2011 . . -. []/ .: . . , . . , . . . .: , 2011. 58 . ( 5 3.02.2011 .) : , . .-. , . , . .-. , . , . .-. , . , . .-. , . , . .-. , . . . , - . - . , . . . , - . - . , . 3 . 1. -------------------------------------------------------------------------------------------------------------------------- 4 2. ------------------------------------------------------------------------------------------------------------------------------ 15 3. ------------------------------------------------------------------------------------------------------------ 29 4. ------------------------------------------------------------------------------------------------ 40 5. ----------------------------------------------------------------------------------- 50 6. -------------------------------------------------------------------------------------------------------- 55 4 1. 1.1. : ;rtA=Avd.drt= v 1.2. : atA=A v d; .dat= v 1.3. - ( ): 00( ) ( )dta t t = + v t v ;00d .tr r t = + v 1.4. [t1, t2] : 2 21 1( )d ; ( )d , ( ) ( )t tt tr t s t t t t A = = = v t v v v . 1.5. , : 2d; ;dn na n a a a aR tt tt = = = + v v, R . 1.6. e | : d d; .d d t tm ee | = = 1.7. - : 2 2 4; ;; ;ns R Ra R a R a Rtm e| e | e= == = = +v , R . 1.1 a, OX . a b=2a 2 c a = ; b c? ? ? - b c. 1.2 b d 1 2a i a , a : ) , ) -. 1.3 , , , a b c d (. 1.1). (-) S a b c d = + . ( ). S a = a c b d Y . 1.1 1 5 1.4 OX - 4 10 S = , - 30. ( ) S. -. ( = y = 23,7 , S = 33,46 , 45.) 1.5 110 F = , 220 F = 340 F = 120 (. 1.2). F . ( ) 310 7 26, 5 H, 19 . F i F = ~ 1.6 1 2 1F 2F (. 1.3). F . 1F 2F. 1.7 () - 1 2F F F = + 1F 2F (. 1.4). . . 1.8 (. 1.5). - 1, 2, 3 1 2 3, , v v v , 1 2 3, , a a a 1 2 3, , F F F , , : ) , ) ) -. 1.9 = 30 - 05 / = v . , - . Av -. 02 sin , 5 /.ggo| |A = |\ .v v 1F 3F2F. 1.2 2 1 3 . 1.5 1F 2F . 1.4 F 1 2 F 1 2 . 1.3 6 1.10 , . 1.6. - Av Av . ( ) ) 2 , 0; ) 2 cos , 0. a o A = A = A = A = v v v v v v 1.11 -, , . ( ) , . A = A v v v = v 1.12 , ? 1.13 , ? 1.14 : (1; 3) a ( 3;1) b. ( )2 3. 1.15 , OX, - (5;5) F. 10 S = . (0,5 .) 1.16 (8, 66; 5) S . , , 10 F = , 86, 6 A= . , - OX F. ( 0 60. ) 1.17 : ) ; ) - ? 1.18 , - -: ) ; ) ? 1.19 , - . , - ? 1.20 , : 1) ;2) , ;3) , ;4) , .constconst constconst constconst const== == == =vv v vv v vv v v 1.21 , 25 4 x t t = + (). v oo) ) v . 1.6 1 7 , 2 - . ? ? ( 29 ; 0 /; 2 / ; 4 ; 4 . ) 1.22 : 2 312 4 x t t t = + (), 2 322 4 x t t t = + (). ? . ( 0, 4 c; 0, 68 /, 0, 72 / . ) 1.23 - 2r ti t j o | = + (), o , | , i j Y. : ( ) y x ; v, a, ; m v a. 2 2 2 22; ( ) 2 ; 2 ; ( ) 4 ; ( ) 2 ; .2y x t i tj a j t t a t tgt| oo | | o | | mo || |= = + = = + = = |\ . v v 1.24 XY sin x A t e = , (1 cos ) y A t e = , A e . : y(x); v a ; o ; S , t . ( 2 2 2 2 0( ) ; ; ; 90 ; x y A A A a A S A e e o et + = = = = = v ) 1.25 25 4 x t t = + (). : (t), ( )xt v ( )xa t , ( ) S t ; x( , v - ( , v 1t =1 2t =6 . ( )3, 0/; 3, 4/x = = v v . 