:

: . , .

- :

. . . , TYPORAMA ISBN: 960-538-496-5 ISBN (SET): 960-538-495-7 , : . .

opusMAGNUM

ISBN: 978-960-538-820-1 ISBN (SET): 978-960-538-819-5 : 14/2-1 Copyright 2008 18, 26335, : 2610 314094, 314206 / : 2610 317244 . 2121/1993, , .

-

:

.

2008

. .., , (CERN) FERMI (FERMILAB) ... , ( ). charmonium , quarks gluons quarks, , , , , . (FERMILAB, CERN) (Northwestern Univ., Bologna Univ., Liverpool Univ.) . 1997 ( , ) CERN, (KM3NeT) (EUROCOSMICS). , . , (HELYCON). ..., , .

..................................................................................................................11 ......................................................................................................15

1

....................................................................................... 17 1.1 ...........................................18 1.1.1 .......................................................18 1.1.2 ...................................................................20 1.1.3 ...................22 1.1.4 ............................................29 1.1.5 .....................................32 1.1.6 ...................................................................36 1.1.7 .............................................................................................39 ...........................................................................................................40 .........................................................................................42 1.2 ..................................................................46 1.2.1 ................47 1.2.2 ..................48 1.2.3 , ...........53 1.2.4 ............................................................55 1.2.5 ........................................................57 1.2.6 ....................................62 1.2.7 ........................................................................................63 , , ......................................46 , , ......................................18

8

:

1.2.8 .................................................................................68 ...........................................................................................................70 ........................................................................................72 1.3 ...........................................................................75 1.3.1 ............................75 1.3.2 ......................................................................76 1.3.3 ...........................................................................80 1.3.4 ..................................................................................83 1.3.5 ..............................................................87 1.3.6 ...........................................................88 1.3.7 .......................................................................92 1.3.8 .......................................................................................................97 1.3.9 .......................................................99 .........................................................................................................101 ......................................................................................103 .....................................................................................106 , , .....................................75

2

......................................................................................109 2.1 ..........................................109 2.1.1 .............................................110 2.1.2 ...........................................................................................113 2.1.3 ..................................................117 2.1.4 ...............................................................................119 2.1.5 ......................................121 , , ...................................109

9

2.1.6 ...................................................................124 2.1.7 ......................................................127 .........................................................................................................129 .......................................................................................131 2.2 ...................................132 2.2.1 ................................................................................................133 2.2.2 1 .............................135 2.2.3 1 ........................................................................139 2.2.4 2 3 .....................................................142 2.2.5 2 3 ........................................................143 2.2.6 .....................................................................................................145 2.2.7 ............................................150 .........................................................................................................153 .......................................................................................154 2.3 ...........159 2.3.1 ......................................................159 2.3.2 ....................................163 2.3.3 .........................165 2.3.4 ..........................................168 2.3.5 ...................................172 2.3.6 ........................................175 2.3.7 ......................178 2.3.8 ............................................185 , , ....................................159 , , ....................................133

10

:

.........................................................................................................191 ......................................................................................194 .....................................................................................199 ..........................................................................207 ..............................................................221

: . , . . , , . Galileo Galilei (1564-1642) . , . Johannes Kepler (1571-1630), , . , , Isaac Newton (1642-1727), , , . , , , , , . , , . , , . . Galileo Galilei . , , ( , ) , . , . , , , , .

12

:

, : 1. . , (). . 2. , . Kepler , . , Kepler . . 19 . , (, , ). , . , . . . , . , . , , . ; , -

13

; . , . . .

. . , . , , . , , , . : , , . , . . , , , . . . . . , , . . . . . . .

16

:

, . , , . . , . . , , . . , . . . , , . , . .

. , , . , , , . : , 1.1. , , ( ), . , , , , . : (.. , , , ), ( , , , ) . , , ;

1.1

18

1:

1.1

, , . : , .

, , .

1.1.1

, . , ( ). . . , , .

1.1

19

( , ), . , , , . , , . . . . , , ; !!! , . ( ). ; , , . . , . . . . 10000 . , . . . , . .

20

1:

1.1.2

. , , .[1] , 1.2.

1.2

, . , (.., 1.2) (.., ). ( ) . , . , , . , , (second) s. , , . 1.2, ,

[1]

4 , , , , .

1.1

21

X, Y Z, . . , , , . .

1.3 0.5

m.

Q XYZ x = 1.0, y = 0.5, z = 0.8. XYZ, 1.3, ' 0.5m;

1.1

, tA ( , x,y,z) tB ( , x,y,z) rA rB , 1.4. r, . -

22

1:

rB rA, r = rB rA.

r =

( xB x A )

2

+ ( yB y A ) + ( zB z A ) .2 2

, ( 1.4), . , 1.5, , . , , .

1.4 . .

1.5

. , . . , , , !!! 1.1.3

. , .

1.1

23

X . . . (s), (m). 1.6 20sec . 1.1 .

1.6 20s . . .

, x, 1.7. , . . () . . () . , .

24

1:

1.2 1.7. , 22, 38 67s.

1.7 1.1

, , , 12s . 1.8. 1.9.

1.1

25

1.8 20s . . 12sec .

1.9 .

26

1:

1.6 1.8 . , , 1.7 1.9 : . ( ) . , 12s . , , . , , 22m 12sec . 20sec , , 1.10.

1.10 20s . 22m . 12s .

1.3 1.3 , , . . , , , .

1.1

27

1.3

( ) . 20sec , , 1.11. ,

1.11 20s . , .

28

1:

. . , , 1.11 1.3. ; , 1.12, . 1.7. , , ( ) , 1.12 () 1.7.

1.12 ( )

1.1

29

: ) ( ), ) ( ) ) ( ). , , , / , / .

, , : .

1.1.4

, . . , , ; . ( ) . : . . :

=

(1.1)

, , (m/s). . ,

30

1:

t1 x1 t2 x2, (1.1) := x2 x1 t2 t1

(1.2)

. , 1.1, 5 . (1.2) 5s . t1 = 0s x1= 0m t2 = 5s x2 = 3m. (30 =)3m (50 =)5s. , 5s 0.6m/s. , .. 5 1.4. . , , , .

1.4 1.2, . (1.2) .

. . .

1.5 1.3, . (1.2) .

1.1

31

1.13 . : [9s, 10s].

5s. , , , , (1.2), . , 1.13. , . , , . , 1.13 9 10 .

32

1:

. , 1.13 1.4. 1.4 , . . , . , 1.13, , , ..., 35, 36, 37, 38 39 .

( ) . t1 t2 ([t1, t2]). , , (1.2):

[t1 , t2 ] =1.1.5

x2 x1 t2 t1

(1.2)

, , (t2t1) , . :v1 = limt2 t1

x2 x1 t2 t1

(1.3)

( ) t1. , , x = x2x1 t = t2t1, (1.3) :v1 = lim x t 0 t

(1.4)

1.1

33

H . . . . ( ). (1.4) !!! . 1.14 ()

1.14 .

34

1:

. . t = 1sec. () t = 1s, [t1 = 1s, t2 = 4s]. [t1 = 1s, t2 = 4s ] = x2 x1 x = t2 t1 t

(1.5)

x1 x2 t1 t2 , x t , 1.14. (1.5) , (tan) AOE :

[t1 = 1s, t2 = 4s ] =

x = tan AOE t

(

)

(1.6)

, , -.

AOE .

[t1, t2] t1 t2 , ( ). , . 1.14, t1. (1.3) : , t = t2t1 . , , t2 t1;

1.1

35

. 1.14. t1!!! . , , t1. , t1: t1, . . , , . (1.3) (1.4) , . . . 1.15 ( ) 160 , . () , (). . (). (V) , , . (V) , .

36

1:

1.15 .

1.6 1.15.

1.1.6

. , . ; :

1.1

37

, , . t ( 1.14) t. , ( ) .

, 1.16. : () .

