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Phy??qu??xp ???m???al?p??pa?a?o???aux?c???c??b?om???cal??ycomp???l'????o ?uc??o?ma?h?ma??qu?aux?c???c???xp ???m???al??M. Ho ?b ?k?, cha???? ?? cou??T?a?aux???????1?Ca??.Sc.M??.??Sc.D???.N. D. N?uy?? (??.??uy??@ul?.ac.b ?)Coll?c??f ?? l'E???????m????? la Phy??qu? ?? P??m???? Ca????a?u??A????aca??m?qu?2002-2003

2

TAB?E DES MATIÈRES3

Tabl????ma??????1I???o ?uc??o?ma?h?ma??qu?51.1Al??b??......................................51.2A?aly??......................................61.3T???o?om?????...................................101.4V?c??u??......................................122C???ma??qu??upo???ma?????l162.1Fo?mula??? .....................................162.2R?ma?qu??....................................172.3P?obl?m??.....................................173Dy?am?qu?193.1I???o ?uc??o?....................................193.2E?o?c?? ??? lo?? ?? N?w?o? p ou? la ?y?am?qu?................193.3?? p o??? ?? l? p o??? ??c??f ............................193.4?? f?o???m???...................................193.5P?obl?m??.....................................204S?a??qu?224.1??? co?????o?? ?'?qu?l?b??............................224.2?? mom??? ?'u?? fo?c?..............................224.3??? l?????? : a?a??a?? m?ca??qu?........................234.4P?obl?m??.....................................235??mou??m???c??cula???255.1??? ?a??abl?? a??ula????.............................255.2R?la??o?? ????? l?? co o??o????? a??ula???? ?? l???a???? .............255.3?? mou??m???c??cula??? u??fo?m? (MC?)...................265.4?? mou??m???c??cula??? u??fo?m?m??? acc?l???(MC?A)..........265.5?a ?y?am?qu? ??? mou??m???? ?? ?o?a??o? au?ou? ?'u? ax? x?.......275.6P?obl?m??.....................................276T?a?a?l,E??????,Pu???a?c?296.1D?????o?? ?? p???c?p ? ?? co?????a??o?.....................296.2P?obl?m??.....................................307Qua???????mou??m???,mom???c?????qu?,lo????co?????a??o?327.1D?????o?? ?? lo?? ?? co?????a??o?........................327.2P?obl?m??.....................................338M?ca??qu????u?????o????qu?ux348.1P???c?p ? ?'A?ch?m???..............................348.2Equa??o? ?? co????u???..............................348.3Th?o??m? ?? B???oull?..............................348.4P?obl?m??.....................................34

TAB?E DES MATIÈRES4

9M?ca??qu????u???????qu?ux369.1Ecoul?m??? lam??a??? ?a?? u? ?ub ?.......................369.2Fo?c?? ?? ??????a?c? ???qu?u??..........................369.3T????o? ?up ??c??ll?...............................379.4P?obl?m??.....................................3710 El?c??o??a??qu?3910.1D?????o?? .....................................3910.2P?obl?m??.....................................4011 Cou?a???co????u?4211.1D?????o?? ?? lo??.................................4211.2P?obl?m??.....................................4412 Op??qu???om????qu?4612.1??? l????ll?? m??c??................................4612.2?a fo?ma??o??? l'?ma??.............................4612.3?'o ??l ........................................4712.4??? ?????um???? ?'op??qu?............................4712.5P?obl?m??.....................................4713 App ????c?49

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?5

2 6?+78 43?5
182u
5 13 54
u4 =u+

2753y5

2 4y+13 =2

3y+144

3 (2?3) + 34
?2 +1 2?6 2 2?5

2?1733x1

5 x+12 <1 24ac

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?6

<0Pa? ?? ?olu??o?- =0??? ?olu??o? ?oubl?x= b 2a >0D?ux ?olu??o??x1 b+ p 2a ,x2 b p 2a Tab.1Solu??o?? ?'u?? ?qua??o? ?u ??co?? ?????Ex.3R??ou??? l?? ?qua??o?? ?u??a???? :(x+ 1)(x+2) = 2?