1.26 23 2 x t t = (). : : ( )xt v ( )xa t , ( ) S t ; 1t =0 2t =5 xa ? ( 0, 75c; 4/ 2.) 8 1.27 ( )00 = v t S, n . . 22 ( 1).( 1)S nn t| | |+\ . 1.28 10 , 30 . , . ( 30.) 1.29 , 100 . 4 , - 10 . . ( 7,5/; 17,5/ .) 1.30 , -. , . t1/t2 , n = 1,2 . ( )( )21 2/ 1 1,8. t t n n = = 1.31 , - v = 20 /. , , . : , . (40/.) 1.32 , 5 /, . , - , 1 /. ( 3 /. ) 1.33 , 1 /, - , 3 /. . g = 10 /2 ( 0,8 . ) 1.34 10 /c, 600 -. ( ). ( 30o , 11, 55/; 13, 3/ .) 1 9 1.35 n , . v . 1 2 v v . 1 21 1,2 2n nn+ +| |= = |\ .v v v v. 1.36 12 /. 6 /, 4 /. . ( 7/.) 1.37 >120 /. 160 /. ? ( >102,9 /. ) 1.38 , , , , , S. , ? - . ( 2S. ) 1.39 . 1.7 , - . , - ? ( . ) 1.40 , , 30, 45 60 . 1:2:3, l1:l2:l3, ; , - l (1o ) (2o )? ( 1:2:3 = 1:2:3, l1:l2:l3 = 1:1,155:1; 1o = 45, 2o = 76. ) 1.41 v0 = 10 / o1 = 15, o2 = 75 . S t = 2,5 . - . 1.7 10 : ) ) . (2 102 sin 25.2S o o t= = v ) 1.42 , 15 /, - 60 . ? ? ( 34,4 ; 40 . ) 1.43 , 30 , , 10 /, -. . ( 2 c. ) 1.44 2 - 30 ( ) . ? - . ( 8. ) 1.45 30 15 /. . . (2 28,5/ ; 4,9/ ; 26,5 .) 1.46 0v h . , : ( ) y f x = ; ( ) t v ; ( )na t , ( ) a tt ( ) a t ; ( ) R t . 32 2 2 222 2 2 2 0 00 22 2 2 2 2 20 00 0( ); ( ) ; ; ; .2ng g t g g ty h x t g t a a Rgg t g tt| |+ |= = + = = = |+ + |\ .v vv vv vv v 1.47 50 R = . 2( ) 10 10 0,5 l t t t = + (). t = 5 , , -. ( )2 2 25/; 1/ ; 0,5/ ; 1,1/ . 1.48 -. m , ) 0,1 , ) 0,25 -. ( ) ) 51,5 , ) 72,5 . a ~ 1 11 1.49 2,0 / - 4,0 /2 , - - = 60 m . , 2,0 ( 14,4 . ) 1.50 , - - , . m . ( m = arctg2 ~ 63,4. ) 1.51 1,34 /2 , - , . - . ( 3,00 /2. ) 1.52 = R 16 , - 22 ) ( t t l = (). : , -; . (22c; 5, 6/ , 45 .) 1.53 = R 1,00 sin l A t e = , l , = 0,80 , e =2,00 1. l A = = l 0 . (2 22,6/ ; 3, 2/ .) 1.54 , - , x o = v , 1 22 c o = . t =0 0x =0 . - , 4 . ( )22 ; 2 / ; 2/. t a = = = v v 1.55 , , -, a p = v , 1 2 3 22 c p = . 04/ = v . : ; 12 ? . ( )2 2( ) (2 ) /c; ( ) 2( 2)/c ; 2, 67; 2c; 1, 33/c. t t a t t = = v 1.56 , , R , - -. t = 0 0= v v . t S. 2 2 20 0 00 20 02 2( ) ; ( ) ; ( ) ; ( ) .( )S SR RR R Rt s e a t a s et R t R R | |= = = = |+ +\ .v v vv v vv v 1.57 2a ti t j o | = + , = o 3 /c3, = | 2,25 /c4. t = 2c. ( l = 5.) 1.58 ( cos sin ) a i t j t o e e = + . t = 0 0 0j = v v . 0, , o e v . x(t), y(t). 02 2(1 cos ), sin . x t y t to o oe ee e e| || |= = + | |\ . \ .v 1.59 , 0v , , 0ktxa ke= v , k . ? 0. Sk| |= |\ .v 1.60 , 1,5 . ( 18; 216. ) 1.61 , 6380 ? ( 464 /. ) 1.62 50 30 /. 57 /. ? ( 1 /, 2,14 /. ) 1.63 , 16 = v , , 15 = l , 25, 5 = v . ( 1,8. ) 1 13 1.64 , m - t 2kt m = , 20, 2 k = . - 2,5 t = c, 0, 65/ = v . ( 20, 7/ .) 