1.7

. , () . : . , [t1, t2] (1.2). 1.16 (1.2) -

1.16

38

1:

. . . , , . . 1.17 , . , 1.1 0, t1 t2 .

1.17

1.8 1.17 : ) t = 0 ; ) ; . , t . , .

1.1

39

1.1.7

( ) . , ( ) . !!! , , . . . 1.18 .

1.18

t1 v1 t2 v2. .

M [t1 , t2 ] =

v2 v1 t2 t1

(1.7)

40

1:

, . t1.

t1 = lim

v2 v1 t2 t1 t t 2 1

(1.8)

: ) ( ), ) ( ) ) ( ). , , , / , / . , . . x1 x2 [t1, t2], : [t1 , t2 ] = :v1 = lim x2 x1 t2 t1 t t 2 1

x2 x1 t2 t1

( ) t1. H .

1.1

41

[t1, t2]: t1 t2 . ( ). t1: t1. . . , . . () . . : [t1 , t2 ] = v2 v1 t2 t1

v2 v1 t2 t1 . : t1 = limt2 t1

v2 v1 t2 t1

42

1:

1.1 .

2000m. , Oregon State University .

1.2

1.19

. . , 1.19(). . , , ,

1.1

43

, 1.19(). , . , 1.19(). , , . 1.20 . , 1.20 :

1.20

) . ) . ) . ) , . ) . 1.19 .

1.3 1.21 , . ) ta .

44

1:

) tb; ) tc . ) ;

1.21

1.4 . (a) (t) 1.22. ) . ) . ) . ) .

1.1

45

1.22

46

1:

1.2

. .

.

1.2

47

1.2.1

. , . . , . x t :

: x = t +

(1.9)

x t = 0, 1.23. ;

1.23

(v) . (x0) .

- : x = v t + x0

(1.10)

48

1:

1.2.2 (1.10), . : . . t = 0 ( 1.1), x0, ( ) , x1. V, v. x1 > x0 V > v. , . 0 :

0 = x1 x0 1

(1.11)

x0 x1 t1. , t1 (1.10) :

x1 = V t1 + x0 t1 = x1 x0 0 = V V(1.12)

, x2. x2 (1.10) x1 x2 v t1, (1.12), :

x2 = v t1 + x1 =

v 0 + x1 V

(1.13)

1.2

49

1(=x2 x1), (1.13) :

1 = x2 x1 = 2

v 0 V

(1.14)

x1 x2 t2. . t = 0 x1 t = t2 x2. , , , t2 (1.10) :

x2 = V t2 + x11

x x t2 = 2 1 = 1 V V

(1.15)

(1.14), :t2 = v 0 V V

(1.16)

x3. x3 (1.10) x2 x3 v t2, (1.16), :

v x3 = v t2 + x2 = 0 + x2 V

2

(1.17)

2(= x3 x2), (1.17) :

2 = x3 x2 = 0 V

v

2

(1.18)

50

1:

3 x2 x3 t3. x4.

1.9 , t3 :

v t3 = 0 V V

2

(1.19)

3, :

v 3 = x4 x3 = 0 V n [2] :

3

(1.20)

v tn = V

n 1

0V

(1.21)

:

v n = 0 V .

n

(1.22)

v , , V n (1.22) n .

v lim n = lim 0 = 0 n n V

n

(1.23)

[2]

( ), (1.21) (1.22) n = 1 , n = 1, n = , n. (1.21) (1.22), 1, 2 3.

1.2

51

, . , . , , , . ; E ; . !!! , n, n . Tn = t1+ t2 + + tn. , , (1.21), n :

Tn = t1 + t2 + + tn = :

0

v + V V

0 v ++ V V

n 1

0V

(1.24)

(1.24) :

v Tn Tn V

0 v 0 = V V V

n

(1.25)

, (1.25) , :

0 v 0 0 V V = V Tn = V v v V v v 1 1 1 V V V

0

n

v n V

(1.26)

, , , .

52

1:

, T, (1.26) n . , v . V

0 T = lim Tn = lim 0 V n n V v v 1 V 0 = 0 V V v v 1 V

v n = V

n v lim = 0 n V V v 0

(1.27)

, , (1.27) . , . 9.01m/s, 0.01m/sec 90m, (1.24) 10 . [3], . , (1.27). . , , .

x1 0 , x0 ( 0= x1x0). v V, V > v, ;

[3]

, . : . 1.1 pokemon.

1.2

53

(1.10), . , t, , x. t x, : : :

x = x1 + v tx = x0 + V t

(1.28) (1.29)

(1.28) (1.29) 1.10 (1.27). ;

1.2.3

,

(1.10) . , (1.10) x ( ) t ( ). ,

x ( t ) = v t + x0

(1.30)

, , (1.30). , , t1 x(t1) t2 x(t2), , x(t), (1.30). t1, (1.3), :

v ( t1 ) = lim

x ( t2 ) x ( t1 ) t2 t1

t2 t1

(1.31)

(1.31) ( t1). , :

v ( t ) = lim

x ( t + t ) x ( t ) t

t 0

(1.32)

54

1:

(1.32) . x(t) t.

:

v (t ) =

dx ( t ) dt

= x ( t )

(1.33)

1.11 (1.33) . : , x = k t + .

, ( , [1.30]), . .

. ( 1.16) ( ) ( ) m m m x ( t ) = 0.094 t + 0.1 2 t 2 + 0.0009 3 t 3 , s s s

.

1.12 m m m x ( t ) = 0.094 t + 0.1 2 t 2 + 0.0009 3 t 3 . s s s . 3s, 8s 50s. : , x(t).

1.2

55

1.2.4

() , (1.32), :

a ( t ) = lim

v ( t + t ) v ( t ) t

t 0

(1.34)

. :

a (t ) =

dv ( t ) dt

= v ( t )

(1.35)

(1.35) (1.33) :

a (t ) =

dv ( t ) dt

=

2 d dx ( t ) d x ( t ) = = x ( t ) dt dt dt 2

(1.36)

.

1.12 .

1.13

a(t) , , . . . , , , : ) , ) ( t = 0) V0 ) 0 t = 0. : ( (1.35)

56

1:

( ), :

v (t ) = t +

(1.37)

t. , , :

dv ( t ) dt

=

(1.38)

(1.38) , :

=a

(1.39)

(1.37) . (1.37) t = 0 . V0, :

= V0

(1.40)

(1.37), (1.39) (1.40) :

v ( t ) = a t + V0

(1.41)

: ) ) ( (1.41)), :

x (t ) =

1 a t 2 + V0 t + X 0 2

(1.42)

, (1.42) , (1.41):

dx ( t ) dt

=

d 1 2 a t + V0 t + X 0 = a t + V0 dt 2

0 (1.42) t = 0.

1.2

57

!!! , :

x (t ) =

1 a t 2 + V0 t + X 0 2 v ( t ) = a t + V0

, X 3m/s2.

1.14

1.2.5

, , ( ) . , ( g) , , . , , . , g. , ( ) .

58

1:

1.24

. , . , , 1971 David Scot, , . , Scot (Galileo Galilei). , .

1.2

59

, , (Isaac Newton) . , , V0(= 10m/s). H . , h ( 1.24), . . , , , . g= 9.81m/s2. , , , (1.41) (1.42). X ( ), ( ), 1.24. . 1.25() 3 . , , a =g = 9.81m/s2. , , X. V0 = 10m/s , (1.41), :

v ( t ) = (10 m s ) ( 9.81m s 2 ) t

(1.43)

(1.43) 1.25(). . t = 0 v = V0 = 10m/s , . .

60

1:

1.25 : ) a(t), ) v(t) ) x(t)

1.15 (1.43) , th, . th, t > th, . . : ) . (1.43) v(th) = 0. ) X.

t = 0 X0 = 0m. (1.42) :

1.2

61

1 x ( t ) = (10 m s ) t ( 9.81m s 2 ) t 2 2

(1.44)

1.25() . . . , th, 1.15. , x(th), (1.44) t = th.

. : (1.44) 1.15.