2(?+3) = 4(5?3)(?5) = (2?+5)

2+905w

2=4w+24y

f(x1 )f(x2 x1 x2oùl??coupl??(x1 ;f(x1 ))??(x2 ;f(x2

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?7

5 4 3 2 1 -1 -2 -3 -4 21-1
f(?) 0 -3 -2 -1 1

654321-1-2

f(x) x 0 (a) (b) (c) 5 4 3 2 1 -1 -2

54321-1-2?

f(?) 1 2 x.

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?8

-30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 2 4 -6-4-2246 f(?) 0 32
30
28
26
24
22
20 18 16 14 12 10 8 6 4 2 -2 -4 -4-224 f(?) 0 (a)(b)

22?3,(b)f(?)=?

2+?+1.

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?9

0(a)=l?mx!a

f(x)f(a)

0(x), ??? la fo?c??o? qu? a??o c?? à u? p o???x?o??omb?? ??????f

0(x). 5 4 3 2 1 -1 -2 -5-4-3-2-112345 x f(x) P 1 P 2

F??.3I????p???a??o? ??om????qu? ?? la ??????? ?'u?? fo?c??o? ?? u? p o???.?'?????p???a??o? ??om????qu? ?? la ??????? ??? ?o???? à la F??. 3. Co??????o?? la fo?c??o?f(x)=

x 2 4 x2 +1. C???? fo?c??o? ??? ?????abl? ??x=2(à a?m?????).S? o? co??????? l????uxpo????P1 ??co o??o?????(2;1)??P2 f(x1 )f(x2 x1 x2 1322
24

12S?ma?????a??o?fa???a????l?p o???P1

l?lo????lacou?b ?p ou?l'am?????u?l?p o???P 2 ,o?co???a??qu?lo??qu?l?p o???P1 p?????ucc??????m???p ou?co o??o?????(1; 3 4 ),(0;1),??(1; 7 4 ), l?? ??o????P1 P2 . Do?c, àla l?m???, la ??o???P1 P2 ??????? la ?a?????? à la cou?b ? ??P2 .?a p???? ?? c???? ?a??????p ?u? ?'?xp??m?? pa?l?m x 1 !x2 f(x1 )f(x2 x1 x2 =f 0(x2 )oùx1 ??x2 ??P2 0(x2 )appa?aî???? lo?? comm? la p ???? ?? la ?a??????à la cou?b ? au p o????'ab?c????x2 . C'???l'?????p???a??o???om????qu??? la ??????? ?? c? p o???.

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?10

??? ?o??? pa?f(x)=f(x0 )+(xx0 )f 0(x) x=x0 (xx0 2 2 f 00(x) x=x0+ (xx0 3 6 f

000(x)

x=x0 1 X?=0 (xx0 f (?)(x) x=x0oùf (?)(x) x=x0 ,???!=123?(fac?o???l l????).Ex.4D???lopp ?? ?? ????? ?? Taylo?, au ?o????a?? ???=0,(1 +?) m?? 1 (1+?) m

αab

s

F??.4A??l? o??????

l'a??l?o?????? 2

[?a?].??pa??a???'u??y???m??'u????àl'au????'??c?u??u??a??l??fo?mul????la Tab.2.Da?? laF??.4, l'a??l?o??????

ca?ba p ou? m??u??.[°]

180°![?a?][?a?]

180°

![°]Tab.2Co??????o? ??? u????? ?? m??u?? ?'a??l?.C??cl????omb???????o?om????qu????c??cl?????o?om????qu?(F??.5)???u?c??cl?c???????l'o???????'u???p ???o??ho-?o?m?.So??ayo????u???a???.????m?-ax?ho??zo??al????o???(ab?c?????po???????)???cho??? comm? ??m?-??o??? o?????? ?? ?ou? l?? a??l??. ?'o?????a??o? ??? a??l?? ?app o???? auc??cl? ????o?om????qu? ??? ?a?? l? ???? a???-ho?lo??? ou ???? ????o?om????qu?. O? y ?????qua???qua??a????o???I,II,III ??IV.??a??l?o??????b?app o???auc??cl?????o?om?-???qu? p o????? u?? ?????? ?? m??u??? ?o?? l'?xp?????o? ?????al? ?? ?a??a? ?'?c???+k(2)k2Z?am??u??p???c?pal??'u? a??l?b??? la m??u?? qu? appa?????? à l'??????all?[; ].

