s s r

rtr s

rs

r

s s r rtr s

t

s ss rst s

tr

rs ss rs

s s rs ss

rss sts

rs rrt

rs rss sts

str

t s

rts

rs

stts rrs

rrs

rt

Prs t

str str sttst

r s

r

s r

s r

rt r rt

s rs rt

rt

rs

strs sttsts

str ss

str t tt

str trr rtr

rs

rs s rs

tr

str rs

s rs

ts tr s ts a b

t r

t s s qrs

rs

s s r rtr s

rs s rs

s s

s rt

P

s s r s s

s s

s s r s s

rs

s s r rtr s

rst

st tt stt tr r trtt s ts s ts st

tsts r t sr s r stts s rss ts

rtr s r t rrr s stts r sr

t r s r stts rts rrs s s ss

rts rts tr rtrs ss rtr sts rs

rs rs trs rs rts tt trr stt rr

tr sts s rtr s s ts s rtrs

r s ts q t r r ss tr rr r

r s ts trs s s s ts s s tr

sttst srs ss q stt q sr ss tr s

ss r s r s s s q ss rtr t

rss s rs r

s ts s ss rst s sr rs r q stt

ss s rr s ts ts s ss rs ss rs

rss sts rrs rrt t r r

s sr s ts sttsts sttsts ss r rst s stts

rrs rts rss s rs tr rs t rrs rts s

rtrt sts s s s rts ts rt ss

s ts rt rt t s rs st t

t r rs sr s strs sttsts trs rqtt ts

s s r rtr s r t s s trt

t s r rs rs rs ss t s ss

r tr rtrs ss trss

s s r rtr s

s s r rtr s

t

s ss rst s

tr

s t st tr ss s s s ts tr

s sttts trs tr s tr s ts ss ts Prtrt

sts t ss sss ss rs str

s rrs tr rts rrs rs sss s rs rt ss

rt t ss st t t st

st ts sr r str rt rr t sr

sr r s rs r r rtr sr sr s

ssr trss t str rs ts

rst r srr t st ts sr s rs s

rsts r rr ts t s str q r r r

r ts stts q s s sts s s qttts s

sr rsss r s tts ss r s str s st t q

s r qtttt s rs ss rtt rr s r

tt srr q rtt ts s trs ss s rrst rs s

rt r ss rtr s s ss s s s

ts ss ts rss s r trs sts Pr

q s s s rs s r t ts rtsts

r ts s t ssr rr qt trtt srt

tr t s trs rts s q s st s

s tr t t

rs ss rs

s s rs ss sr rsss trs s s ts r

s rr qt r tr qt s

r r s Pr tr r tr rss

ss rr rs s r r rs ss trs r tr 14a

r r sr Pss s rr s st rs ss ts

strs q stt s st tr s Prtrt

st trts st s trs rrs rs ss ts r

t ss t ss sst t sst tr r

s s

s r q rt 10 m ss qr r q rt qst rrs s rt r tr s s trs

rs r r trq s rs sts trs s st

s s r rtr s

s ss rst s

r

rt tr rt trt rrr rt tr

t 1/299.792.458 s

t s s r 9.192.631.770 rs r rrst trs

tr s s rs st t t s

ss r ss rtt tr qr stt sttt

tr Pss s rs r

rrt

tr

r ts rrt tr stt q t s

trs rs rts rt t s rr

sr sts st tr tr s r

tr sss s trs r 2 107 t r tr

rt

qt

tr

qt tr sst q t tts ts q

ts ts stts 0, 012 qrs r

trtr r 1/273, 16 trtr tr t tr

ts

s

ts s r t q t

r rt rq 5401012 rt rt s

ts rt ss r 1/683 tts tt

s r srr

Prs sst tr s

r s

r M L/T 2 1 kg m/s2 = Newton (N)r M L2/T 2 1 N m = Joule (J)Pt M L2/T 3 1 J/s = watt (W ) L/T m/sr L/T 2 m/s2

s M/L3 kg/m3

ss s s rs ss

sts s sts rs

st sts s rs s ts sts s s s rs

ts ts s ts 10 s rs sr s qqr ss103 s 1 milisegundo 1 ms 106 W 1 megawatt 1MW

s s rs ss

r s sr sr ss s trs

s rt s rt rt x r s X rst tr s t t s r s s tr

rt r r rrt tr rtr

ss tr t rtr

rt q rst r tt q r sr

trs rtt srs s tr v rr sr t trs st rrr x tr t t, sv = x/t

s s r rtr s

rss sts

t r

1018 t 1015 t 1012 109 106 r 103 102 t 101 101 102 t 103 106 109 1012 tr 1015 t P1018

Prs ts sts t 10

s rs ts s r q s r r s

rs st st qqr r s t q s r s rs

q s r sr r sttst r rrst

r rr r s s st sr rt s rr qt

r s s r rr r q sr t r

rt r rt s r tr tr rt s

s r rr r r rr r rr r s r

stt r rr rr st q r q s rt q sts

tt r rst r r rstr

ssr rr q s r q rrst r s Pr

s ss ssr ssr rr s r qqr

rss sts

r s sr r r s q s

rr r s rs q s r t st t rt r rss

q s s r rrstr s s r s ss s tr st s rss q

s t rt str rrts s rs s sss rss s s

rss sts s qt rss r s

rt r rss sts st rtt rs

r q qt s rs r r rss sts Pr

ss q s q rt s 64, 5 mm s q s rss 6 4s ts 5 rs s st rt st trst s rs s st s rss s sts sr sst

srs ts r tr rrts

s rss sts sr trs r s sts rrs

rs

rs st s rs sqr rr rs rt r ss

t qqr t s s s l = 32, 5 cm l = 0, 325 mrrst s t rss sts 5 = 0, 5 10 = 0, 05 102 =

s s r rtr s

s ss rst s

s r tr

0, 005 103 ss 1 rs st 26 = 2, 6 10 = 0, 26 102 = 0, 026 103 ss2 rss sts 0, 00034606 = 0, 34606 103 = 3, 4606 104 ss 5 rsssts

r rt rs st t rs st Prtt l = 32, 5 cm l = 32, 50 cm s rts s rr t 3 rss sts qtq s s rs t 4 rss sts

st r st tr rss sts Pr l = 3, 25 m t 3rss sts qt q l = 3, 025 m t 4 rss sts

s trt s tt s r r q 5 = 5, 0 = 5, 00 = 5, 000trtt q s rsts s s sr rr q 5 cm 6= 5, 0 cm 6=5, 00 cm 6= 5, 000 cm q sts s t 1 2 3 4 rss sts rstt trs rs rs s rt

rs rrt

rrt sr t rss rs r r

tr q rt sr ss

r sss rs s rrs rrt s

t rs r sr sr t s rs sr srt

s rr 423, 0012 = 423, 001

t rs r sr sr rs s rs

sr srt s srr 245, 6 = 246

s rs srt sr s s srt str qsqr

trs rss rts r t rs t sr rs

2, 0502 = 2, 1

s rs srt sr s s srt s str

rs str tr rs t rs t sr rs

st s r r s 4, 3500 = 4, 4 ; 1, 25 = 1, 2

rrt sr t st st s r rrt rr

t t r t rs r rrr t s tt

ss r t t r s qr rrt sr rs

rs rss sts

str

s s str s rs sr s rss sts

rst tr rs r r rs Pr st

r s

s s r rtr s

rs rss sts

12, 56 + 0, 1234 = 12, 6834 = 12, 68

r 12, 56 t qtr rss sts t rs st 6 q s s tss r 0, 1234 rst t qtr rss sts s trs st 4 s s s ss t rs st rst str s s r r rs ss 12, 56Prtt t rs st rst str s s tss

t s

t sr s rss sts rst tr s

r rss sts r r qt rss

sts Pr st r t

3, 1415 180 = 5, 65 102

r 180 rst trs rss sts s r 3, 1415 rst rsssts rst tr s trs rss sts

rts

r ss s rst rt r rss sts

s rt Pr r ln(5, 0 103) = 8, 52 st 2 rss sts rt rtt ss s rt s r ln(45, 0) = 3, 807st 3 rss sts rt rtt ss s rt