1.16

, t0 , X0 = 0m. (1.44), , t0. , x(t0) = 0m, :1 2 0 = (10 m s ) t0 ( 9.81m s 2 ) t0 , 2

. . x0 = 0m . . .

, t0, . :

1.17

1 2 0 = (10 m s ) t0 ( 9.81m s 2 ) t0 . 2 x = 0 m , t = 0s.

62

1:

1.18 t0. . : (1.43) 1.17.

1.19 t0 .

, , . . , , 10 , (1.44) x(t) = 10m. , , t1 t2, .

1.20 t1 t2 .

; !!! , , . ;

1.21 , . 1.25() 10m. H . . . 1.2.6

( ) .

1.2

63

2m/s2. 90m 0.01m/s. ; ; . x t. , x t, .x = 1 m 2 m 2 t x = 0.01 + 90 m 2 s2 s

.

1.22

1.2.7

. , , . . , , , . ! . , . , , . .

64

1:

1.26 . , V1 . t1 t2.

, . . . . A [4] , , 1.26. , V1 V2. . , , :

[4]

. . . , .

1.2

65

V1. V2, . . : . . . . X . X X, . , X, ( 1.26), , V1, . , X, V1 . X . t1 y1 x1, t2 y2 x2. , , :

y2 y1 t2 t1

, . :V1 = y2 y1 y2 y1 = V1 ( t2 t1 ) t2 t1

(1.45)

66

1:

, :

V2 =

x2 x1 x2 x1 = V2 ( t2 t1 ) t2 t1

(1.46)

t1 x1 , t2 x2 . , , : x2 x1 t2 t1

, . , , V2, :V2 = x2 x1 t2 t1

(1.47)

( 1.26), , x1 ( , , t1), x1 ( t1) y1 ( t1) :

x1 = x1 y1

(1.48) (1.48)

1.23

x2 = x2 y2

(1.48) (1.48) (1.47), (1.45) (1.46)

V2 =

x2 x1 ( x2 y2 ) ( x1 y1 ) ( x2 x1 ) ( y2 y1 ) = = = V2 V1 t2 t1 t2 t1 t2 t1 t2 t1V2 V1

(1.49)

1.2

67

, V1 ( ) , V2 ( ), V2 ( ), V2V1. (1.49). ( ) 20 km/h, 8 km/h. 1.26, ( x), ( ) ( x). : V1 = +20 km/h V2 = 8 km/h. (1.49), :

V2 = 8km h 20 km h = 28km h , X ( V2 ) 28 km/h. , ; : V1 = +20 km/h V2 = +8 km/h.

1.24

. : . , V2 = 0km/h. (1.49).

1.25

10 km/h X 1.26. 20 km/h X. . : V1 = +10 km/h V2 = 20 km/h. (1.49) V2.

1.26

68

1:

1.27

, ( , ) , , , 1.27. 30 km/h, 60 km/h . . X . , . (1.49), V1, V2 V2 :

V1: V2:

V2 : x :

V2 = V2 V1 +60 km h = 30 km h V1 V1 = 90 km h

1.27 .1.2.8

. . , . , . (A. Einstein).

1.2

69

1.28

. ; , . . . , , . , ( ) . . -

70

1:

. . : ( ) ; , ; 1.28. !!! ( ) ...

:

v (t ) =

dx ( t ) dt

= x ( t )

:

a (t ) =

dv ( t ) dt

= v ( t )

.

a (t ) =

dv ( t ) dt

=

2 d dx ( t ) d x ( t ) = x ( t ) = dt dt dt 2

, , , :

1.2

71

x (t ) =

1 a t 2 + V0 t + X 0 2 v ( t ) = a t + V0

, , ( ) . V1 ( ) , V2 ( ), V2 ( ), V2V1. ( )

1.5 . , . , . 500 km 20 km/h, 500 km 30 km/h 500 km 50 km/h. . , .

1.6 1.5 s 14 (3m ). ) . ) 2.33 m/s. ) 2.33 m/s.

72

1:

1.7 () 240 km/h. . ( ) . 1.75 m.

1.8 , : . , . , . , , .

1.9 , , 120km/h. , , , . : ) , ( ), ) ( 0.5s), ) ( 100km/h 5s). , , .

1.2

73

1.10 . 343m/s 1.5 . . ( .)

1.11 10m 0.9m/s2. . ( Indiana Jones .)

1.12 15m/s 25m/s 20m. : ) ) .

1.13 10s 3m/s2. , 4m/s2. 2 , : ) ) .

1.14 . , 1.1 . .

74

1:

1.15 a1 5 . a2 = 5m/s2 10s 5m/s. a1 15s .

1.16 VT = 20m/s.

1.29

, . 15m 50m . , V, , ; .

1.3

75

1.3

, , ; ; , .

(.. ) .

1.3.1

. , , , . .

76

1:

; : , , ; , , . 2000 . , , . .

1.1 . (384322 ..) 2000 .

: !!! , 1.3.2 , . , , 1.3.2 1.3.3. 1.3.2 , . , . , . , . -

1.3

77

( ) . , . , . , , , . . , . ( ) . . . , , , , : ) ) () . . , : . ( 1.30) . , . , ; , . . , . , , , .

78

1:

1.30

1.31 . . . .

. . , . , , .

1.3

79

, 1.31. ( , , , ) . , , . , . , . . .

, , , , . ;

1.28

80

1:

1.3.3 , (GalileoGalilei). , , . , , , . , , . , , , . , . . 1.32 . : , .

1.29 ; , 1.31, . . . , .

1.3

81

, . (V) 1.33() . , V. , 1.33(). , ( 1.33[]).

82

1:

. . . , , , , . , , . . , () V.

1.3 Sir Isaac Newton (16431727) . .

(t) , y x . y x . . , , 1.33().

1.3

83

, , , . .

1.31 .

1.30

. 1.34.

1.34

1.3.4

. , , , ( , , .., ) . (Isaac Newton) Principia Mathematica Philosophiae Naturalis ( ) 1686. . .

84

1:

: . , . , . , , . .

, . : ( ). m Fx X, , ax, :[5]

Fx = m a x

(1.50)

, .. , F , , F. . , . .

[5]

O Fx =

d ( m vx ) . , dt

, , (1.50).

1.3

85

. : . . , (1.49) . , . ( Leibniz[6]). , . : . , . , dv , : a x = x dt , , () . : ( SI ) (kgr). . , . . . : . .

[6]

O 16651666. . , , . Gottfried Leibniz . .

86

1:

1.35

, . , , . : . . , , . , , . . 1.35 , F1 F2, , R, . X. , F, : F = F1+F2R. SI Newton (N).

, . , , . . , . . .

1.3

87

1.3.5

. , . : ( ). , . ( ) ( ) . . ( 1.33) . . . 1.34. , ( ). [7]. .

[7]

. , . , , , . . , , , 9.81m/s2. 4.4 103 ( ) 3.37 102m/s2 . .

88

1:

1.3.6

. m Fx X, , ax, :

Fx = m a x

(1.50)

Fx ax . , ( ) , (1.50). . . , m , , (1.50), . .

. ( ) 1kgr, . (1.50) . m0 ( 3kgr, m0). . . ( ) F , a0, a1, a2, , aN.

1.31 :

mk = m0

a0 k = 1, 2, , N ak

(1.51)

1.3

89

: , ao, , F, : F = m0 a 0 . k , ak, F: F = mk a k . .

, . , , , m0 ( 4 3kgr), . m0 . m0

;

1.32 m0 a = m0 0 k = 1, 2, , N m0 ak

: (1.51) m0 . : mk = mk

(1.50) . 1 (Newton) 1kgr 1m/s2. . / / . , m1 m2 r, , , :B =G m1 m2 r2

(1.52)

(1.52), , . G (6.67 1011Nm2kg2).

90

1:

. , . , , m h . , 1.36, . , , .