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?11

1 1 sin cos tg

αcotg

III III IVa y x opq co? ?? co??= co???? 2+co?

2=1p ?u? ê??? ??mo?????à l'a??? ?u ?h?o??m? ?? Py?ha?o?? ?a?? l? ???a??l?aop.Nomb???????o?om????qu?????a??l????ma?quabl???aTab.3?o???l???al?u??????omb???????o?om????qu????qu?lqu??a??l????ma?-quabl??.A??l??a??o c?????? ?omb??? ????o?om????qu?? ??? a??l?? a??o c??? à l'a??l?b?xp??m?? ?? fo?c??o? ????omb??? ????o?om????qu????b?o?? ?o???? à la Tab. 4.Ex??c?c??

Ex.5Plac???u? l?c??cl?????o?om????qu?l?? ?ma?????? a??l???o??l??m??u????o??:95 6 934
3 +k(k2Z).Ex.6Exp??m?? ?? fo?c??o? ??co?xou???x:co? (25x),??? (11+x),??? ( 13 2 +x).

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?12

???co???co??

0°010+10

30°

1 2 p 3 2 p 3 3 p 3 6

45°

p 2 2 p 2 2 11 4

60°

p 3 2 1 2 p 3 p 3 3 2 3

90°10+10

2

180°0-101

270°-1010

3 2 Tab.3A??l?? ??ma?quabl??.Ex.7Do???? l?? ?al?u?? ?xac??? ??co? 40
3 416
.Ex.8O? ?o???co? 5 p 5+1 4 , calcul?? :co? 45
,co? 65
,co? 95
.1.4V?c??u??No??o?????c??u?????c??u???? u?? ?????? ma?h?ma??qu? ca?ac??????? pa? u?? a F??.6 ~b(F??.7(1)),o???p o???l'o???????u??c??u? ~bàl'?x???m????u??c??u?~a(F??.7(2)).?????a??la??o???mo ???pa?la??a???u??'u???c??u?.?a?omm?~c=~a+ b(F??.7(3)).?'a?????o???c?o???ll????commu?a????:?ll?p ?u??'??c?u???a??u?o????qu?lco?qu? (~a+ ~b= ~b+~a).?a ?ou???ac??o???c?o???ll??? ?am???à u?? a?????o???c?o???ll?.E? ???, ?ou???a??? l???c??u? ~b?u??c??u?~a?qu??au?àa jou???à~al'opp o???? ~b:~a ~b=~a+( ~b).?'opp o??

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?13

A??l??opp o???:l?u??omm??au? 0

??? ()=???co? () = co???()=??co??()=co??

A??l???uppl?m???a????:l?u??omm? ?au?

??? () = ???co? ()=co???()=??co??()=co??

A??l??a???-?uppl?m???a????:l?u? ??????c? ?au?

??? (+)=???co? (+)=co???(+)=??co??(+)=co??

A??l?? compl?m???a???? : l?u??omm? ?au?

2 2 ) = co?co? ( 2 2 )=co??co??( 2

Tab.4A??l?? a??o c???.

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?14

a b c b aa b ab=+ (1) (2)(3) F??.7P???c?p ? ?? l'a?????o? ??c?o???ll?.?u ??c??u? ~b, ?o?? ~b, ??? l? ??c??u? ?? mêm? ??a???u? qu?