1.65 = R 50 33 2 0, 01 () t t m = + . , 10. t = ( 2 2 20, 3/ ; 0,5/ ; 0, 58/ .) 1.66 , 50 -1, - , 1680 . . ( 4,7 /2.) 1.67 0,01 /2. 80? - ? ( 23,8 ; 0,45. ) 1.68 - | = ot, o = 2102 /3. t, m = 60. ( )3(4/ ) 7 c. t tg o m = ~ 1.69 - e0 = 5 / | = a bt, a = 4 /2, b = 2 /3. ? ( 5,3. ) 1.70 , , - k | e = , 32 k = . 0 t = 09 e = . - . ( 23 /; 3 / .) 1.71 , 0e e om = , o -. 0 t = 0 m = . : ( ) t m ; ( ) t e ; 14 ( ) t | . 00 0( ) (1 ); ( ) ; .t t tt e t e eo o oem e e | e oo | |= = = |\ . 1.72 - v, . 1.8. A, B, C D. (2v; 2 v ; 0; 2 v .) 1.73 (.1.9). A B Av Bv . .( ) ( ) ( )2 2B A B. v v 2v 1.74 , , - v = 2 / (. 1.10). 2 , 1 v1 = 1 /. ( 3 /, 5 /. ) 1.75 0,50 R = - 1, 0/ = v . : ; , - . ( 2 /2, ; 4 .) .1.9 - - A B - - - B C D .1.8 v A . 1.10 2 1 v 2 15 2. 2.1. ( ): ddpFt =, ma F = . 2.2. 22ddrm Ft =. 2.3. : ima F F = + . 2.4. : | |20; ; 2 i F ma F m r F m e e = = = v . 2.5. : ;i iic c iimrr ma Fm= = . 2.6. : 21d; d .dt tpF p F tt = A = 2.1 , , 150 . - , : , 150 ? 100 ? 2.2 F . ? 2.3 m1 m2 = 2m1 . , ? . ( )1 22 , a g a g = = . 16 2.4 4 , 0,3 , . 2.1. -, 2,5 . ( 1,225 H. ) 2.5 m, - , . - . - , ? ( 2mg. ) 2.6 - (. 2.2). - F. F , : ? 2.7 m1 = 2 m2 = 4 . , - 10 , - , 2,5 . . ( 5 .) 2.8 k1 k2 -, . k , ? 1 21 21 2; .k kk k k kk k| |= = + |+\ . 2.9 ( ) - . - ? : . (2g.) 2.10 , 400 . 30 , 300 ? ( 0,6 . ) 2.11 , , 0,5 /2. - . - , 0,1? ( 1,5 /2. ) 2.12 , . 2.3. a , , . 2.1 o a . 2.3 F o . 2.2 2 17 o = 60 , , ? ( ~ 7 /2. ) 2.13 0 , 1 - . 2 ? ( 2 = 20 1. ) 2.14 = 420 -. m , , n = 5% ? 240 .1m Mnn| |= = |+\ . 2.15 F m = 2 , , = 20 /2? k = 0,01. ( 0,6 . ) 2.16 5 , , 4 . , , 0,1. ( 0 /2 4 . ) 2.17 , , , . , - , k? ( k > tgo. ) 2.18 5 30. : ; ; 2 . ( ~ 43 ; 5 /2; ~ 10 /. ) 2.19 , - , m (. 2.4). F ? . ( tg ; cos a g F mg o o = = .) 2.20 m , , k. , , o . o a. 2.4 18 2.21 , - , . = 14 k? ( k = tgo = 0,25. ) 2.22 , - 45, t1 = 1,3 . t2 , k = 0,4? 2 112c.1kt tk| |+= ~ |\ . 2.23 v o . v , k (< tgo)? ( )2 2 22 sin (sin cos ). k o o o v 2.24 m = 50 , - F = 300 . - , . = 30. , g = 10 /2. ( ~0,2 /c2, 583 . ) 2.25 m, , F. - F , k < tgo? tg tg.1 tg 1 tgk kmg F mgk ko oo o| | +s s |+ \ . 2.26 o, - , m (. 2.5). - k (k > tgo). - , , , ? ? 2 2 2tgcos sin ; arcsin . F mg k Fkoo o || || |= = | |\ . \ . 2.27 F, - . , - k. -? ( ) cos sinFa k kgm o o| = +