1.36

( , , ) , . R, [8]:

B =G

M m

( R + h)

2

(1.53)

6.37 ( 5.98 1024kg), , h , (1.53) [9]:[8]

. (1.53), : ) , . ) . , , . . () . h10m (1.54) 99.9999%.

[9]

1.3

91

B G

M m

( R)

2

(1.54)

. , , . (1.54), :

B = ma M m M g a =G 2 M m ma = G 2 R R B =G R2

(1.55)

, (1.55), , , G. . , , .

( (1.54)). . ; ( 6.37 106 . .) : , a, B = M a : M m B = G R2 1 , x = 1mm = a t 2 , t. 2 , , ( ) ( ) ( ) .

1.33

B = mg

(1.56)

92

1:

1.34 ( ) . : (1.54). 7.36 1022kg 1.74 106m .

1.37

, . 1.3.7

1.37, , . , ( ) . .

1.3

93

, . 1.37; . , .. , F , , F. . 1.37 ( 100) . , . , a1 a2 , .

; a1 a2 50kg 2000kg. , 100. : . (1.50) ( m1 a1) ( m2 a2).

1.35

t , . . .

94

1:

1.36 50kg , 2000kg. : . , t 1.35. , t (.. 2s) . , .

. . 1.38 . . . 1.38() 1.38() . , , . , , . . , , F. , , F. , . , , .

1.3

95

( , .) . , , B F. , . 1.38() . ( ). .

: . , . . . N .

F . : , . S = B T. . , , S, . F , , , .

1.37

96

1:

a, 34. . 33. a.

1.39 a

() 1.39 . , a , , . , :

T B = ma m . (B = mg) , :

T = m (g + a ) > m g , , .

1.3

97

= F , F , , , . F . , : ( ) , mg , , .

, a, (a = g/2); : : T B = m a T = m (g a ) < m g

1.38

1.3.8

. , 1.40 : F Fk . , , , (), .

98

1:

1.40

. , 1.40 (.. ), . , , , , . ( [1.40]) . , . :

T = Fk

(1.57)

, , . . .

1.3

99

1.3.9

, , . . . , . , . , , , . , . mA = 10 mB = 20kg, , , 1.41. . t = 0 h = 1m . . 1.41 : , , , BB = ( 20 kgr ) ( 9.81m s 2 ) = 196.2 N. , , , BA = (10 kgr ) ( 9.81m s 2 ) = 98.1N. , , FB. ,

FA.

100

1:

FA FB , , . , , .

FA = FB = F

(1.58)

( , FA FB ). : :

BA FA = mA a ABB FB = mB a B

(1.59) (1.60)

a a . , . , , .

1.39 : ( ) , .

, , , . ( [1.41]):

a B = a a A = a

(1.61)

(1.58) (1.61) (1.59) (1.60), :

BB BA = a ( mB + mA ) a =g mB mA = 3.27 m s 2 mB + mA

101

, , , 3.27m/s2. , , . , , , , , .

. : 0.5m 3.27m/s, .

1.40

, . : ( ). m Fx x, , ax, :

Fx = m a x , .. , F , , F. . . , . .

102

1:

. N . m1 m2 r, :

B =G

m1 m2 r2

, g, , M, , R , G. .

g =G

M R2

103

1.17 12kg. , () 30m/s2;

1.18 80km/h , . 5 45km/h. 100N ( ), .

1.19 , 1.42.

1.42

F . . . .

104

1:

1.20 1.43 , m1, m2 m3 . . a1, a2 a3 : a1+a2+2a3 = 0. m2 = 2m1 m1 , : , 2 3, 3.

1.43

1.21 m1. m2 m1, 1/5 . m1/m2.

105

1.22 , r. r/2, , 1.44. 4r . .

1.44

1.23 m1 = 2200kg , m2 = 310kg m3 = 260kg , 1.45. 1.9m/s2, : ) , Fth, , ) , 1, ) , 2, .

1.45

106

1:

1.1 110km/h. 0.14s . 60kgr .

: 13100

1.2' X 108m/s. 1018s. Y . H 9.11 1031kg.

: 9 105

1.3 . , . , 532. 60kg ( ), . ;

: 8.87m/s2

1.4 50kg 40km/h. .

: 490

1.5 450 . 32kg. .

: 4.26m/s2

107

1.6 74kg . : ) , ) 2.4m/s, ) 2.1m/s, ) 1.7m/s2 ) 1.7m/s2;

: ), ), ): 725, ): 851, ): 599

1.7 1, 2 3kg, , , 1.46. 12 . . : 10.

1.46

1.8 (, 3kg).

: 6

1.9 11kg 1.8kg. 2.3m/s2. (1) (2); ; .

: 1 = 29.4 2 = 25.3

108

1:

1.47

1.10 , m, 1.48, . , a. .

1.48

1.11 0.42, ( ) . ( ) 60km/h 1.5m.

: 4.12m/s2, 86.8m/s2

. . . , , , , , . , . , , . ( ) ( ). 1 . , , .

2.1

1 ( ) ( ).

.

110

2:

2.1.1

. 2.1, . , ( ) . [1], 2.2.

2.1

2.2

[1]

M , .

2.1

111

. . , 1. . , , 2.3. . , , 30 . 2.1 2.2 , . , 2.1 2.2 . . , , .

2.3

112

2:

. , .

2.1

113

, . , . , , . , , ( ), . ( 2.4) 2.935951m 2.629045m. {2.935951,2.629045}[2] . , R:

R = 2.935951 i + 2.629045 j

(2.1)

i j [3] .

2.1.2 . . R (2.1) , , : R=+ : (2.2)

= 2.935951 i = 2.629045 j

(2.3)

[2] [3]

H X . i j .

114

2:

(i j) R . R , , , 2.4. R (2.2) ( , R) :

R = 2 + 2

(2.4)

2.1 , (2.2), (2.3) (2.4), 2.4.

2.4 K

2.1

115

2.4 R :

= R cos = R sin cos = =R sin =

+ 2 2

(2.5)

R

=

2 + 2

R .

(2.5). (2.2) , , :

2.2

R = R cos i + R sin j

(2.6)

2.4 ( [2.3]) 2.3 (2.6). , , . . . A B. 2.5() :

A = x i + y j B = x i + y j

(2.7)

, , , , :

= A + B = ( x + x ) i + ( y + y ) j

(2.8)

116

2:

, :

= B A = ( x x ) i + ( y y ) j

(2.9)

, , 2.5(). 2.5() 2.5().

2.4 A B . .

2.5

2.1

117

2.1.3 R , 2.6.

2.6

:

R = k + m k = ai R = ai + b j m = bj

(2.10)

i j , k m R . , , :

k =a m=b R = a 2 + b2 , , ( 2.6). R . (2.11)

118

2:

R . k m , (2.6) (2.11).k = k cos k sin m = m sin + m cos

(2.12)[4]

(2.10) (2.12) (2.11), :

R = k + m = ( a cos + b sin ) + ( b cos a sin )

(2.13)

( 2.7) , R. , . , tx ty, :

T = tx i + t y j

(2.14)

2.7

[4]

k Y.

2.1

119

, , , , . R ( 2.7). R R T.

2.5(2.15)

R = R T = ( a t x ) i + ( b t y ) j

R . , , (2.13) (2.15), :

R = ( a t x ) cos + ( b t y ) sin + ( b t y ) cos ( a t x ) sin (2.16)

2.1.4

2.1.1: . 2.8() . 2.8() ( ), 2.8() ( 2.1 2.1.1). , . ( 2.8[] 2.8[]) . 2.8() 2.8() x(t) y(t), .

120

2:

2.8 : () , () , ()

, , x(t) y(t) ( : m m m x ( t ) = 1.736 t y ( t ) = 9.848 t 4.905 2 t 2 ), s s s (2.2). , , :

= x (t ) R (t ) = x (t ) i + y (t ) j = y ( t )

(2.17)

2.6

m : x ( t ) = 1.736 t s m m y ( t ) = 9.848 t 4.905 2 t 2 s s 1.162s 1.796s.