Pr ts s ss rtt q s rs s ts s ts s rss sts

q rrt s t st rst

rs

ts rss sts st s rs rs

12, 5 cm

0, 00020 kg

3 108 m/s 6, 02 1023 1, 03 106 s 30008, 00 cm/s

rrt s rs r s rss sts

25, 38 cm

2, 361 m/s

9, 563 103 s 3, 45 g

7, 96 106 m 0, 0335 J

3857 N

s rs tts sr s rss sts

s s r rtr s

s ss rst s

235 m+ 32, 2 m 2, 052 m

2, 02 105 g2, 1 cm3

ln(351, 0)

s s r rtr s

t

stts rrs

rrs

r t rs q r rr rst rt rr rst s r rr r s sr q r rr sr tr rt ts r r

rr r rr rr sr st rt rr st abs t r r tr r xmed r rr xverd st

abs = xmed xverd Pr tr rr rt rel t r tr rr st abs r rr xverdst

rel =absxverd

rr rt t sr rst trs rts t q r 100% str r rr rr rt st rr

s tr s ts rrs sr sss trs sts trs

rrs rssrs rr t rt r str rr

s s tr s rr ss rrs t sr ts

rt s s s

rrs stts s rrs sstts s ss r ts ts r

sr s ss sts q s s ts st sst

tt r r r t s ts rrs

sstts s

r t trs t s r q trs t sr t r stt t rr r r tr ss

ts ts rt rr t q r trtr t

s tr t r t rsst r r r s t q t

rtr tr

rrs trs s ss rss rts ss trs q

r rt srs ssss q s s s rss

P tr rs rs tr s s strts qs rs s

s s r rtr s

stts rrs

s ts rss trtr ts rs t trs rs

rr srr q st sts ts rtr s

rt

rt u rtr ss srs rs q sr rttrs sr rt r rst

rt t t t r rr r s r r

r t t rss rtt srt

s ts rt sr rs s trs t

rts rts s rts uA s qs stsr ts sttsts trs r rr rrstr rts q

rtrs sttsts trs t s r t

sr sr strrt s rts uB s s r t ts ts s rs ss sr s rs sr q s ts

r rtrs sttsts trs s rs r s rs r

t rtt r s rrs trs rs ss

rt s rs r rts r ts s rr trs

s r rss rr s s ssrs r

t rs ss r rt s rt

sr srs rr tr r ss s s

s s r rt ts trs

r t rr q s r sss ts sr Prtt rt

rst sr rr s rst

tr rtt r r rtr

r st r sq tr r tr rt ssttr s

t rt st tt rss rt tr

rt tr rt tt t t tr

sr ss q t rt rst r rst

t st rs s rt tr qs

q trr r trr r s

Prs t

rs st ss strt r r

r r srs sttst r r rss sts rs

sr rt rst st r rs rs r rs s

rs sr srt r rs trs ts s r r

Pr tr t t r q r

t s rr ss r rr t r s r r

r r rr s tr r rs r s r r

s ssrt r t r strt str sr

r r rs r rr t sr srt trs rr

st rr rt

r tr r tr rs t r srs

rts sr t tr tr qt trr r rr

t sr tr s r ss tr Pr tr rs sr tr

s r ss rt t str s rts sss srs

s s r rtr s

str str sttst

rts sr

Psss ts ts str r tr rs t

t rs t tr r tr rr rt u q rr st r r rr t Pr tr

q rt st r s r ssrt r rr

rs

str str sttst

str sttst t s rtrs

rr rs sr r str s q r rr

stt rt s t r ts st t st trs

r sttst rtt r sttst q t r t r rs

rtr str rrstt

s sr rs rs r r rrs rqs

tr r t rts trs rs tr ss

str rqs trs ss sr rst r r

str r str t t s rrsts s

ts s rs s 32 s tr tr rs r r rs r rst str srs q s s t

s t q rtt ss strr str rqs srs st

rts trs ss s rqs s tr ss sr strs

str t str rqs sr t s r

s t tr r tr str rrstt t

s ss strs str

srs q r trs ss str rs q r

s str t rrt ss rss s ts rr srr s

s sr trs stt s trs ss s strtr rrss

t t r q r s t t sss s str

trsrs r s str tr str t sq

q t t r sr trt tt ss str rq

t q r sr str r r s sss

s rts tr r s tr t ts

s s r rtr s

stts rrs

s ts s s str rqs s ts st trs

sss str rstt ss str

s s sss s s rs rrs rtr t ss rrs

ts rs sr t rss r s

tr t ss tr r st r s r s rrs

str rrst st ts s tts q s rt

rt str str t sq tr

s s s srs s str s sts strs

t t r s str r str rrst

str t sts q s s st ts rs Pr

tr r s q r r q s s s s

r rt r s

str Pss t sts q s ts s ts q

s s trs Pr tr r ts q ss r

tr t r t rts ts

str ss r t r strs rs r r s

n > 30 Pr s ss rtrst rtr str sttst ss tr s t st s rsss s rrs sss

tr rt r s rs ss

str t tt st str ss r q s s ss strs rs qqr r s

s s r rtr s

r s

r s

r s s r rtt rs s rts ts

s r s q r trt r r stt r sr ss

r n r tt s trs ts xi s r X t r sr r

x =ni=1

xin

s s st s rtt s ts s s

s tr r t tr trs ts r

r st stt r srs s s s tr r

n r tt s trs ts xi s r X t s sr r

dm =

ni=1

|xi x|n

s t rqtt r rr rs s rs rs

r rs r rs s s

r

sttst t r s rqtt r srr t srs

t str r r

rr s r str s s r xit r n trs ts s r X r

2 =1

n

ni=1

(xi x)2

x r t s n trs ts q s rs s t trr r t r sss ss

ss str ts t r str r

S2 =1

n 1ni=1

(xi x)2

x r str q t sr q s q r ns r tt trs ts str q q

ts

S2 =n

n 12

ttt 2 s r n n 1 sstt r s n 1 n S2 rq str stt q s rt r r rs rs n st st ss

s r

s r s r rt s qrt rtr srs

s s s tr r t tr n trs ts

s s r rtr s

stts rrs

s r stt rt r t r s

r r r t Pr s r r qr

st r ss r q s r s r

=2 =

ni=1

(xi x)2

n

s r str

S =S2 =

ni=1

(xi x)2

n 1

q sr tt srt st r

=

1n

ni=1

(x2 2 xxi + x2i

)=

1n

ni=1

x2 2 x 1n

ni=1

xi +1

n

ni=1

x2i

=

n x2n

2 x x+ x2 =x2 2 x2 + x2

=x2 x2 q stt t sr t q q sq t

x2 =(1

n

ni=1

xi

)2

x2=

1

n

ni=1

x2i st

ni=1

(xi x)2 = n(x2 x2) = n

i=1

x2i 1

n

(ni=1

xi

)2

s r

s r s r rt s r tr

s r str S q r qr r tt n s trsts st

m =Sn=

ni=1

(xi x)2

n (n 1)

st rss stt r r rt x r s r q sr rss s trr s r str S qrtt t r tt n trs ts rs s r m trs r r tr 1/

n Pr sr s t rss srs r s trs rtrs s r sr t

r rr s rts trs s rts rs s rtrs s

r rt s r m sr t q s q q st

s s r rtr s

s r

r m

i xi x2i

ts

xi =

x2i =

r s r

r m

i mi (g) m2i (g

2)