\; cos.sinFkmg F oo |= | . o . 2.5 F 2 19 2.28 , , o = 23 -. , . 0, 2.2k tgo| |= = |\ . 2.29 m - k. , ? 2.1kmgk| | |+\ . 2.30 m1 = 50 m2 = 100 , , T = 5 , . - F , ? ? ( 7,5 ; 15 . ) 2.31 m1 m2 , , , . o = 30, k = 0,2. - , m1/m2 = 0,25? ( 0,6 /2. ) 2.32 R = 4,85 - . , t0 = 2 0 = 1,2 /. t , k = 0,02? 20 040 01 20c.t kgtRte e| || | |= = | |\ .\ . 2.33 m = 200 , L = 3,0 , R = 1,0 . . ( )12 2 2117, 8 ; 2,1.21g mgL RR Lt | || | |= = | |\ . \ . 2.34 R = 500 v = 360 /. , m = 70 , . g = 10 /2. (2,1 ; 0,7 ; ~1,6 . ) 2.35 m Y sin x A t e = , cos y A t e = , , . , . ( F = m2; . ) 20 2.36 - m = 1,0 ( ) ( ) ( )2 31 2r t a t i a t j t t = + + , 1 = 30 /2, 2 = 1,0 /3, t = 10 . : ( ) F t, F0 0o , ; , 90. ( ) ( ) ( )o1 2 0 02 3 ( ) , 85H, 45 ; 20 . F t m a i a t j F t o = + ~ = 2.37 m , , ( )21 2F t ti t j o o = + , 1 2 . - ( ) tv ( ) S t . 2 31 2;2 3t ti jm mo o | = +

\ v3 41 2.6 12t tS i jm mo o |= + |. 2.38 m = 1 |.|

\| =ttF F 10 , 05 H F = , 10 c t = . t = 10 . 2; 25 .2 F ttm t| || |= = | |\ . \ .v v 2.39 m, , F = t, . y = y(t). : 0;mgt yo| s =

\ 3 2: ( ) .2 2mg t gtt y tmoo |> = |. 2.40 m F = F0sint, F0 . S(t) - . ( )02( ) sin .FS t t tm e ee| |= |\ . 2.41 4 /. , - 3 /? . ( g/4. ) 2.42 m v0. t = 0 - . , - F r = v, v(t) - v(s), Smax. 00 0 max( ) ; ( ) ; .rtmm rt e s s Sm r | |= = = |\ .vv v v v 2 21 2.43 , v0, h v. , - . ( )000.lnht| | | = | |\ .v vvv vv 2.44 , , 0. , , - . - . ( ) 0 02 ; . g 2.45 , o = 45 k = x, = 0,5 1, x , . vm Sm, . ( ) m msin cos 3, 7/; 2tg 4, 0. g S o n o o n = ~ = = v 2.46 m = 2 (t) = 58t+4t2(). 2 4 . ( 16 /; 48 /; 16 . ) 2.47 m - v0. p A t . ( ) . p mgt A = 2.48 , m, v0. . ( 2mvosin; 0. ) 2.49 100 , , - 20 /. p A p A , : ) ( ); ) ( ). ( ) p A = 4 /, p A = 0; ) p A = 2 /, p A = 2 /. ) 2.50 -, 20 20 /, . , , - 0,1 . ( 8H. ) 2.51 10 45 ( ) . - 22 , , 0,2 . g = 10 /2. (0,4 . ) 2.52 , 10 , 400 / 300 . ( 20 . ) 2.53 m l , - . t = 0 . F(t) , . 222( ) ; 2 .3mg mF t t p gll| |= = |\ . 2.54 , k = 0,5. . ( 2g.) 2.55 , m. , k? . ( ) ( ) . k M m g + 2.56 o , , . - a , k ( k < tgo )? sin cos sin cos.sin cos cos sink kg a gk ko o o oo o o o +| |< < |+ \ . 2.57 o , , . - a , k ( k > tgo )? tg tg.1 tg 1 tgk kg ak ko oo o| | +s s |+ \ . 2.58 m = 60 R = 3,0 , = 1 / , . , , -, , . (2/ 4 45 F m R e = = .) 2.59 , = e 6 /c, m = 0,5 2 23 ' v = 50 / . , , r = 30 : ) ; ) . ( ) 6,2 ; ) 8,4 ; 5,5 . ) 2.60 m = 1 ' v = 30 /. F F , , -: ) 60 m = ) . . ( ) F =1,7 210 ; F = 3,8310 ; ) F =3,4 210 ; F = 0. ) 2.61 m = 2000 = 60. : , v = 54 /; , - , , - . ( 3780H F = , ; ) 420 /, . ~ v 2.62 = 5,0 / , . m = 60 v0 = 2,6 /. , , t = 0,5 . ( )202 1 ( ) 4, 2 H. F m t e e = + = v 2.63 =2,0 / , -. m = 0,5 , - v0 = 1 /. ( - ) , r = 50 . ( )2 202 1 ( / ) 2,8H. F mr r e e = + = v 2.64 , , . , , h -. = 60, - v = 900 /, S = 1,0 . 22 sin7, .Sh T i Tt m | |~ = |\ .v 24 2.65 h = 500 . , , s - . 2, 2 / 24 , - . 3 s h h g e e| |~ = |\ . 2.66 , o , - v. ( tg 2 sin / g o e m = v .) 2.67 0/ P P c = A (0 ) , . 