2.1

121

: ( ) . : : x(t) : y(t) .

m (.. x ( t ) = 1.736 t s m m (.. y ( t ) = 9.848 t 4.905 2 t 2 s s . (2.17). 2.1.5

tA R A ( = x ( t A ) i + y ( t A ) j) , tB, R B ( = x ( t B ) i + y ( t B ) j) o 2.8. = R B R A () , 2.8(). tA tB :

= ( x ( t B ) i + y ( t B ) j) ( x ( t A ) i + y ( t A ) j) = ( x ( tB ) x ( t A ) ) i + ( y ( tB ) y ( t A ) ) j

(2.18)

(2.18) () t= tB tA. ( U, Ux Uy):

U=

y ( tB ) y ( t A ) x ( tB ) x ( t A ) = i + j t t tUx Uy

(2.19)

Ux Uy 2.8() 2.8() .

2.7

122

2:

, , ( 1.1.5 1), Ux Uy , , [tA, tB]. U ( 2.8[]) [tA, tB]. , (Ux Uy) . t = t B t A . 2.8() 2.8() . AB 2.8() 2.8(). , . [5], 2.8() ( ). (2.19) U : t = t B t A 0

V = lim

t 0

x ( tB ) x ( t A ) y ( tB ) y ( t A ) = lim i + lim j t 0 t 0 t t tVx Vy

(2.20)

V, (2.20), ( lim ) t 0 , 2.8(). , Vx Vy, .

[5]

. 2.8() 2.8() , .

2.1

123

V , .

2.8 () X ;

2.8

Vx = 1.736 m s, , , m m Vy ( t ) = 9.848 9.81 2 t. s s 1.162s 1.796s. .

2.9

. , (2.20) x(t) y(t).

t = t B t A Vx = limt 0

x ( tB ) x ( t A ) t y ( tB ) y ( t A ) t

= =

dx ( t ) dt dy ( t ) dt

(2.21)

Vy = lim

t 0

, R(t) :

R A = R ( t A ) = x ( t A ) i + y ( t A ) j R B = R ( t B ) = x ( t B ) i + y ( t B ) j V = limt 0

= RB R A

(2.22)

R ( t B ) R ( t A ) dR ( t ) = lim = t 0 dt t t

124

2:

(2.21) (2.22), : V (t ) =

dR ( t ) dt

=

dx ( t ) dt

i +

dy ( t ) dt

j

(2.23)

2.10 :

m m m R ( t ) = 1.736 t i + 9.848 t 4.905 2 t 2 j s s s 2.8. t = 0.3s. t = 0.5s. t = 0.9s , t = 1.7s . : .

2.1.6

2.9 , 2.1.1, . , , . , .

V ( t ) = Vx ( t ) i + Vy ( t ) j

(2.24)

Vx Vy ax ay ( 1.1.7 1).

t = t B t A

a x = lim

Vx ( t B ) Vx ( t A ) t Vy ( t B ) Vy ( t A ) t

t 0

= =

dVx ( t ) dt dVy ( t ) dt

(2.25)

a y = lim

t 0

( ) :

a = ax i + ay j

(2.26)

2.1

125

2.9

(2.26) (2.25), :

t = t B t A a = lim Vx ( t B ) Vx ( t A )t 0

t 0 t dVy ( t ) dV ( t ) dV ( t ) a= x i + j= dt dt dt

i + lim

Vy ( t B ) V y ( t A ) t

j

(2.27)

126

2:

: . . t :

R (t ) = x (t ) i + y (t ) j

(2.28)

x(t) y(t) . ( 1.2.4 1), :

1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2

(2.29)

Vx 0 , Vy 0 x0, y0 , t = 0 . ax ay .

V ( t ) = Vx ( t ) i + Vy ( t ) j :

(2.30)

Vx ( t ) = Vx 0 + a x t Vy ( t ) = V y 0 + a y t

(2.31)

:

a = ax i + ay j

(2.32)

, :

a (t ) = a x (t ) i + a y (t ) j

(2.33)

, .

2.1

127

. . 2.1.7

. . 2.10, . R :

R = rx i + ry j + rz k

(2.34)

i, j k , .

2.10

, .

128

2:

rx = x ( t ) ry = y ( t ) rz = z ( t ) 2.1.6 : , . . t : (2.35)

R (t ) = x (t ) i + y (t ) j + z (t ) k x(t), y(t) z(t) , Y Z . ( ), :

1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2 1 z ( t ) = Vz 0 t + a z t 2 + z0 2 Vx 0 , Vy 0 , Vz 0 x0, y0, z0 t= 0. ax, ay az .

V ( t ) = Vx ( t ) i + Vy ( t ) j + Vz ( t ) k :

Vx ( t ) = Vx 0 + a x t Vy ( t ) = V y 0 + a y t Vz ( t ) = Vz 0 + a z t :

2.1

129

a = ax i + ay j + az k , :

a (t ) = a x (t ) i + a y (t ) j + a z (t ) k

. . t :

R (t ) = x (t ) i + y (t ) j x(t) y(t) . , :

1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2 Vx 0 , Vy 0 x0, y0 , t= 0, . ax ay .

V ( t ) = Vx ( t ) i + Vy ( t ) j :

Vx ( t ) = Vx 0 + a x t Vy ( t ) = Vy 0 + a y t

130

2:

:

a = ax i + ay j , :

a (t ) = a x (t ) i + a y (t ) j , . . t :

R (t ) = x (t ) i + y (t ) j + z (t ) k x(t), y(t) z(t) , Y Z . ( ), :

1 x ( t ) = Vx 0 t + a x t 2 + x0 2 1 y ( t ) = Vy 0 t + a y t 2 + y0 2 1 z ( t ) = Vz 0 t + a z t 2 + z0 2 Vx 0 , Vy 0 , Vz 0 x0, y0, z0 t = 0. ax, ay az .

V ( t ) = Vx ( t ) i + Vy ( t ) j + Vz ( t ) k :

2.1

131

Vx ( t ) = Vx 0 + a x t Vy ( t ) = Vy 0 + a y t Vz ( t ) = Vz 0 + a z t :

a = ax i + ay j + az k , :

a (t ) = a x (t ) i + a y (t ) j + a z (t ) k

2.1 r0 = ( 0.5m ) i + ( 0.6 m ) j t = 1.4s, v = (1m s ) i + ( 0.3m s ) j. , r ( t ) .

2.2 :

r (t ) = x (t ) i + y (t ) j x ( t ) = A cos (t ) y ( t ) = 2 A sin (t ) A = 4 m = s. 0, 0.5, 1, 1.5s. .

132

2:

2.3 t = 0

r0 = ( 2 m ) i + (10 m ) j + ( 5m ) k

v 0 = ( 0.5m s ) i + ( 2 m s ) j ( 0.8m s ) k

a = ( 0.5m s 2 ) j + (1.1m s 2 ) k . 10s.

2.4 , , 2.4m/s 1.1m/s2 3s. 5.7m/s. .

2.5 ;

2.6 , 260 i m s a = 0.38 i m s 2 + 0.72 j m s 2 24s. : ) ) .

2.7 :

r ( t ) = ( 3.2 t + 1.8 t 2 ) i + (1.7 t 2.4 t 2 ) j m, ( t ) . .

2.2

133

2.2

1.3 1 . .

.

1, 2 3 ( , , ) 2.2.1

, 2.11.

2.11

134

2:

, , . , . ; . ( 2.11[]) .

2.11 ; . , ( ), .[6] () . , , ( G) B F . G N.

2.12

[6]

, . , , . 2.11() . .

2.2

135

, : . . 2.11 , 2.12, () , , . . [7] , , : B = Bx i + By j ( 2.12), :

2.12

Bx = B cos By = B sin .

:

S x = Bx F S y = N By S = S x i + S y j . 2.2.2 1

, , ( ) .

[7]

.

136

2:

2.13 . O , = 90.