~b, ?? mêm? ????c??o?, ma?? ?????? opp o??.Mul??pl?ca??o?pa?u??cala???Pou?mul??pl???u???c??u?~apa?u??cala???(?omb?????l),?l?u???mul??pl???la??a???u? ??~apa?. ?a ????c??o? ??? co???????,ma?? l? ???? ??? ??????? ??<0.Comp o?a?????'u???c??u??? ??c??u? p ?u? ê??? ??al?m???ca?ac??????pa? ???

aa a xy xy F??.8

compo?a?????a??u???p ???o??ho?o?al?o???.Ap???a?o????plac?l'o???????u??c??u?~a????l'o???????u?y???m??'ax??x??yp ??p ????cula?????u??p ???,l??comp o?a???? ??~a, ?o????ax

??ay a x =aco?a y

=a???S? l?? ?al?u?? ??? comp o?a???? ?u ??c??u? ?o?? ?o?????, ?l ??? p o???bl? ?? ?????m???? la??a???u???la ????c??o?.E????, ?'u??pa??, pa?appl?ca??o??u ?h?o??m??? Py?ha?o??,o? aa

2x +a 2y =a

2, c'???-à-????a=

q a 2x +a

2yD'au??? pa??, pa?m? l?? ??la??o?? ?a?? u? ???a??l? ??c?a??l?,l'u?? p ??m?? ?'?c??????=

a y ax )=a?c?? a yax

S?c??o?1. I???o ?uc??o?ma?h?ma??qu?15

?'???p ?c??o??u???????ax ??ay ??? p o????f. ?o??qu'?l ??? ?????? ???? l?? ?al?u?? ??c?o???a????,ax??????a??f.?acomp o?a???ay

?? lo??u?u? ??al? à 1 ?? ?????? ?u??a?? l?? ?al?u??c?o???a???? ??x. D? la mêm? ma?????, o? ????? u? ??c??u? u???a???~?y

, ?????? ?u??a?? l???al?u??c?o???a??????y.Tou???c??u?~ap ?u??'?c????,?a??l???p ???o??ho?o?almu????c?? ??c??u?? u???a????,?ou? la fo?m?~a=ax

~?x +ay ~?yS? l? ??c??u? ~b?'?c??? ~b=bx ~?x +by ~?y , alo?? l? ??c??u?~c=~a+ ~b?'?c??? :~c=(ax +bx {z}cx ~?x +(ay +by )|{z}cy ~?y??? comp o?a???? ?u ??c??u?~c?o?? ?o?c l?? ?omm?? ??? comp o?a???? ??? ??c??u??~a?? ~b.Ex??c?c?? Ex.9(K&S2-3)?? ??c??u?a u?? comp o?a????u??a??l'ax? ???xqu? ?au?ax =10??u??comp o?a????u??a??l'ax????yqu??au?ay

=+3.(a)D???????u??y???m??'ax???? p o????o???? l? ??c??u?~a; (b) calcul?? la ??a???u? ?? la ????c??o? ?? c? ??c??u?.Ex.10(K&S2-2)Da?? la F??. 2.18, qu?ll? ?o?? ê??? la ?al?u? ?? l'a??l?p ou? qu? l???c??u?

~C= ~A+ ~Ba?? (a) u?? ??a???u? m???mum ; (b) u?? ??a???u? max?mum ? (c) Qu??au?j

~Cjlo??qu?=90°?Ex.11(K&S2-12)Da?? l? ca? ??? ??c??u?? ?? la ?u?? 2.22, ??alu?? la ??a???u? ?? la????c??o? ?? (a)D=A+B+C; (b)E=ABC.Ex.12D????m???z l?? comp o?a???? ?u ??c??u?

~F?a?? l?? ?y???m? ?? co o??o?????(~?x ;~?y 0x 0y )(F??. 9). O? ?o???F=5,=60°??=40°. e' x y ?x ?y F F??.9

S?c??o?2. C???ma??qu? ?u p o??? ma?????l16

?Acc?l??a??o?????a??a???:a=l?m?!0 ????V??????moy????:?= x x ?x??R?la??o? acc?l??a??o?-p o????o?:a= 2x 2

Mou??m?????c??l????u??fo?m?(MR?):8

a=0?=?0 (co???a???)x=x0 +?0 (??0 a=co???a????=?0 +a(??0 )x=x0 +?0 (??0 1 2 a(??0 2 ?????? MKSA :

Acc?l??a??o?am/?