2 3 24 / 0, 34%; , ,, .R GMT R M i G ii c t | | = ~ | \ . 2.68 , n m = 60 , , . 2 3 2 24 cos / 0,1%; , ,, .R MT R M i ii c t m | | = ~ | \ . 2.69 1, 2, 3, 4 - 30 . - . ( 3 . ) 2.70 , 2 , 4 , 6 8 , 2 . . ( 1,72 . ) 2.71 2 , 4 6 - , , 6 /, 3 / 2 /, . . (0.) 2.72 m = 5 20 2,5 / , : ) ; ) . . ( ) 0; ) 2,5 /. ) 2.73 , , - m1 m2 . - . ( )( )21 221 2.m ma gm m| |= | |+\ . 2 25 2.74 , - m1 m2, 0P. , - ( ) P t - cr . ( )20 1 2 0 0 0 1 2( ) ; / 2, ( ).cP P m m gt r t gt P m m = + + = + = + v v 2.75 m , v 3v. P. (m1 = 3/4; m2 = /4; P = 0.) 2.76 . , 25 . , 8 /. , - 100 ? ( 8 /; 2 /.) 2.77 , 10 / - 60 , -, , 45 -. . (v1 = 27,3 / 7,3 /; v2 = 14,1 /. ) 2.78 m1, v - m2 u . ? ( )( )2 21 2/ . m m u + v 2.79 , , - , . - ? ( 26,6. ) 2.80 , , m. - v . u - . , : ) ; ) . ( ) ( )) ; ) .m M mu m M mua m M m M+ + | | |+ \ .v v 26 2.81 60 o = . - , v0 = 180 /c. , u , -. = 50 , . 0 0cos cos1, 8/.1u o on n| |= ~ = |+\ .v v 2.82 , v o , - , -. n , k. : , - ; o , . ( ) 1) sin (1 ) ; 2) arctg . kg k o n o + s v 2.83 R = 6370 - g=9,83 /2. (5,981024 .) 2.84 (27,32 ) (384,4 . ). (5,981024 .) 2.85 , (149,50 . ) 365,26 . (1,991030 .) 2.86 1v , , , . ( ) 17, 9 /. gR = ~ v 2.87 2v , - , . ( ) 22 11, 2 /. gR = ~ v 2.88 3v - , , . . 2 2 2 2 23 1 1 1 1 002 ( 2 1) 17/, / , / ,). V M R V M RR | |~ + ~ = = | |\ .v v v 2.89 , . - , ? (Fc/F=2,1.) 2 27 2.90 227,70 365,26 . (1,08108 = 0,63 . .) 2.91 h R , ? ( ) ( )2 1 . h R = 2.92 -? . ( )32 1 25 T R M t = = . 2.93 , , ( ) V1 = 0,8 /. r1 = 61012 . V2 = 50 /. r2 ( ). 12 211 2 121110 .2V VrM r | || | | = + ~ | |\ .\ . 2.94 , . , R g? - . ( maxgR = v .) 2.95 , - r1, r2. . ( )( )31 22 , 1 2 . T M a r r t = = + 2.96 m R . : F(z) z - ; F1(z) F2(z) - z F(z); zm, F . ( )3 2 3 22 21) . 2) ; . 3) .2mmMz mMz mM RF F F zR zz R | | |= ~ ~ = |+\ . 28 2.97 , - - a. G . 4.3G atp| |= |\ . 2.98 V0, . R g, , . . ( )20/(2 / 1). h R gR V = 2.99 , , p = 3,3 /3. ( )3 / 1,8 . T t p = = 2.100 . h , V. ( )2/( / 1). h R gR V = 2.101 , . , ' ? ( )2 43( / 2 ) 4, 2 10 ; 3,1/. r M T t = = = v 2.102 , , - . , = 45 . 2 00 2 1 sin cos 17/c .M M RR M R m m| | | || | || = + ~ | | |\ . .\ .\v 3 29 3. 3.1. : d ;dAA F r P Fto= = = v . 3.2. : E A = K U A A A + A = + 3.3. : 2 22 11 1grad ; d dlF U U U U F r F l = A = = = 3.1. x = 10 + 2t + t2 10 , 60 . 10 . ( 600 ; 60 ; 110 ). 3.2. , 12 r i j = + 22 3 r i j = , , 3 4 F i j = + ( ). -. - -, . ( 2 1( ) 17 A F r r = = ). 3.3. , 5 1 /, 2 , . : ; , -; . ( 20 ; 0, 2 , 0; 1,27 ). 3.4. , 5 1 /, . 2 , . -: ; , -; . ( 8,66 ; 2 , 1 , 1 ; 0,83 ). 30 3.5. = 1 o = 30 v0 = 20 /. - , g = 10 /2. ( mgv0sino = 100 ; 0). 3.6. 2 / 10 . . ( 0). 3.7. 60 3,5 2,8 . - (1 .. = 735 ), . ( 1 ..). 3.8. , , l , . , , k1 k2. ( A = mgl(k1 + k2)/2). 