3, 002 9, 012

3, 015 9, 090

2, 915 8, 497

2, 998 8, 988

ts

mi = 11, 930

m2i = 35, 587

s r r ss t

m =1n

1n 1

[ni=1

x2i 1

n

(ni=1

xi

)2 ]

Pr ts rts s rrrr t n = 5 rr r s r m rtr q s s strs s t sr sstts rtt q r r r s

tr m

trs s r r qtr s ss t s

rsts trs r 3, 002 g 3, 015 g 2, 915 g 2, 998 g r m rt u(m) ss s s trs ss m t r q rt u(m) s s r m s rssrrtt r ss t

r m ss m

m =4

i=1

mi4

=3, 002 g + 3, 015 g + 2, 915 g + 2, 998 g

4= 2, 982 g

rt u(m) sr s r m s strs ss rt s strs ssrs r s r mss r q r rt u(m) sr

s s r rtr s

stts rrs

u(m) = m =14

1(4 1)

[4

i=1

m2i 1

4

(4

i=1

mi

)2 ]=

1

2

1

3

(35, 587 g2 1

411, 9302 g2

)

=1

2

1

3(35, 587 g2 35, 581 g2) = 0, 022 g

q sr rtt t rssr rt s 1 rsst Prtt s rrr rt r 0, 02 g s t t r m = 2, 982 g rt u(m) = 0, 02 g r r ss t t st rsst s srr q rt tr st s s

q rt trs ss s rt ss s ss s

r ss rst trs ss s ss s q r ss

sr rr t r s ss s rs

m = 2, 982 g u(m) = 0, 022 g rst m = m u(m) = (2, 98 0, 02) g

rt r rt

t t rt sr rss

rtt r stt r q rr ts s ts rt

sr ss stt s rr rt uc s rrst rr tr s ts s rt sr sss ts rt s

rrs s s ts sttstt ts s trr s s ts

sttstt ts rt srs q r s srs ss ts

rts sttstt ts r s ss st ss s sr trt q

t rt sr s s ss r

s trss t r rt ss r t rtr

s rs ss rts sss ss s rts s rs

s rt t s rs ss tr s r r rt

r s rt t s rs ss s s rrr

ss rss r rt q ss rr

s t q s ss s ss s s

rs s tr qt s r s s t s

q stt f r s s F st stt x r s tr X st

f = f(x)

str rtt r stt f stt x rtx = u(x) r s X s r r rt f = uc(f) r s F

q x s t r r n s r s X s tr q r x s r stt r r rr X ss s s rs sf x sr rs t P r f = f(x) x = u(x) r sttq s s x s rr f(x) r rt tt f(x) t x = x r uc(f)/u(x) sr rt tt r ss t r f(x) ss t st

uc(f) df(x)dx

x=x

u(x)

s s r rtr s

rt r rt

rtt r r rt f r rt x

q rt r rt r s s F q st st r s tr X trtt rqt s rst r s s s ts r s

q stt f r s s F s s stts x y s rstsrs ss tr X Y

f(x, y) = x y q ss q q s s X Y s sttstt ts st X t Y ss s trs r rt uc(f) F rst s s trs rt u(x) u(y) s s X Y rstt tts trs u(x) u(y) r uc(f) sr ts ts strr s trs rr s s rs rts s r rr r st

s s rt Pr tr ts sr sst s s trs

t q s ss s rt r r trr s ts

tr s r Pr srr r rrt

sr s s trs rts u(x) u(y) r rt uc(f) s rtr q s rrrr t s r q st

=

1n

ni=1

(xi x)2 =

(xi x)2

s r rss srs s s tr r sr

rr stt r r rt ss s r s qs

rss r uc(f) = f

u2c(f) =

(fi f

)2=

[(xi yi

) (x y )]2 = [(xi x) (yi y)]2

=(xi x

)2+(yi y

)2 2 (xi x)(yi y)

u2c(f) = u2(x) + u2(y) 2 (xi x)(yi y) u(x) = x =

(xi x

)2 u(y) = y =

(yi y

)2s s ss rs x y

rstt trr tr rt q tr r

s qst s s x y s ts ss strr q tr r str ss rst sr st q r r s ts s

rss rtr s ss r q sr

s s r rtr s

stts rrs

r rt

f(x) = ax a = stt f = a x uc(f) = au(x)f(x) = xa a = stt f = xa uc(f) = a xa1 u(x)f(x) = ex f = ex uc(f) = exu(x)f(x) = ln x f = ln x uc(f) = 1xu(x)f(x) = sen x f = sen x uc(f) = cos xu(x)f(x, y) = ax by a, b = stt f = a x b y uc(f) =

a2u2(x) + b2u2(y)

f(x, y) = xy f = x y uc(f) =y2 u2(x) + x2 u2(y)

f(x, y) =x

yf = xy uc(f) =

1

y2y2 u2(x) + x2 u2(y)

rsss r s s s rs s rts s s rs

ss s f q ss s rs ss tr ts

u2c(f) = u2(x) + u2(y)

ts q r rs ss s F q ss s rsss tr ts X Y q rr q rtr q

u2c(f) =

(f

x

x=x

)2u2(x) +

(f

y

y=y

)2u2(y)

st q ssr trr t r r sr s r d r rsstr t q s rs f(x, y) r s rs sr ts ss stt tr r ss s ss rq s rs ss X Y sr s tt tr s s rts sts ss

rr s rts s r ts s rs ss s F q ss srs ss tr ts strs

ss t t s r stt f = f(x1, x2, ..., xN ) r s s F s s stts x1, x2, ..., xN s N rs ss tr X1 X2XNsttstt ts ss s rt s sr trs

ts q r

u2c(f) =

Ni=1

(f

xi

xi=xi

)2u2(xi)

xi s rr stt is r s tr Xi q s rs rsf

xi

xi=xis s ts ss r s rs ts xi

r q rss r r stt rt ss s rs

ss tr sttstt ts st s ss rtrs rqtt rs

ts rt s qs s stt ss rr s s rr s

str N rs ss tr sttstt ts q r ss srt rtt q r q st

u2c(x1 x2 x3 ..... xN ) =Ni=1

u2(xi)

s s r rtr s

s rs rt

s s s rr t s N rs ss tr sttsttts q t sr t rtt q rtr r s

rst rss rt rt srt trs s rts rts s

ts s

u2c(f)

