 7a%&a %&+a"( a 6; '

#$" '$!&( Vaa- %+ a'$"$a')! E! 'a%( afa&/( 'a"'+"a $"!+$/( /a"(.

Resouci*(

aGR = ; ' . 1 1;; ' = ;8;

La a'$"$a')! *+$ a'&5a %(0$ $" '+$,( $% "a &(:

&(0 :9R ? = V?;8; V = ? . ;8; G1?

an

: 0 58? % .s1

0G

SI8 ,+$%&( *+$ /aa $" a#(:

R = 6; ' . 1 1;; ' = ;86;

&(0 :9R

C(( $" ,(0"$a !( #'$ !a#a $%,$'&( a "a /$"('#a# #$0$$(%&(a"a '(( '(!%&a!&$ 8;6 %G:

&(= 8;6G?;86; =45?6 % . s

P(f$%(: A. >aa2(


129/137


E3$''( $%+$"&( N 9;U!a +$#a #$ 1 ' #$ a#( %$ ,(!$ $! (/$!&( '(! +!aa'$"$a')! a!2+"a #$ ;8? a#%?. Ha""a $" &$,( *+$ &a#a "a +$#a $!#a ?; /+$"&a%.

Resouci*(

N( $%&$ $'+a')! a"2+!a *+$ !(% #$&$!$ $" !5$( #$ /+$"&a%. P$(%a0$(% *+$ u(&


130/137


Resouci*(

= ?4 a# &= %gf = 1; a#%

aG

^) 0 ô [ . !

D$%,$3a$(% #$ "a $'+a')! a!&$( g(:

ô 0 ^) F [ . ! 1G

"( ""$/a$(% a "a $'+a')! ?G:

_ 0 ô . ! . [ . ! ?G = gf c . &G . & Z . c . &?

= gf . & c . &? Z . c . &?

?4 = 1; . c . ? Z . c . ? ?4 1 = c . 9 ;8 . c . 9

4 = 9 c 48 c 4 = 48 c [=156 r&'. sV("/$(% a "a $'+a')! 1G:

ô= 1; 18 . = 1; 681 = 15K r&' . s1

0G

& = 18 %gf = ;g( = 189 a#%

^) 0 ô [ . ! ; = 189 c . 18 189 = 18 c [==45? r&'9s

P(f$%(: A. >aa2(


131/137


'G

C(!('$$(% $" $%,a'( a!2+"a #$%'&( 7a%&a *+$ %$ ,aa:

= 189 . 18 Z . 486G . ?8?_ 0 ^o . ! . [ . !_= 8 89? ==5K= r&'.

R$'(#$(% *+$:

1 aa2(


132/137


^) 0 ^o [ . ! ; = ?@ c . ?@ = c [= 8542 r&'9s

0G

_ 0 ^o . ! . [ . ! = ?@ . Z . ;84@G . ?

_= 1;@ @ = @ =156 r&'

R$'(#$(%:

V+$"&a = R$/("+'(! 1 aa2(


133/137


:i(e&0 ^ . R :i(e&= ?81 . ;8?; =858 % . s1

E3$''( $%+$"&( N 9 F+$!&$ #$" E!+!'a#(: D,&(. #$ F #$" INS E" E%'(a". R$%("+')!: A.>aa2(


134/137


Lo(-i!u'= . R = ?@ . R = ? . 814 . ;8; =851K %

'G

^85@0 ^o [ . ! ^85@= ; @ . ;8 =156 r&'9s

#G

C(( "a a'$"$a')! a!2+"a $% '(!%&a!&$8 "a a'$"$a')! &a!2$!'a"&a0! "( $%. La a'$"$a')! &a!2$!'a" "a 'a"'+"a(%:

&!0 &i(e&0 [ . R = @ . ;8; =858K %9s

La a'$"$a')! !(a" &$!$ %+ $'+a')!:

&(0 :9R 1GPaa & = ;8 %^ 0 156 r&'9s

:i(e&0 : 0 ^ . R= 18 . ;8; =85846 %9s

S !(% /a(% a 1G:

&(= ;8;4G?;8; =8586= % . s

E3$''( $%+$"&( N 94 F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(


135/137


_ 0 _o ^o . ! . [ . ! 1G

:) 0 ^) . R :) = . ;8? = 15? %9s

&i(e& 0 &!0 [ . R

E! $" M.R.U.A.:

:) 0 :o & . ! & 0 :) F :o 9 ! ?G

L"$/$(% a 1G &(#(% "(% #a&(% $*+/a"$!'a%:

= g( . 4 Z . a"!$a"R . 4?

?; = V(R . 4 Z . Vf V(&GR . 4?

?; = V(;8? . 4 . Vf V(4G;8? .

?; = ?; V( . 186 V(4 G;8?

?; . ;8? = ?; . ;8? V( ? . 186 V( G

4 = 4 V( ? . 186 V( G

4 = 4 V( 8? ? V( 4 8? = ? V( ;8 = ? V( :o 0 854 %9s

E! "( $f$$!&$ a "a a'$"$a')! &a!2$!'a" /a""a(% a "a $'+a')!:

&i(e&0 &!0 :) F :o 9 ! &!= 186 ;84 4 =85= %9s

E3$''( $%+$"&( N 9 F+$!&$ #$" E!+!'a#(: $3$''(%f*.'(. R$%("+')!: A. >aa2(aa2(


136/137


Resouci*(

g( = 9;; $/(".! . ?@ a# $/(". . 1 ! 6; % = ;@ a#% c= @ a#%?gf = ;

R$'(#$(%:

gf = g( c . & ; = ;@ @G . & @ . & = ;@ != 18 s

E! "( $f$$!&$ a" ! #$ /+$"&a% #$0$(% 'a"'+"a ,$( $" $%,a'(

a!2+"a #$%'&(:

_ 0 ^o . ! . [ . ! = ;@ . 1; Z . @G . ?= ;;@ 18@ =

_= ?68@ = KK5?1 r&'

La ,(,(')!:

1 aa2(aa2(


137/137


R = Z . D = Z . 1?; = 6; V( = ;g( = ; &= 1; %Vf = 9; 7 . 1;;; 1 . 1 76;; % = ? %

S$25! $" M.R.U.A.:

:) 0 :o & . ! ? = ; a . 1; &= 5 %9s.

E" $%,a'( $'(#( "( ,(#$(% 'a"'+"a:

e 0 :o . ! . & . ! V( = ; e= Z . ?8 . 1;?= 1 %

0G

La a'$"$a')! &a!2$!'a"8 a&8 %$ 'a"'+") $! $" a,a&a#( a!&$(:

&!0 & 0 5 % . s

'G

& 0 :9R & = ?G?6; = 18 4 % s

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