2.11, F . , , . , 2.13(), [8], . , : ) , , . ) , m, . 2.13(). . , F T.[9] , . : S x = S y = 0[8]

( ) , . , T, , ( ).

[9]

2.2

137

, , , < m ( s ).

S x = Bx Ts = B cos ( 90 ) Ts = 0 Ts = B sin

Ts = tan N S y = N By = N B sin ( 90 ) = 0 ( ) N = B cos

( )

( )

(2.36)

(2.36) , m 0, , : ( 1.3.5 1) ( (2.36[]) . , . , tan m N . , s, m. Ts s N ( ), , . , ( ), , . , k, . . (1 ). , :

S x = Bx Tk = B cos ( 90 k ) Tk = 0 Tk = B sin k

Tk = tan k N S y = N By = N B sin ( 90 k ) = 0 ( ) N = B cos k

( )

( )

(2.37)

(2.37) k .

138

2:

k k. :

Tk = k N

(2.38)

[10] , .

2.13 k < s . , , k, a. , :

S x = B cos ( 90 k ) T = m a T = B sin m a ( ), . , k. > k (2.38).[11]

2.14 , . k > k, k = tan k . : 2 (2.38).

[10]

. . , .

[11]

2.2

139

: , . : Ts s N . s . , . Tk = k N , k ( ) . . , . , .

; : .

2.15

2.2.3

1

2.14. . .

140

2:

2.14

2.15() , . F1 , ( ) F2. , F1 F2 , F2, . F1 = F2 . ( ) . , . , , [12] . -

[12]

, .

2.2

141

. 2.15() , . . .

F1 cos 1 = F2 F1 sin 1 = B

() ( )

(2.39)

30kgr (F1) 1000, ( F1) . (2.39[]) ( 10m/s2), :

m 30 ( kgr ) 10 2 s = 0.3 = 17.46 sin 1 = 1 1000 ( N ) ( F2); 2.15() . tan 2 =B . 2 F1

2.16

2.17

(30kgr) 1000. : , . 2.15() . , :

B = F1 sin 1 + F2 sin 2 F1 ( sin 1 + cos 1 tan 2 ) = B F1 cos 1 = F2 cos 2

142

2:

2.18 1 = 72 2 = 18, F1 F2 (30kgr) .

2.2.4

2 3

F, a, ( ) :

F = ma = m

dv d2R = m 2 dt dt

(2.40)

v R . ( i, j k), :

F = Fx i + Fy j + Fz k v = v x i + v y j + vz k R = xi + y j+ z k (2.40) , : (2.41)

dvx d2 x = m 2 dt dt dv y d2 y Fy = m a y = m = m 2 dt dt dv d2 z Fz = m a z = m z = m 2 dt dt Fx = m a x = m

(2.42)

, , . (2.42), 1.3.4 1, . . F, G = F. , .

2.2

143

2.2.5

2 3

2.16, m1 = 50kgr, , m2 = 15kgr. , , , 45o. ( ) = 0.5, , a, . ( , g = 9.81m/s2.)

2.16

. , : 1. , , , .

. : t h. x. , x = h. .

2.19

2. . , , .

144

2:

3. . 2 . , . , , 2.16(), . 4. . , , 2.16. 1: , a, . 2: 2.16() 2.16() . , , F ( 2.16[]) F ( 2.16[]). ( ), , F = F. . 3: , F (W = m2 g). a. F , F = F, :

F W = m2 a F m2 g = m2 a F = m2 ( a + g ) ( 2.16[]):

(2.43)

) F, ) B ( B = m1 g ) , ) N ) T (T = N ) . . , ,

2.2

145

( ), , . , . . , :

m1 g sin F N = m1 a m1 g cos N = 0

(2.44) (2.45)

4: (2.43), (2.44) (2.45) . a, :a =g

m1 sin m2 m1 cos m1 + m2

(2.46)

.

2.20

2.2.6

2.1.1 . 2.1 V0, . , . , , . ( ) , 2.17.

146

2:

2.17

, (1 ) . , (2 ) . . , . 2.1.6 2.1.7 , . , . : ) : . , V0 x :

V0 x = V0 cos

(2.47)

t ( ) :

x = V0 cos t .) :

(2.48)

2.2

147

. , ay. 2 , ( 1.3.6 1) ay .

ay = g

(2.49)

t :

Vy = V0 sin g t

(2.50)

gt (2.50). : .

2.21

t :

1 y = V0 sin t g t 2 2

(2.51)

(2.48), (2.49), (2.50) (2.51) , . 2.17, , , . , , . . . , Vy = 0: ( [2.48], [2.49], [2.50] [2.51]) (2.50) . , th, , :

148

2:

0 = V0 sin g th th =

V0 sin g

(2.52)

(2.51) . (2.52) (2.51), . (), .

V sin 1 V0 sin (V0 sin ) h = V0 sin 0 g = g g 2g 2 2

2

(2.53)

2.22 10m/s 80 , .

, R, , . 2.17, , , . , (2.51) y = 0, . (t = 0), . , .2 V0 sin + (V0 sin ) t1 = =0 g 1 2 0 = V0 sin t g t 2 2 V0 sin (V0 sin ) 2 V0 sin = t2 = g g

(2.54)

t2. (2.54) (2.48), :

R=

2 V02 sin cos g

(2.55)

2.2

149

, (V0), = 45 . : 2 sin cos = sin ( 2 ) (2.55) :

2.23

R=

V02 sin ( 2 ) g

(, , 2.4). , , . (2.48), , , . , , (2.48), , t, x :

t=

x V0 cos

(2.56)

(2.51) . (2.56), ():

y = x tan x 2

g 2 (V0 cos )2

(2.57)

2.1 2.1.1 = 80 V0 = 10m/s. 2.1 ( ). (2.57) (2.57) .

2.24

150

2:

2.2.7 .

2.18

2.18, . h ( ) ( ) V0. , , R, , VT, . . . . , ( [2.18]) :

Bx = B sin By = B cos

(2.58)

2.2

151

( ). :

V0 x = V0 cos V0 y = V0 sin

(2.59)

, ax ay .

Bx = B sin = m a x a x = g sin By = B cos = m a y a y = g cos :

(2.60)

Vx = V0 cos g sin t 1 x = V0 cos t g sin t 2 2 : (2.61)

Vy = V0 sin g cos t 1 y = V0 sin t g cos t 2 2(2.62)

( 2.18) h. (2.62) y= h. :

V sin t = 0 1 1 h = V0 sin t g cos t 2 2 V0 sin + t2 =

(V0 sin )g cos

2

+ 2g cos h + 2g cos h(2.63)

(V0 sin )g cos

2

152

2:

2.25 t2 ( h = 0). t2; : :V0 sin (V0 sin ) + 2 g cos h .2

. t2 , t1 . , R, (2.61) t1.1 R = V0 cos t1 g sin t12 2

(2.64)

t1 (2.63). (2.61) (2.62):

Vx = V0 cos g sin t1 Vy = V0 sin g cos t1 t1 (2.63).

(2.65)

2.26 :) , h. ) , h. ) , h = 0.

2.2

153

1 ( ) , ( ) 2 F, a, :

F = ma = m

dv d2R = m 2 dt dt

3 F, G = F. , . Ts s N , . s . , . Tk = k N . . k ( ) . . , . , .

154

2:

2.8 4kg . 30 45 . .

2.9 30kg ( 0.25) , 45 . : ) , 0.08m/s2.

2.10 675m/s 511m . ;

2.11 :

F = ma 1 x = x0 + v0t + a 0t 2 2 x = x0 + v0 cos 0 t 1 y = y0 + v0 sin 0t gt 2 2 2 v R = 0 sin 2 0 2g : ) ; ) ;

2.2

155

2.12 10m/s, 30 . 18m . .

2.13 h. x . h, x . .

2.19

2.14 2.20 45 60 . ) , , . ) 100m/s 1.3km, .

2.20

156

2:

2.15 ( ) 90000 60 . , . .

2.16 10kg ( ), 2.21.

2.21

2.17 , , . ( .)