3.9. F = kx, , - , k = 0,1 /. 10 . ( 5 . ) 3.10. m = 1 F = kt, t , k = 0,1 /. t = 10 . ( )2 48 12, 5 . A k m t = = 3.11. , , ? l, h, - k. ( A = mg(h + kl). ) 3.12. = 500 , F = 3,6 . , l = 5 - v = 6 /? ( ) ( )2arccos 2 60 m Fl = v 3.13. - 100 /. - ? ( 141,4 /. ) 3.14. , v, - S. S1 - v1 = 2v? . (S1 = 4S. ) 3.15. , -, . 3 31 30 /, . -. ( 21,2 /; 42,4 /. ) 3.16. - . N , - n = 11% . - . ( )21/(1 (1 ) ) 4,81 5. n N n = = = 3.17. = 100 - R = 50 . - F = 22,5 . - , - ? ( )2 / 15 /c. FR m = = v 3.18. R , - = oS2, o . - , F(S). ( )2( ) 2 1 ( / ) . F S S S R o = + 3.19. l . , , - : ) ; ) . ( ) mgl/2; ) mgl/4. ) 3.20. - m . (U = mgh, h ) 3.21. , , . m = 200 , h = 5 . g = 10 /2. ( = 5 . ) 3.22. l, U- , - , . 3.1. . , , - , , - . ( = mgl/4. ) 3.23. 1 2 . . - , g = 10 /2. ( 10 . ) F - . 3.1 32 3.24. , - 10 , 5 ? g = 10 /2. ( 2,5 . ) 3.25. 10 , 1 100 . ( 50 . ) 3.26. 1 5 . , 1 ? (15 .) 3.27. = 1 , k = 20 /, u = 0,2. , , , -? ( = (ug)2/2k = 0,1 . ) 3.28. 1 , 0, 1 /. -, 50 -? g = 10 /2 . ( 10 . ) 3.29. v0 = 20 /. -. . ( )20 0/ 2 20 ; 2 14 /. h g gh = = = = ~ v v v 3.30. , , h = 40 , . - . ( )8 / 3 32 / gh = ~ v 3.31. k m - . . . , U A , U A U A . . 3.32. = 5 , h = 20 , 16 /. , . ( 18 .) 3.33. , , ( ) h = n (n < 1). , c , . ( c = (1n)/(1+n). ) 3 33 3.34. 0,5 2,25 3 /. - , ( ) . ( 0,5 . ) 3.35. m = 20 (0;0) - U = o(x2 + xy + y2), o = 0,1 /2. (5;10) (). - , = 26,5 . ( 30 /. ) 3.36. = 20 , - 1 /, . , k = 10 /. ( 10 . ) 3.37. l = 3,6 30. , g = 10 /2. ( )6 /. gl = = v 3.38. l, , . -, . ( )/ 2. gl = v 3.39. , (. 3.2). - h, Sm, Sm. ( h = H/2; Sm = . ) 3.40. 2.46, - . ( ) m msin cos 3, 7/; 2tg 4,0. g S o n o o n = ~ = = v 3.41. l = 1,35 , . , - n = 1/3 . ? ( )(1 ) 3/c. gl n = = v 3.42. 30 - = 30 . l = 3 , = 100 . , ? g = 10 /2. ( 1000 . ) S h H . 3.2 34 3.43. - , . , k. (o = 2arctg k). 3.44. ( ) h , , . , , ? ( = 2mgh. ) 3.45. 2.45 . ( m = 20; m 0= v gX .) 3.46. m = 10 h = 40 m = 10 . 0, - , ? - . 2m0m1.2( )XXh X| |= = |+\ . 3.47. m k , - . , . , , . 2.mgXk| |> |\ . 3.48. l = 7 , - . - k = 0,1. . v , n = 7 . g = 10 /2. 20,5 /c.( 1)kgln n| |= = |+\ .v 3.49. m1 m2, 1v 2v , . A. - A . 2 1 21 21 2( ) .2( )m mKm m| |A = |+\ . v v 3 35 3.50. m1 = 4 m2 = 10 , v1 = 12 / v2 = 4 /, . Q, - , : ; . ( 365,7 ; 91,7 ). 3.51. m1 = 4 - m2 = 2 . = 6 . 1. 1 2119 .m mK Km| | += = |\ . 3.52. , v1 = 2 / v2 = 4 /, . 1 2K K , v = 1 / . 125.4KK| |= |\ . 3.53. m1 m2, , . 20 - . , 1 2m m . 2120.mm| |> |\ . 3.54. m1 m2 = 1,5 m1 - l = 1 , . o = 60 . , h - . ( )( )22 11 cos8 c.1 ( /lhm mo| |= = | |+\ . 3.55. , - - . 3.56. , v, . ? , . (v1 = 0, v2 = v.) 3.57. m1 . . m2. (m2 = 3m1.) 36 3.58. m1 m2. - ? (m2 > m1.) 3.59. m1 v1 = 3 / - m2, v2 = 1 /. - . - m1/m2. 1 1 22 12 1.3mm| | = = |\ .v vv 3.60. m1 = 2 - m2 = 6 . , 2 0K K . 2 1 220 1 24 3.( ) 4K m mK m m| |= = |+\ . 3.61. m1 m2. - . m1/m2, n = 3 : ) ) . 121) 21mam nn| += =