2

V???????m/?

Po????o?xm

T?mp???

S?c??o?2. C???ma??qu? ?u p o??? ma?????l17

?'u? ?au??u? au mom??? où ?l qu???? l? ?ol (??????? ?'?chapp ?m???), (b) l'acc?l??a??o?a?à c? mêm? mom???.???l????, ap??? l'a?o?? ??mo?????,la ??la??o??

2=? 20 +2ax.

D???a?c? ?'acc?l??a??o? (?)Hau??u? ?????cal? (h)

Homm?0,51,0

Tab.5Ex??a?? ?? la ?abl? 1.4 (?? m?????)

1C?laco??????,pa??x?mpl?,à??ol??l???mp???a??lalo??u mou??m??? p ou?laco o??o????x??p?o c???? à ?a ?ub????u??o? ?a?? la lo? ?u mou??m??? p ou? la co o??o????y, ?o?? l'?xp?????o? ma?h?ma??qu??? ???f??m??a plu? qu? l?? ?a??abl?? ?'??pac?x??y.

S?c??o?2. C???ma??qu? ?u p o??? ma?????l18

P?ob.5(K&S1-71)Da?? l? f?u?ll??o? ??l??????'homm? qu? ?ala?? ??x m?ll?o?? ?? ?ol-la??,l?colo??lAu????al??capac?????'u??u?homm?.Aucou???'u??p??o ??,?l??????'a???ap ?? u? homm?qu? ?'??fu?? ?a?? u?? ?o??u?? ?? ?p o??. ?a ????a?c? ??????ux ??? ??100 m au mom???oùla?o??u?? comm??c? ?o? acc?l??a??o?. C???? acc?l??a??o? ??? co???a????? ?au? 5 m/?

F??.10P?ob.7O? f?app ? u?? ball? ?? ?????? à pa?????? la l???? ?? fo?? (???u?? ??x0 =0)??y 0

=1m au-????u? ?u ?ol ?a?? u?? ????c??o? fa??a?? u? a??l?=9°a??cl'ho??zo??al?.(a)Calcul??lap o?????u???(ab?c????x?

?up o????'?mpac?).?aball????-?ll???ouou?? ?a lo??u?u? ?u ????a?? ?au?xT

=23;77m. N??l???? la ??????a?c? ?? l'a??.(b)Aqu?ll?????a?c?ylaball?pa???-?-?ll?au-???u??ul?? ???l??????ou??àu?? ????a?c?xF

=11;89m ?? ?a hau??u? ?au? 0,915 m.

F??.11

S?c??o?3. Dy?am?qu?19

~F??c???a???pou?fou????u??acc?l??a??o?~aàu?ob j????ma???m????o???? pa?

F=m~a3.S? u? ob j?? ?x??c? u?? fo?c?

~F?u? u? ??co?? ob j??, l? ??co?? ?x??c?, ?u? l? p??m???u?? fo?c? ??al? ?? ??a???u? ma?? opp o??? ~F.

?,???????comm???a??lafo?c??o?al?qu?c????p ???o???ouc??ob j???x??c??u?u??bala?c?.C????fo?c??????al???opp o???àlafo?c?

~S?x??c??pa?labala?c??u?lap ???o???ou?u?l'ob j??.D?faço??????al?,l? p o??? ??c??f ?'u? co?p? ?? ma???m?? acc?l??a??o?~a??? ?o??? pa?~w

?=m~?m~a3.4??f?o???m????o??qu'u??fo?c????appl?qu??àu?ob j??au??p o??u?u???u?fac???qu?c????fo?c???pa???lafo?c???f?o???m?????a??qu?max?mum,

~f (max),?lcomm??c?à??mou?o??.Exp ???m???al?m???,o?mo????qu?quotesdbs_dbs47.pdfusesText_47

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