2.22

2.2

157

2.18 2.23 F . .

2.23

2.19 () 0.8. ( ) .

2.24

158

2:

2.20 F 2.25, m=5kg, .

2.25

) . ) . ( 30, F , 0.8 0.4.)

2.21 , m1 m2, 2.26. : ) m2 > m1 ) 1 2 .

2.26

2.3

159

2.3

. , , . , .

2.3.1

( 2.17). , . , , ( ). , cartoon 2.27. ( 2.28), ( ) .

160

2:

2.27

, . , , . , , . . .

2.28

2.3

161

. 2.29() r = 20m. , . , i, j k , . , R, . ( , , 2.29[]) :

R = r cos i + r sin jx y

(2.66)

( x = r cos y = r sin ) 2.29[] 2.29[]. . , , r.

162

2:

, :

= lim

d = t 0 t dt

(2.67)

, , . ( 2.29[]) ( : ) ( 2.30). 2.29[] 2.29[] . , , . , , . . , .

2.30

2.27 2.29[] / 2.29[] .

2.3

163

2.3.2

. 2.31 , R, t, , . , t , R . , :

V = lim

R t 0 t

(2.68)

.[13] , . . t0 , R, , S, . , :

V = lim

t 0

R S = lim t 0 t t

(2.69)

2.31

[13]

, 2.2.

164

2:

( ) :

S = r

(2.70)

(2.67), (2.69) (2.70), :V = limt 0

r = r lim = r t 0 t t

(2.71)

2.28 (, ) . : (2.71) .

, ( ) (2 ).

=

2 T

(2.72)

(2.71):

V=

2 r T

(2.73)

: : () , V , , : =V 2 = r

2.3

165

2.3.3

t. , (VA VB ) [14], , , , 2.32(). , V = (VB VA), . V 2.32(). V .

V = 2 V sin

2

(2.74)

V VA VB.

2.32

[14]

. , .

166

2:

, , . ( ) .

a = lim

V t 0 t

(2.75)

( [2.75]) . : a = a = lim

V V V = lim = lim t 0 t t 0 t t 0 t

(2.76)

(t0) , () . , VA VB ( 2.32[]) V VA. , . 2.32 . , r, ar. t . , : 0 t 0

lim sin

= 2 2

(2.77)

(2.74) (2.77) (2.76), :

V = lim a r = lim t 0 t t 0

2 V sin

2 = lim V = V t 0 t t

(2.78)

(2.71) (2.72), (2.78) :

ar =

V2 4 2 r = 2 r = r T2

(2.79)

2.3

167

: ) ; ) ; ) ;

2.29

: , . :

ar =

V2 4 2 r = 2 r = r T2

, m ar Fr, :

Fr = m a r

(2.80)

(2.80), Fr ( ) . , , , . , (2.79) (2.80) :

Fr = m

V2 4 2 r = m 2 r = m r T2

(2.81)

, , ( ). :

Fr = m

V2 4 2 r = m 2 r = m r T2

168

2:

2.3.4

2.3.1 . ( ) ( m) h ( [1.50] 1.3.6 1):

Fr = G

M m

( R + h)

2

(2.82)

R .[15] , Fr, . , :

Fr = G

M m

( R + h)

2

=

m V 2 = m 2 ( R + h) R + h) (

(2.83)

2.33

[15]

.

2.3

169

; ; ; : (2.81) (2.82). . () 2.84 108 m 2.66 106 rad s. ( G = 6.673 1011 Nm 2 kg 2 ) .

2.30

2.31

( ) . . 2.34() . ( 2.34[]) r. . , , . ; , .

2.34

170

2:

2.34[] t. . () . , , , . , :

N cos = B

(2.84)

:

Fr = N sin =

m V 2 r

(2.85)

(2.84) (2.85) . , = m g, :

r=

V2 g tan

(2.86)

. , , , . (2.86) . , , (2.86), ! , .

2.3

171

. . r, V. , . , . , , ( ) 180 ( 2.35[]) . 2.35 t. t, , . , .

2.35

172

2:

. F , . . , :

B = m g = F cos

(2.87)

. :

F sin =

m V 2 r V2 gr

(2.88)

(2.87) (2.88), :

tan =

(2.89)

( [2.86]), . 2.3.5

, . , , 2.36, m . . , , L. ( B = m g ) (T), , .

2.3

173

2.36 , , . ( r) ( t). :

Fr = m g cos + T :

(2.90)

Ft = m g sin

(2.91)

, V, .

2.36

, , , . , , . : Fr (2.90).

174

2:

ar :

ar = :

V2 L

(2.92)

Fr = m g cos + T =

m V 2 L

(2.93)

Ft (2.91). . at () :

at = :

dV dt

(2.94)

Ft = m g sin = m

dV dt

, :

a=

dV dV V2 = et + er dt dt Lat ar

(2.95)

et er . , . , .

2.3

175

2.3.6

2.37 V(t) t. , , u(t) . , . . 1.2.6 1 , . 2.1.3 , . ( ) , , , 2.37. , .

2.37 u(t) V(t). , .

176

2:

R ( t ) , R ( t ) R ( t ) , t . : () . :

V ( t ) = lim

R ( t + t ) R ( t ) t

t 0

=

dR ( t ) dt

(2.96)

() . :

u ( t ) = lim

R ( t + t ) R ( t ) t

t 0

=

dR ( t ) dt

(2.97)

2.37 :

R ( t ) = R ( t ) R ( t )

(2.98)

(2.98) , :

dR ( t )

dt dt dt V ( t ) = V ( t ) u ( t ) V ( t ) = V ( t ) + u ( t )

=

dR ( t )

dR ( t )(2.99)

, u(t) V ( t ) , V ( t ) :

V ( t ) = V ( t ) u ( t )

(2.100)

(2.100) .

2.3

177

.

u V :

2.32

u = 2 km h i + 0.2 km h j V = 0.9 km h i + 12 km h j i j , V .

, (2.100), . , . , L, V . u. . . , . L ( ) V, L/V . , 2 L/V. . , , , . , , , . u () . V ( ) ().

!!!

178

2:

( [2.99]) V+u. : L/(V+u). B A V ( ) (). ( [2.99]) Vu. : L/(Vu). , , :

T=

L L 2 V 2 L + = L 2 = 2 V +u V u V u V

1 u2 1 2 V

2.33 ; . , , . . . , , ( ). , Lorenz, Einstein. 2.3.7

, (2.100) . ; ( ), ;

2.3

179

, . . . . . . , 2 . . , , , ( ) : , . ( , ) . , ( ) . , . ( ) . . ( ) ; . . , . .

180

2:

2.38 m , a. , () , , . : , , a. ( = m g) (). , a. , . , , .

2.38 a. : ) ) .

2.3

181

(2.101).

2.34

T cos m g = 0 a tan = T sin = m a g

(2.101)

. . , () FI, 2.38(). : FI =ma ( ) ( ), :

T cos m g = 0 a tan = T sin m a = 0 g

(2.102)

.

( a) ; : F = m a , m .

2.35

1.3.7 1 . .

182

2:

2.39 a

2.36 2.39 a. F ( ). :

F = m (g a )

(2.103)

, , (2.103), ( a = g), . , , .

2.37 ( ) ; : a > g (2.103) . . , , . Atwod, 1.3.9 1.

2.3

183

2.40 twood

, , , mA mB (mA < mB) , . a, (a < g, a = g). 2.40, . . , , , . 2.40 , , . , ( ), ( ), . , 2 , :

184

2:

: :

mB g mB a FB = mB mA a FA + mA g = mA

(2.104) (2.105)

: . 3 , ( ).

FA = FB = T

(2.106)

(2.104), (2.105) (2.106) , :

mB mA (g a ) mB + mA 2 mB mA (g a ) T= mB + m A

=

(2.107) (2.108)

. (2.107) (2.108) . , . , , . , , : . , , , .

2.38 , a > g, (2.107) (2.108) , , , . , ( ) . 2.41.