\; 121 1) .1 2mm nn | = = |+ . 3.62. m1 = 5 m2 = 2,5 . . 0 1, 2 = 5 . (0 = 5,62 , 1= 0,62 .) 3.63. m1 = 1 m2 = 2 , , -, U = 3 . , ? 211 1 22 2 ;( ) m Um m m|= =

+\v122 1 22 1 .( ) mUm m m |= = |+ .v 3.64. 900 , , , 2 . , - 9 ? ( 100 / 200 /). 3.65. . - v1 v2, o. 3 37 v1' v2'. |, . 1 21 2arccos cos . | o| || |= | |' '\ . \ .v vv v 3.66. . . - , , o . .2to| |= |\ . 3.67. m1 m2 2to = . , . 0 1 10 1 22.K K mK m m| | = |+\ . 3.68. 1 2 o = 60 - . 1 2m m . 121 2cos 2.mm o| |= + = |\ . 3.69. , v0 = 2 /, - o = 30 - . v1 v2 | - 0v 2v . 1 0 2 0cos , sin , .2to o | o| |= = = |\ .v v v v 3.70. m1 = 2 1 = 100 m2 = 3 . o, - , 2 = 50 . 1 1 1 2 21 1 1 22 ( )cos 0,53.2 ( )m K m m Km K K Ko| | += = | |\ . 3.71. m1 m2 - H. - . - 38 , . h ? 21 21 2.m mh Hm m| || | | = | |+\ .\ . 3.72. M, - , - m. - 0. h, - v. 2( ).2M mhMg| | += |\ .v 3.73. m M h, -. v , ? ( )( )2 . gh M m M = + v 3.74. m = 10 , - M = 100 , v = 5 /. k = 0,2. - , , . A , . S, . 2 22114 ; 0, 53.2( ) 2 ( )mM MmA SM m kg M m| |= = = ~ |+ +\ .v v 3.75. M1, M2. -. m v . - . . 22 12 1 2.2 ( )M mAM M M| |= |+\ .v 3.76. m, v1, ' M, . ' , , v2. Q, . ( )22 21 2 1 2.2m mQM| || |= | |\ . \ .v v v v 3 39 3.77. m = 100 . , v = 100 /, . Q1, , Q2 = 90 . , - . ( ) 1 2 22 4 240 . Q mQ Q = = v 40 4. 4.1. : ddzzLMt = . 4.2. , : z zI M | = 4.3. : 2 2d dVI r m r V p = = . 4.4. : 2cI I ma = + 4.5. : l 2/12 ml () - R () 2/ 2 mR R 22 / 5 mR 4.6. : dzA M m = . 4.7. , : 22IK e= . 4.8. : 2 22 2c cm IK e= +v. 4.1 p 0L - . L ', - 0r, . | | ( )0 0, L L r p = + 4.2 m v - R. L(z) - ' , 4 41 , z . -, L ? , - '. ( )2 2. L m z R = + v 4.3 , (4, 3, 0) , (3 6 ) H. F i j = M d . ( )0, 33 ; 4, 91c M k d = = 4.4 ( ) d ( ). , -, , - . 4.5 4 0,3 2,5 . , , 4.1; , . ( 1,225 H ) 4.6 m = 0,5 - , (. 4.2). . k , o = 45. ( ) ( ) ( )1 sin 1, 4; 1 cos 2, 9 k T mg o o = ~ = + ~ 4.7 m = 1 . - 1F, l = 20 1F 2F 5 . L, - = 2 /2. 221FL lma| |= = |\ . 4.8 l . - - (R/r) = n. l, F . 4. 3 l . .4.2 . . 4.1 42 F (. 4.3)? 0 00 0,, arccos(1/ ). o o o oo o o n< >| | |= =\ . 4.9 F o , m k ? 0 00 0, , arccos(1/ )sin coskmgF o o o no o| |= < = |+\ . 4.10 l m , : ) ; ) . 2) ;12mla|