f2=

Ni=1

u2(xi)

x2i

s rs rt

ts ss rts s rts sss s rtrsts trss

s rs rt ssr r s rt

rss rs sr rs r s s sss rtrsts

q sr ss Pr ss st srr rt q

stt x s t rtt n trs s r s tr Xs s r r stt ss r r

x ss rt rt uc(f) r st rt u(x) ss r trss rt uap(x) ss s rtrsts trs r ss s stt f r s s F sr

f = f(x, xap) = x+ xap

xap stt rr s rtrsts trss r r f = x + xap r s s F rrst r stt ssr q q ts

u2c(x+ xap) = u2(x) + u2ap(x)

uap(x) rt ss s rtrsts trss r s stt x r s tr X st s tr ss rt tr rss ss s u(x) = 0 q rt r s uc(x+ xap) sr rt rt trs ruap(x) st uc(x + xap) uap(x) Pr tr q r s tr X s rrr ts s t rs q rs

r s r sss ss u(x) >> uap(x) q sr strstr rt uc(x+xap) st trs rt tr u(x) st uc(x+xap) u(x)

s r stt f r s s F s stts xi srs ss tr Xi i = 1, 2, ..., N q s q rss rrs s rts u2c(xi + xap) = u

2(xi) + u2ap(xi) q ssttr rt

u2(xi) q st

u2c(f) =Ni=1

(f

xi

xi=xi

)2[u2(xi) + u

2ap(xi)

]

rt

r rt uc s rst t r rssr rt rst r s ss s strs s rs rs s

sr ts s ssr r rt q r tr s rt

rt ue ss s tr r

s s r rtr s

stts rrs

tr str sttst rs q sr rt tr sr

rt

ue = kuc

uc rt k tr r ss str sttst rr tr rt r r q r k s s 2 r rt r r s t s str sttst st r rrs r

95% r str ss r q st r strt tt r r r s s tr r k rr tr srs 2 3 rqr r tr rt sr

rs

ssq s rrs ts r s tr s r st s

rssr sstt tr

t rr tr s ts r q st t

r s r s r ttrs

rr qt rt

rt r 1 m r

rrt tr trtr t 350 C rtrr trtr 200 C

tr s ss rt 2000 stts Pr trr tr s stts rtr s trt n = 100 trts t s tr rst tr str

tr trs

r stts

tr s stts s

stt tr s stts s

s r str trs s

t r r q r stt 35 s s trs t q q sr t sr s tr

h0 = 2, 0 m s trs t r s rtr rs 0, 01 s Pr str r str rqs stt tr strstr

tr ss s r trr str

t s trs t q s r rrst r r srs s s

rq str rt r s q s tr tr

t t+

s s r rtr s

rt

str rstt s 35 s trs t q sr t

s s r rtr s

stts rrs

s s r rtr s

t

strs sttsts

str ss

s s rsts sttsts s rrs sts trrt s ts tr

str ss s q str rtr t r r s

n > 30 ss ts s rrs str ttrs s r Pr r ss tr stt rr t rtr

str s rts stt x r s tr X ss ts q q r s rs t str rqs s s

sts st rss r

y = ymaxez2/2

ymax =1

u2 r str z =

x r sttst r x

s s s ts r s r tr str str str s rts xi r s X ssrr q x s r s ts rstt rtr t ymax rr 2 tr 1/2ymax str t q st r tr rr tr t s r sr stt srs tr r

s s s str rtt r str

ss s s s rsts rtrs sttsts trs r str

q t q r n s t t s rss srts r s r t s ss rsts rss ts

tt s r s rs s rtrs r q s stt s ts sss rtrs rtr str s tr

tr str ss str q rt s r ss tst st r 68, 27% t q 68, 27% srt tt q s t t t s ss t s r

s r rt uc rts 68, 27% t r r tr tr z = 1 z = +1 str q s q s t r r s str

qt s s tr rt st str ss sr

s t sttst r r rt ue qt tr r k r sttst z r rt ue sr t rt uc r r rr z s r s st

ue = zuc

s s r rtr s

strs sttsts

str s rts xi r s X rststr tr ss

Pr q s t rt uc r z = 2 ts rt ue = 2uc r 95, 45% s rsss s sss rt rts sss s tr

rt s st rt tr t s r

uc m t q ue zm r r z ssr r rs trss str s rs ts ss r tt s rsts s

0, 67 50%

1, 00 68%

1, 28 80%

1, 64 90%

1, 96 95%

2, 00 95, 4%

2, 58 99%

3, 00 99, 7%

3, 29 99, 9%

rs ts z tt s rsts s q srss tr r k s ss strs r r s

rtt rsstr q t rt r str ss

st s rtr r r r s str t r

r qqr r s str t tt st s

str t tt

s s str q r sn < 30 str ss sr t sr r stt rt rst

r qs strs trt s XX r q rs qss tt s sst q sr s

rr r r rts ts qt trss r rr ss

s s r rtr s

str t tt

r tt str sttst q r rr s rsts sss rss

r

f(t) = y0

(1 +

t2

n 1)n/2

= y0

(1 +

t2

)(+1)/2

y0 stt q t r s n rss q r r f(t) t s tr t r sttst

= n 1 rtr sttst r rs r r s

q rss t st t rt

str t tt str st r str ss r s s rs t q rr rs t s trs r rt rss str q r str t tt rtr st r rs r t r r st rtr sttsts r str ss str

rs str t tt r rts rs r

str t q rtt str ss r tr r

k s rt ue t r r sttst t r rs r tr rss s q r str ss rt ue sr t rt uc r r r sttst t s r s st

ue = tuc

s s rsss s s ss rt

rts sss s tr rt s st rt

tr t s r uc m t q ue tm

s rss s t rt

r rs r t eff s r r s q ttrt r

eff =u4c

Ni=1

u4ii

s s r rtr s

strs sttsts

N r tt ts rts ss ui rt ss is t i r rs r ss is t rt uc rt Pr rt r rs r

q st Ai = ni 1 r rt r rs r t st Bi r eff r tr s tr r tr tt rr rt r r q t

rt str ss sr t s trr str

t tt sr s rr

r sttst t ssr r rs trss r rs r eff str s rs ts t tt s rstss r rts rs r rs r srs q