\ 2)3ml ||. 4.11 m - R , : ) ; ) ; ) . ( () mR2; ) 2mR2; ) mR2/2. ) 4.12 m - R , : ) ; ) -; ) ( ) mR2/2; ) 3mR2/4; ) mR2/4.) 4.13 R - , : ) ; ) . ( ) 2mR2/3; ) 5mR2/3.) 4.14 R - , : ) ; ) . ( ) 2mR2/5; ) 7mR2/5) 4.15 R = 30 r = 10 = 1 . J , . 2 22 2( )5 10 2m R rJ | | += = |\ . 4 43 4.16 1,4 . , , , 0,9 2. ( 1,0 2,1 ) 4.17 - 1 l , 5 20 (. 4.4). , . ( 7,6 2.) 4.18 m1=10 - . , m2 = 2 . - , ? 2 21 222,8 /c2ma gm m| |= = |+\ . 4.19 m = 200 , - m1 = 150 m2 = 250 . - , ? , . 2 2 12 11, 96 /c/ 2m ma gm m m| | = = |+ +\ . 4.20 1 = 1,0 m2 = 0,5 -, , . 4.5. T1 2 . m1 k = 0,2. = 3,0 , . 2 2 11 21 20, 98 /c ; 2, 94 H; 4, 41 H/ 2m kma g T Tm m m| | = = = = |+ +\ . 4.21 R = 1,0 t , = 400 . , - F. F, - k = 0,5, t1 = 2t ? . ( F = 200 .) .4.4 . 4.5 44 4.22 m = 50 R = 20 - n = 480 /. , : t = 50 ; N = 200 . ( = 1,0 .) 4.23 , - , l m. . , , . xa gl| |= |\ . 4.24 , , l m. . , , . 22m xa gM m l| |= |+\ . 4.25 l . . . 2xa gx l| |= |+\ . 4.26 m R . M, , t. - k? 2 3;3 4kmgR RMkget| |= = |\ . 4.27 R, , - . , k? - ? ;3Rtkge | =

\ 2 26m RA e |= |. 4.28 R , r. : - F(); 4 45 h , ; . ( ) (17cos 10);7mgF o o| =

\ 10;17h R = 1017gRe |= |. 4.29 . : t , , h = 10 l = 2 ; k , . 2 224 c; 0, 025.22l h hkl ghh lt| |= ~ = ~ = | |+\ . 4.30 . : t , , h = 1,25 o = 30; k , . 1 3 11, 0 c, tg 0,19.sin 3hkgt oo| |= = = = |\ . 4.31 . -: t , , h = 1,75 o = 30; k , . 1 14 21, 41c, tg 0,165.sin 5 7hkgt oo| |= = = = |\ . 4.32 o , . . sin 1 2, 1 ,1 2 5ga i i io | |= = = = |+\ . 4.33 o -, . , k . 1 2tg , 1 ,1 2 5k i i i o | |> = = = |+\ . 46 4.34 - . . - ? 4.35 m = 60 - k = 1,0. R r n = 2, I = mR2, = 0,7. F = 0,3 , - o = 45 (. 4.3). | , r = 1,5 . 2 2( cos 1)0, 6 /c , 20 /(1 )F aam Rn o |n | | = ~ = ~ |+\ . 4.36 , 1,0 , - , 80 , . - , . (2,0 .) 4.37 1500 /. , , 500 . ( 3,18 . ) 4.38 J = 50 2 = bt + t2, = 2 , b = 16 /, = 2 /2. (t) t1 = 0 , t2 = 4 , t3 = 8 . ( (t) = 2J(2t b), 1 = 3,2 , 2 = 0, 3 = 3,2 .) 4.39 40 2 - 20 . 10 . ( 500 ) 4.40 80 30 n = 10 /. , : ; , . ( 1. 7,1 ; 2. 28,4 . ) 4.41 m = 10 v = 800 /, - n = 3000 /. d = 8 , K . (2 2 2 2( / 4)(2 ) K m n d t = + v = 3,21.) 4 47 4.42 = 4 , v = 1 /. ( 3 ) 4.43 800 , , 14 . . ( 5 /.) 4.44 , - v. m. m0. 201( 6 ) 12 2K m m| |= + = |\ .v 4.45 m , o . . 2 2 25sin14K mg t o| |= |\ . 4.46 , h = 1 . -. ? ( 12 4, 43/c gh = = v ; ) 23,13/c gh = = v 4.47 , h = 1 . . ? ( 12 4, 43/c gh = = v ; 22 3, 61/c3gh |= = |.v 4.48 , - h = 1 . - . - ? ( 12 4, 43/c gh = = v ; 2103, 74 /c7gh |= = |.v 4.49 o . : , , - v0; . 2034Hg| =

\ v; 1tg3ii k o|> |. 48 4.50 l = 15 , , . , , v : ; . ( 14 /, 1,05 /; 14 /, 2,1 / ). 4.51 R, 2R ( ), - . 60 . , . 15347gR| |= |\ .v 4.52 r, - , R = 2r, , . . , , -, , . - ? ( )41 ( / ) 17 R r i n = + = 4.53 R = 3 n1 = 1 /. = 60 . 2 , ? J = 1080 2. . 22 11 1,5 /mRn nJ| || |= + = | |\ . \ . 4.54 R = 1,5 = 180 - n = 10 /. - = 60 . v , ? 20,94 /c2nMRM mt| |= = |+\ .v 4.55 m2 = 240 R = 2 1 = 80 . - . , e - , v = 2 / . 21 220, 4 /c(2 )mm m Re| |= = |+\ .v 4.56 -, . m - 4 49 , n = 4 ? . 41202tmn| |= = |+\ . 4.57 l, - ,