r t s s s rs t r str t tt q s rs z r str ss q r rs rt t

r rs

r eff

t68,27% t90% t95% t95,4% t99% t99,7%

1 1, 837 6, 314 12, 71 13, 97 63, 66 235, 8

2 1, 321 2, 920 4, 303 4, 527 9, 925 19, 21

3 1, 197 2, 353 3, 181 3, 307 5, 841 9, 219

4 1, 142 2, 132 2, 776 2, 869 4, 604 6, 620

5 1, 111 2, 015 2, 571 2, 649 4, 032 5, 507

6 1, 091 1, 943 2, 447 2, 517 3, 707 4, 904

7 1, 077 1, 895 2, 365 2, 429 3, 499 4, 530

8 1, 067 1, 860 2, 306 2, 366 3, 355 4, 277

9 1, 059 1, 833 2, 262 2, 320 3, 250 4, 094

10 1, 053 1, 812 2, 228 2, 284 3, 169 3, 957

11 1, 048 1, 796 2, 201 2, 255 3, 106 3, 850

12 1, 043 1, 782 2, 179 2, 231 3, 055 3, 764

13 1, 040 1, 771 2, 160 2, 212 3, 012 3, 694

14 1, 037 1, 761 2, 145 2, 195 2, 977 3, 636

15 1, 034 1, 753 2, 131 2, 181 2, 947 3, 586

16 1, 032 1, 746 2, 120 2, 169 2, 921 3, 544

17 1, 030 1, 740 2, 110 2, 158 2, 898 3, 507

18 1, 029 1, 734 2, 101 2, 149 2, 878 3, 475

19 1, 027 1, 729 2, 093 2, 140 2, 861 3, 447

20 1, 026 1, 725 2, 086 2, 133 2, 845 3, 422

25 1, 020 1, 708 2, 060 2, 105 2, 787 3, 330

30 1, 017 1, 697 2, 042 2, 087 2, 750 3, 270

35 1, 015 1, 690 2, 030 2, 074 2, 724 3, 229

40 1, 013 1, 684 2, 021 2, 064 2, 704 3, 199

45 1, 011 1, 679 2, 014 2, 057 2, 690 3, 176

50 1, 010 1, 676 2, 009 2, 051 2, 678 3, 157

100 1, 005 1, 660 1, 984 2, 025 2, 626 3, 077

1, 000 1, 645 1, 960 2, 000 2, 576 3, 000

rs ts t tt s rsts s r rtsrs rs r eff q sr ss tr r k sss strs qqr r s

s s r rtr s

str trr rtr

str trr rtr

s strs trr rtr r s trs strs sttsts ts ts

t tt r s trsr sss str

s s ts r s rts q r s s

ts s trs s rss s rt rtt s Pr

str sss s rts r rs r r rss

q s str trr rtr sr t ts ss

s t r sr sr s ss ssrr q t r

str qqr t tr rs rtrs tr a a+ s s tr t rrt s t rs qtr

rtr t rs rs s sss ss rt ss

rss s str rtr str s srr q

r ymax =1

a+ a q r r y(x)x sr

tr r =a + a+

2 r stt tr r sr r

tr sttst rt r tr ss str rtr

=a3

a =a+ a

2

rtr q t tr tr a a+ trs a r

y(x) r sr s s rstt ymax =1

2a = a + a

strs trs rtr r trr

s ss t s t r sr sr ss ssrr q

t r str s r tr stt tr rs tr a a+ s s tr t rrt r tr s s s ss

ts ss s s s qtr rtr rt

rs rs s q s r rr r s s

r tr tr tr s rs sts s sst

strr sss ss rt ss rss s str trr

str ss r r r str trr

s s r rtr s

strs sttsts

ymax =2

a+ a =1

a r s q s r str rtr

r tr sttst rt r tr ss str trr

=a6

a s rtr q

rtr rs 0, 1 A s r r rrt i rt tr tr s sst strr rtr

rst rt uap(i) ss s rtrsts trss rtr q t s r st s rss

rrtt r rrt rt tr

tr rrt tr rtr

rtr sr str trr rr rrst s s s

rs st 2a = 2 0, 1 A a = 0, 1 A ss r q

=a6=

a6=

0, 1 A6

= 0, 04 A

tr s sst strr rtr q r rrt

tr i = 0, 75 A r rrt rt tr sr rss rrtt

i = (0, 75 0, 04) A

qtr rs 0, 02 mm s r r tr d st s rsts trs r 8, 40 mm 8, 42 mm 8, 40 mm 8, 44 mm 8, 42 mm r d rt u(d) ss s s trs tr d s t rtr r S =

1

4d2 r S r s t rt r uc(S) rt u(d) rt uap(d) ss s rtrsts trss qtr rt ue(S) r q t 95% s rss rrtt r r s t

r d tr d

d =5

i=1

di5=

8, 40 mm+ 8, 42 mm+ 8, 40 mm+ 8, 44 mm+ 8, 42 mm

5= 8, 4160 mm

s s r rtr s

str trr rtr

r d

i di (mm) d2i (mm

2)

8, 40 70, 560

8, 42 70, 896

8, 40 70, 560

8, 44 71, 234

8, 42 70, 896

ts

di = 42, 080

d2i = 354, 146

s r r tr d s t

rt tr sr s t s r

d q s strs s s rt s strsssrs r s r d ss r r s r sr

u(d) = =1

5(5 1)

5i=1

d2i 1

5

(5

i=1

di

)2=

1

5

1

5

(354, 1464 mm2 1

542, 0802 mm2

)

=1

5

1

5(354, 1464 mm2 354, 1453 mm2) = 0, 00297 mm

r S r s t sr

S = 14d2 = 1

4(8, 4160 mm)2 = 55, 629 mm2

qtr sr str rtr rr rrst s rs st

2a = 0, 02 mm ss r q rt trs qtr sr

uap(d) =a3=

0, 02 g

23

= 0, 00577 mm

rt r uc(S) r q sr

u(S) =

(S

d

d=d

)2[u2(d) + u2ap(d)

]=

(1

2 d

)2[u2(d) + u2ap(d)

]

=

(1

28, 4160 mm)2

[(0, 00297 mm)2 + (0, 00577 mm)2

]= 0, 08579 mm2

str ss q r s n = 5 t rt ue(S) sr str t tt str q t = 2, 776 r 95% r r = n 1 = 5 1 = 4 ss s rt sr

ue(S) = tuc(S) = 2, 776 0, 08579 mm = 0, 2381 mm2 rssr rt s 1 rs st t s rrr rt r ue(S) = 0, 2 mm

2 s srr q rt st rr

s q rt trs ss s rt ss

s r r S = 55, 629 mm2 rst trs ss s ss s q r ss sr rr t r s Prtt rsst

sr

S = S ue(S) = (55, 6 0, 2) mm2

s s r rtr s

strs sttsts

rs

s rt ss s rs rs s str

rtr trr

ttr s

tr

Pqtr

Pqtr t

trs

rtr

r r rs 1 mm s r r rr x rt y rt rt rst s s trs

x (mm)

y (mm)

s rr x rt y rt rt

s rs s x y s rts u(x) u(y) sss s s trs rr x rt y rt rt rtr r S = xy r S r rt rt rt r uc(S) s rts u(x) u(y) s rts uap(x) uap(y) sss rtrsts trss r r rt ue(S) r q t 95% s rss rrtt r r rt rt

s s r rtr s

t

rs s rs

tr

r r q str r tr s rs s

s s rs st rs tr s r r r

s qts t tr ss r t sr r r rstr s

s rts Ps r q r strt t r rr

ss q s s s s r s rr srs r r

ts s ts rts trr r q r s st s rt ss

sss ts r ss r r rtr rr st s ss

srr trs s rt s sr q sr

rrt tr s rs s rtr s s rs s

s s rt s s r qr ss sss t

str rs

s rrs ss q sr ss str rs s

r tt s s st q r s rs s

srr s s rs s rs rrsts ss rsts s

rt sss r t st r s rs s

ss rtr rt r r t

s q t rr

s sr ss r r s t

srs tr rs ts sts s rs 1 2 4 5

s t rs sr tr

sr ss ts q rs s ts sr r s rt rs

s ts q rrst s s rts r r t t s

rst q st rs s s s ts sr t ss

trs r sr r r

sr ss ts q rst r q t r rtt s s

s ssr s s rrsts s ts rts sr s s r

s

s s r rtr s

rs s rs

ts s ts rts str rs r rst trr r st

rs ssr sr ts s ts t ss q r ss r t

r ss ssr q r t rr tr t t

rt

str str r rtt rtr r

r s qs rrs rt rt rs sr t rt

s s rrs rr sr sss rrs r stt s

rts sss t rt rstt rss

s rs Ps tr s rsts sts s r sss

s rt s rts trs t rt s sts trs

strs sttsts tr r n s r r r r t rt sr t s s rs s

r r

rst r r r rt y = ax+b s tsr b r a

str s rts rtr r rt s

ts qrs rrst s s rts s srs s rts s

rt r rt t rrst r q r sr r

r x r y

s rs

s rs s qs s qs s rs s st rs r

t

y = ax+ b

t r t r t r rrs

rt s y/x qt q t r t trs rt y r x = 0 sr srs s ts q rt trr ststs rtr s s rts

s s r rtr s

ts tr s ts a b

ts tr s ts a b

r sr q sr rrt tr

s rs s rtr rt rs s rs rr

s s qs ts r st sts ss qs sr rr tt

sr trs rs ss r tr sq rst

s ts r trr st r rtr s rts q ss rtt

r rtt tr q sts s s s ts trs trtr s s

s s s

t r

ss t rr q s t r r ts rts (n > 10) st rqr s ss t s s stt s rtrs

rt q r s st s ts rts rtr tr r (x , y) sssts strs sr r

x = 1n

ni=1

xi y = 1n

ni=1

yi

s s rs s s rs x y rstt

rt rt rt q ss r st t r qtr qrts

s st rt t s ts rts st

trr qrt t s Pr s str rt q r s st s ts

rts s t s sr t (x , y) s r trsrtrs r tr t (x , y) t q rt 84% s ts q r trr qrt s qt s qrt

tr s ts a b t r

rt tr sss s t amax rt s r rt ss rt trt y r x = 0 tr t r bmin t s t (x , y) r r t q rt 84% s ts q r trr qrt s qt s qrt rt tr sss

s t amin rt s r rt ss rt trt y r x = 0 tr t r bmax srs q r ts rts ts rt 68% s ts rts q sstt t s r st s sss srs

s s r rtr s

rs s rs

rt q r s st sr s ts rts rt q r trr

tr s rts s ts r a r b rt ss rts u(a) u(b) s ts r

a =1

2(amax + amin) b =

1

2(bmax + bmin)

u(a) =1

2n(amax amin) u(b) = 1

2n(bmax bmin)

s s ts rts t rts rs rts s sr s r

t r s r t s rt t sss ss s rt

s rs rr rr t

t s s qrs

st rs t s s qrs rtt s trr t

r t rtr ss t s

st

R (a, b) =1

n

ni=1

(yi ycalculadoi

)2=

1

n

ni=1

(yi axi b)2

ss s rrs str s s (xi, yi) str q t q s ts a b r tr s rs yi s s rs y

calculadoi (xi) s r ss

q trs rs s rrr s rs a b q sts s s

af(a, b) =

bf(a, b) = 0

af(a, b) =

1

n

a

[ ni=1

(yi axi b)2]= 2

n

ni=1

(yi axi b)xi = 0

bf(a, b) =

1

n

b

[ ni=1

(yi axi b)2]= 2

n

ni=1

(yi axi b) = 0

ni=1

yixi + a

ni=1

x2i + b

ni=1

xi = 0 ni=1

yi + a

ni=1

xi + nb = 0

ss t q

ni=1

b = nb s sst qs r a b ts

a =

nni=1

xiyi ni=1

xi

ni=1

yi

nni=1

x2i (

ni=1

xi

)2 b =ni=1

x2i

ni=1

yi ni=1

xiyi

ni=1

xi

nni=1

x2i (

ni=1

xi

)2

s qs sr s rt r rr rsss s rrs

s s s s rs s s rs x y s s qs rr q trs

s s r rtr s

ts tr s ts a b

a =

nni=1

xiyi n2 x y

n

ni=1

x2i n2 x2=

n

[ ni=1

xiyi n x y]

n

[ ni=1

x2i n x2]

a =SxySxx

Sxy =ni=1

xiyi n x y

Sxx =

ni=1

x2i n x2

q s

Syy =

ni=1

y2i n y2

Pr tr s s s rs s s rs x y s s q rst

b =

n

ni=1

x2i y nni=1

xiyi x

nni=1

x2i n2 x2=

ni=1

x2i y ni=1

xiyi xni=1

x2i n x2

str tr n y x2 rr q s s qts

b =1

Sxx

[ni=1

x2i y ni=1

xiyi x+ n y x2 n y x2]

=1

Sxx

[y( n

i=1

x2i + n x2) x

( ni=1

xiyi + n x y)]

=1

Sxx

[ySxx xSxy

]= y Sxy

Sxxx

b = y a x s rs 2x

2y s rs x y s ts rts r cov(x, y) tr sss rs s ts sttsts rtts q sr s trs SxxSyy Sxy

2x =1

nSxx

2y =

1

nSyy cov(x, y) =

1

nSxy

s s r rtr s

rs s rs

tr t sttst rtts rqtt t t s s qr

t rr r trs Sxx Syy Sxy

r = a

SxxSyy

t rr r rst rt s rsts st s rs ts rrs tr r r r r tr 1 s sts rt r rst t s rt r rst t s r st

t r sr st rr i st sr rrst r srs s rt t rt r rt st st

i = yi ycalculadoi r s rrs S sr

S2 =Syy aSxy

n 2 = Sxx(1 r2n 2

)

r s rrs sr n 2 s n 1 rq st r trss s ts a b qt str ts st r srs q q s r2 = 1 t r s rrs r st ts sts rts tt rt st ss r s rrs St trrt rtt r s rt sr r s rts

r y trr s r q sss rts st s sts q tr tr st tr rrs y q st s s t

t s rts u(a) u(b) s ts st r sr s rtr q

u(a) =

S2Sxx

=|a|r

S2Syy

= |a|

1 r2(n 2)r2 u(b) = u(a)

1n

ni=1

x2i

Pr ts rts s rrrr t r

n = 5 r r s rs s ts r a r b s rts sss rtr s qs s rs tts s t sr sstts

rtt s rsss s rs s x y s s Sxx Syy Sxy

r a b

i xi yi x2i y

2i xiyi

ts

xi =

yi =

x2i =

y2i =

xiyi =

r s ts r a r b

t s s st stt Pr tr s

st st s rs t str s sts x t st t t r r xt r s ts rts

s s r rtr s

ts tr s ts a b

s r s t s s qrs tr s ts r a r b rt q r s st s ts rts r rt rrst rx t t rr r s rts u(a) u(b) sss s ts ra r b r s rrs S

x (m) 5, 90 5, 10 3, 90 3, 10 1, 90

t (s) 1, 0 2, 0 3, 0 4, 0 5, 0

rs s sts x t st t t r

str r x t r s s

r xx s ts rts rs s r t q rs st s ts rs r t s s qrs

s s rtts r st strs

r a b

i ti (s) xi (m) t2i (s

2) x2i (m2) tixi (m s) 1, 0 5, 90 1, 0 34, 81 5, 9

2, 0 5, 10 4, 0 26, 01 10, 2

3, 0 3, 90 9, 0 15, 21 11, 7

4, 0 3, 10 16, 0 9, 61 12, 4

5, 0 1, 90 25, 0 3, 61 9, 5

ts

ti = 15

xi = 19, 9

t2i = 55

x2i = 89, 25

tixi = 49, 7

r s ts r a r b

rtr s rr s sts s

t = 15, 05

= 3, 0 s x = 19, 95

= 3, 98 m

s s r rtr s

rs s rs

Stt =ni=1

t2i n t2 = 55, 0 5 3, 02 = 10, 0 s2

Sxx =ni=1

x2i n x2 = 89, 25 5 3, 982 = 10, 048 m2

Stx =ni=1

tixi n t x = 49, 7 5 3, 0 3, 98 = 10, 0 m s

a =StxStt

=10, 010, 0

= 1, 0 m/s b = x a t = 3, 98 (1, 0) 3, 0 = 6, 98 m

r = a

SttSxx

= 1, 0

10, 0

10, 048= 0, 99761

s rs a b r r t sr ts rtt t r t str rt tr r x t q r s st ts rts srts u(a) u(b) sss s s s ts r a r b rstt sr trs rtr s qs st

u(a) = |a|

1 r2(n 2)r2 = | 1|

1 0, 9976123 0, 997612 = 0, 04 m/s

u(b) = u(a)

1n

ni=1

t2i = 0, 04

55

5= 0, 132665 m

r q r s rrs sr

S =

Sxx

(1 r2n 2

)=

10, 048

[1 (0, 99761)2

5 2]= 0, 1264 m

s tr s rrts ssrs s r q

a = (1, 00 0, 04) m/s b = (7, 0 0, 1) m

rs

t t r r ss M rs s rs trt s s sss t r ss r

q r rt x r r tr ~F sr r rstrr ~F rr rt r x st ~F = k~x k stt rtrst stt st Pr tstr

ss tr rssr r

rstrr

~F r rs rts rt r x str rst t r

x (cm) 1, 00 2, 00 3, 00 4, 00 5, 00 6, 00 7, 00 8, 00 9, 00 10, 00 11, 00

F (N) 8, 11 21, 8 29, 1 40, 5 49, 8 60, 1 69, 7 80, 6 88, 8 102, 0 108, 1

rs s rts r x s rs rstrrs ~Frrsts

s s r rtr s

ts tr s ts a b

r F x r s ts s r s t rtr s ts r a r b rsts rts rt q r s sts ts rts sr q r F = ax+ b tr rt rrst r F x r q rt q F = kx rst rtrr r stt st k s s rts tr sts r a r b ss rr

tr r sst s r sr t ts s

rs trt stt t sst st r r r sr

t rt r r tr t r rts sts

rr sr tr r stt t

x (cm) 7, 0 10, 0 11, 0 16, 0 18, 0 20, 0

t (s) 0, 61 0, 87 0, 96 1, 40 1, 57 1, 75

rs s sts x rr sr tr r trs t t rrsts

r x t r s ts s r s t ss qrs tr s ts r a r b rsts rts rtq r s st s ts rts sr q r x = at+b tr rtrrst r x t r q rt q x = x0+ vt rst tr t rt r r trr r v s x0 rr tr r s s rts tr s ts ra r b ss rr tr t rt r

s t r q r q r

~A rr r~F sr r st

~A =1

M~F

stt rr 1/M st ss M r Pr tstr s t stt r

~A r rs rts r

~F s r ss M rt rtr r ss rss sr stt

str rst t r r A F r sts rsts rrs rr s r s t s s qrs

tr s ts r a r b rsts rts rt q r s sts ts rts sr q r A = aF + b tr rt rrst r AF r q rt q r trr r ss M r s s rts tr s ts r a r b ssr s t

F (N) 0, 150 0, 260 0, 350 0, 450 0, 550

A (m/s2) 0, 25 0, 01 0, 45 0, 02 0, 68 0, 03 0, 90 0, 04 1, 12 0, 05

rs r

~F q t r ssM s s rs ~A rrsts

s s r rtr s

rs s rs

s s r rtr s

t

rs s rs

s t str s s str r r rrst rqtt

rs s rs ts s sss s s s sr ts

rs t s ss s rs t 1/x 1/x sr ts r sss s sr s s rs s sts t t

s s str t s tr s qs q s t

t q rrs rt ss s r s s str r

t r q s t r ss rt r

rs t s t r r s rs

sr t tt ss rs ss rts r rt s s

ts sts tr t rtt r t

s s t y = f(x)

s s

sss s t st r r

y = kxm

k m s stts rt s s s q ts

log y = log k +mlog x

s s r rtr s

rs s rs

Y = B +AX

Y = log y B = log k A = m X = log x Ps strr r q s r tr tr s rs A B rtt rtr r Pr s ts rs r s rstts rt

s s x y tr r rtt y = f(x) t

r s r tr

rt rt q r s st s ts rs r str

ss s r s rs s ts r A r B sr ts rtt rt st

A =Y

X=

log y2 log y1log x2 log x1

Btrs rt Y r X = 0

s tsA B r sr ts t t t s s qrs tr s sts ss rtr s rs ts A B s trr stt k trs r k = 10B t m rtt m = A

s s strs r s rs x y rtt s r str r st

s r tr rt q r s st s ts r s

ts r A r B rtt rt tr t t t s sqrs

Pr r r s rrt rt x y srs sss rts st str t r s ts rrsts s

rts x y s r ts r r str rtstr st r rt q r s st s ts rts s ts r

A r B sr ts rtt rt r rr r Pr tr q s t s s qrs r sss ss ts ts

A = 2, 5 B = 0, 65 ss k = 10B = 4, 5 m = A = 2, 5 rr y = 4, 5x2,5

s s r rtr s

s rt

s r

r s r

ss ss rr s s rt t ss t

s trr sst trs q rqr rt ts s ts

rts trtt sss s sr sss q s s ss ss s

ss rts

s rt

str s rt st r s rt st rt rts

rrs s rs s rts s rs tr s a s tss s r s 10 ts rt s r tr ts 10sts 10n 10n+1 n r tr str s str srt s 10 ss s t x 10n r tr 10n 10n+1

t rt rt t r strr s rt s 10

ss r Lx st t x 10n r rr 10n L10 st tr 10n

s s r rtr s

rs s rs

10n+1 s rt s rt r tr sss sts s rs srts s ts rrsts rt st

L10log 10n+1 log 10n =

Lxlog ( x 10n) log 10n

L10log

(10n+1

10n

) = Lxlog x+ log 10n log 10n

L10log 10

=Lxlog x

log x =LxL10

q rt trr rt qqr r x s 10 rtr s strrt s tr e = 2, 781... s s a qqr ts rstt

ln x =LxLe

logax =LxLa

rtt r q r s rt rs r 100 = 1 s r t 10 t ss st rt rs ts s sss sr ts r strr s rt

P

str s str rtr s rt t

r r s s ss s ss rt rt s

r rt Prtt ss rr s rs s rs t t

rtt sr s ss s ss r rt

sss rs s s s rt sr ts r rrstr ts

rts sss s rs ss Pr s s rs tr rs 0, 1 10 rr s s rs tr 0, 1 1 s s rs tr 1 10 s ss t s rt ss sss rr rr tr

0, 1 t s tr 0, 2 s rr tr 1 t s tr 2 ss r t rst 10 sss q t str ss 2 5 10 rts s s rts rs rst ss s t rs ss s rtr q r s s t 10 q r

Ps tr q r r r s rts s rs

2 0, 0148 s ss 10 qqr tr q s qr Pr ss s srr q qqr rt t rt L10 30 mm ss s rts L2 L1,48 s r ts

log 2 =L2L10

9, 0 mm30, 0 mm

0, 30

log 0, 0148 = log(1, 48 102) = log 1, 48 2 = L1,48

L10 2 5, 0 mm

30, 0 mm 2 1, 83

s s rts rstr rts ts s st s rrs

tr rts rts trs ts s ss s ss rsts

s s r rtr s

s s r s s

sts rs s s

s s r s s

srs q q s rs t Y = log y t X = log x s srts ss s q q s str r s rs rs (x, y) rtt ts rst rt s r q rr r

rr s t s Pr s ts rs s

rstts s s x y tr r rtt y = f(x) rtssr t

rt rt q r s st s ts rs r str

ss s r r t r A sr t rtt rt stss s ts rtrrs (x1, y1) (x2, y2)

A =Y

X=

log y2 log y1log x2 log x1

r t m t rtt m = A

stt s ts (x1, y1) = (0, 23; 0, 030) (x2, y2) = (1, 4; 7, 0) ss rtrrt sr rt st q ts

m = A =log 7, 0 log 0, 030log 1, 4 log 0, 23

0, 84 (1, 52)0, 15 (0, 64) 3, 0

s s r rtr s

rs s rs

r

r k q t r B sr tr rtt rt stss st q

m log x = 0 x = 1srs q st rst

log y = log k + 0 k = yss s q rt rt x = 1 st rst s r r k = y srt trs trs rt x = 1 rt st r srr q ss rr r y = k = 3, 0 Pr tr q rt rt x = 1 st rst s r trt sr r r k trs q s t rtrr (x1, y1) rt st ss s q r y1 = kx

m1

k =y1xm1

s t s ts ss t (x1, y1) = (1, 7; 15) t

s k =15

1, 73,0 3, 0 q r r trrt Prtt q r s rs x y st y = 3, 0x3,0

rtt r q t s s qrs t r sr s

q r r qqr s ts A B sss ts s s s ts rsrs sr ss t rs

s s

sss s t st r r

y = kemx

k m e = 2, 781... s stts rt s s s q ts

s s r rtr s

s s r s s

log y = log k +m (log e)x

Y = B +AX

Y = log y B = log k A = m (log e) X = x s s strr r q s r tr tr s rs A Brtt rtr r Pr s ts rs r s rstts

rt s s y tr r rtt y = f(x) t

r s r tr

t rt rt q r s st s ts rs r

str ss s r s rs s ts r A r B sr tsrtt rt st

A =Y

X=

log y2 log y1x2 x1

trs rt Y r X = 0

ts s ts A B r sr ts t t t s s qrs t r rtr s rs ts A B s trr stt k r r k = 10B t m rtt

m =A

log e

s s r s s

s str str rtr s rt t

r r s s s s s rt sr

ts r rrstr ts rts sss s rs ss

ss s rt r rt qt q s rt st

rt tr Prtt ss rr s rs s s rs

r t t rtt sr s ss

s s r rtr s

rs s rs

tr rs tts

srs q q r t Y = log y rt qt q r t X = x r ss s q q s str r s rs rs (x, y)rtt ts rst rt s r q

rr r rr s t s Pr s ts rs

s rstts s s x y tr r rtt y = f(x) rts sr t

sts rs s s r

rt rt q r s st s ts rs r s

tr ss s r r t r A sr t rtt rt stss s ts rtrrs (x1, y1) (x2, y2) sstts q

stt s ts (x1, y1) = (0, 3; 6, 0) (x2, y2) = (1, 3; 200) ss rtrrt sr rt st q ts

A =log 200 log 6, 0

1, 3 0, 3 2, 300 0, 77

1, 0 1, 53

ss q r stt m r m =A

log e 1, 53

0, 43 3, 6 r k q t r B sr tr rtt rt stss st q

m (log e)x = 0 x = 0srs q st rst

log y = log k + 0 k = y

s s r rtr s

s s r s s

ss r k = y sr t trs trs rt x = 0 rt st r srr q ss rr r y = k = 1, 9 Pr tr q rt rt x = 0 st rst s r trt sr r r k trs q s t rtrr (x1, y1) rt st ss s tsy1 = ke

mx1

k =y1emx1

s t s ts ss t (x1, y1) = (0, 3; 6, 0)

ts k =6, 0

e3,60,3 2, 0 q r rt r trrt Prtt q r s rs x y st y = 1, 9e3,6x

t s rs rs t s s qrs t r

sr t q r r s ts r A r B sss ts s s s ts rs rs sr ss t rs

rs

t q r s t t rt rt

r t q r rr rt y srr q ss t

y = y0 + v0yt+1

2gt2

g r r t v0 = 0 tr y = h rtr s y0 = 0 q trs

h =1

2gt2

h (m) t (s)

0, 200 0, 1593

0, 250 0, 1831

0, 300 0, 2046

0, 350 0, 2238

0, 400 0, 2432

0, 450 0, 2593

0, 500 0, 2750

0, 550 0, 2912

0, 600 0, 3056

0, 650 0, 3202

0, 700 0, 3336

0, 750 0, 3468

0, 800 0, 3599

s q t r rts sts q r

stt rst q s tr r rs

rr q tr t q r ss rtr t

rs stt s ts q t t rs r

rts sts q r str rst rt t

r rq sss ts tr rt q r s st s

s s r rtr s

rs s rs

ts rts r r g rs stt rst r ss rr q t q r t

t rt rt r stq

tr r tr s trs rss rsss ss st rsss

s s st t ttr s t r rs

trstr s q r rr sts rsss

sst r t t tr trss r tr t

t R0 t t R ts s trssr tr sssr t s tr r

R = R0et

t r t tt q s rsss tr t tr

t (mm) 1, 22 2, 21 3, 10 4, 32 5, 31 6, 53 7, 28 8, 50 9, 91

R (CPS) 9, 32 8, 83 8, 46 8, 27 8, 38 8, 15 7, 90 7, 63 7, 46

s ts r rts sssrs

t str r tr r rr tr trt

137Cs55 s t t ts r s Pr rss sssrs s ss tr str rst rt

t ss r rq sss ts tr rt q r s

st s ts rts t t Al r

s s r rtr s

r

rs rrt r ss rsr rs s

rs s Prs rs r rrt rs 3a

st r st rs sr t rt s r

PP t t rss rtt srt

rrt rrt trt rt r trt

r r Prs rtt srt Prt

ts tr rrs r r t

rss rt srt r rts

rr tt ts sttst Prt rst Prss

sst st sttst

s r r str ttrtt t r

rrt rrrs Pr s Ps r

t tt r s tr r r t srt

t t t rss rtt srtst rs P

t s Prt ss rsr rs s

t st tttrrstqrsttt

t s sts s s r

s s s P

s s r rtr s