2.3

185

2.41

2.3.8

. ( ) , . , , 2.42, . m, . () ( ) . . : ;

186

2:

2.42

() , (), , R R (R< R) , 2.43. : . . , , , , 2.43.

2.43

2.3

187

() () 2.43. , R. , :

2.39

tan =

2 rg

(2.109)

, , 2.44. , FI, .

2.44

188

2:

() , . , . , , , . , , . , FI , , , R , , ( 2.44) :

FI = m 2 R

(2.110)

, , , . , :

T sin = m 2 R T cos = m g

(2.111)

, , ( [2.109]) . , . , . , .

2.3

189

, . ; ;

2.40

2.45

. R= 6.37 106m = 7.292 105rad/s. m ( ) , 2.45. [16] . r, :

r = R cos

(2.112)

[16]

. , . , .

190

2:

, , . ( B0 = G

mM , M ) R2 , . , , , ( 2.46[]).

2.41 .

, , , (.. ) , T, . , , , . , , , . . . , FI, 2.46.

2.42 , , .

191

2.46 ()

. , :

= lim

t 0

d = t dt

, , . .

192

2:

: , V , , , =

V 2 = . r T

, . :

ar =

V2 4 2 r = 2 r = r T2

, , , ( ). :Fr = m V2 4 2 r = m 2 r = m r T2

: Fr Ft. , :

a=

dV dV V2 = et + er dt dt Lat ar

193

et er .

u(t) V ( t ) , V ( t ) , :

V ( t ) = V ( t ) u ( t )

. ( ) . . . . : FI = m a

194

2:

2.22 , ( ) , , 8.2 108. 5.3 1011m, .

2.23 132 . . .

2.24 15m . 1m/s, 1.5m/s. ) . . ) ( ), ;

2.25 m . .

2.47

195

2.26 2.48 F. 0.5, 0.75. .

2.48

2.27 , , r m ; ( : m = 1kg, = 10kg r = 0.5m.)

2.49

196

2:

2.28 2.50 . m r v. . : ) ) .

2.50

2.29 ( ). . ( 6.38 106m 5.975 1024kg , G = 6.672 1011m2/kg2.)

2.30 . . m v0, 2.51, t = 0. , . s. : )

197

, ) ) 10s .

2.51

2.31 1200km/h. . ( 2.52) . .

2.52

2.32 65km/h 120m. , , , 0.65m/s2. 90.

198

2:

2.33 940g 1.3m. ) 120, . ) .

2.34 TSS (Tethered Satellite System) SA 500kg, , 20km . 250km , 92.6% . TSS . 230km, 93.2% . TSS . ( TSS .)

2.53

2.35 40km/h 130m, 60km/h. ;

199

2.1 m F. v ( t ) = ( b t 2 ) i + ( c t + d ) j , t , , b, c d . b, c d, . : b, c d m/s3, m/s2 m/s. : a (t ) = ( 2 b t ) i + (c) j

2.2 21km/s. , . 0.035km/s2, . 250s, . : 22.6

2.3 , 1.9m . 4.5m, . : 7.23m/s

2.4 v0, h , 2.54. a .2 : a = tan 1 gh 2v0

2.5 5m 5m , 2.55. -

200

2:

2.54

; ; 1m . : ) 12.7m/s, ) 51.3

2.55

201

2.6 , 1968, . , . . (g = 9.786m/s2) . , Robert Beamon 8.90m , . , 25 . Beamon , g = 9.81m/s2, . ( ). : 8.88m

2.7 2.5kg 1.6m/s. , F1 F2. , F1 F1 = 15 j. 3s, {4.8m, 10.8m}. F2. : F2 = 9 j N

2.8 3.1kg . , a = 0.91 i 0.27 j. : F1 = 1.2 i 2.5 j , ; : 4.2 i + 1.66 j N

2.9 3700kg , 2.56.

202

2:

1100 25 . : ) ) 0.16m/s2. : ) 1990N, ) 1400

2.56

2.10 15kg , 2.57. . : 528

2.57

203

2.11 10kg , , 2.58. . : 1 = 139, 2 = 98

2.58

2.12 2.59, a1 a2, . . : a 2 = 2 a1 , a1 = m1g ( m1 + 4m2 )

2.59

204

2:

2.13 22kg 35 . 0.68, ; . : 342N

2.14 2.5kg 3.1kg , 2.60. 0.51, 0.23. . : 1.63m/s2 3.29

2.60

2.15 75m . : 97.6 km/h

2.16 7.2 kg . : ) 0.95m/s2, )

205

14m/s, ) 9s 14m/s. ( ) 450. : ) 77.47 ) 70.63 ) 59.4 52.69m/s2.

2.17 m1 m2, 2.61. F . . : F = ( m1 + m2 ) g tan

2.61

1.1 . : {0.5, 0, 0}. , Q, {1.0, 0.5, 0.8}, Y : {1.0(0.5), 0.5, 0.8} = {1.5, 0.5, 0.8}.

1.3 1.1 12s 22m.

1.4 x, t. (1.2). , , ,

( x2 x ) ( x1 x ) ( x2 ) ( x1 ) = , ( t2 t ) ( t1 t ) ( t2 ) ( t1 )1.5

.

x2 x1. (1.2) :

( x2 ) ( x1 ) ( x2 ) ( x1 ) = ( t2 ) ( t1 ) ( t2 ) ( t1 )1.6 . , , . V V , , .

1.7 : , , .

208

:

. , . , .

1.8) t= 0 ) .

1.9 2.

1.10, , (1.28) (1.29), (1.27). x x1. (1.27) x x (1.28), : x x1 = v 1 0 V v

1.11x (t ) = t + v (t ) = dx ( t ) dt = d d ( t ) + ( ) = dt dt

1.12 . t, , . , t , t .

v ( t ) = ( 0.094 m s ) + 2 ( 0.1m s 2 ) t + 3 ( 0.0009 m s3 ) t 2

1.13a ( t ) = 2 ( 0.1m s 2 ) + 2 3 ( 0.0009 m s3 ) t

209

1.14x (t ) =1 ( 3m s 2 ) t 2 2 v ( t ) = ( 3m s 2 ) t

1.15 .

1.16 .

1.17 .

1.18 .

1.19 . .

1.21 , , 10m/s. 1.25() . , , . 10m : . . , 10m/s ( ) . .

210

:

1.22 :

1 2 ( 2 m s 2 ) t 1 2 2 2 ( 2 m s ) t = ( 0.01m s ) t + 90 m 2 x = ( 0.01m s ) t + 90 m x =

( m s ) t ( 0.01m s ) t2 2

t 9.48s 90 m = 0 1 t2 +9.49s

. t1 (.. ), .

x = ( 0.01m s ) t1 + 90 m = ( 0.01m s ) 9.48s + 90 m = 90.0948m

1.23 1.26.

1.24V2 = V2 V1 = 8km h 20 km h = 12 km h , .

1.25V2 = V2 V1 = 0 km h 20 km h = 20 km h

1.26V2 = V2 V1 V2 = V2 + V1 = 10 km h 20 km h = 10 km h

1.27 : +90km/h

1.28 , . , , , , . , .

211

1.31 .

1.32 .

1.33 .

1.34 (1.54) m:

M m R2 M m B = G 2 R M m G 2 6 2 22 R2 M ( 6.37 10 ) ( 7.36 10 ) B R = = 2 = 0.165 2 M m R M B 1.74 106 ) ( 5.98 1024 ) ( G R2 B = G

1.35 .

1.36 . , V = a t . .

1.37 .

1.38 .

212

:

1.39v ( t ) = a1 t a1 = a 2 v (t ) = a 2 t

1.40 .

2.12 2 R = ( 2.935951) + ( 2.629045 ) 12

= 3.941026

2.2

2.3cos = R = 2.935951 3.941026 0.75, sin = R 0.661, 2.1 R = R cos i + R sin j

2.4A+B =

( x + x )

2

+ ( y + y ) , A B =2

( x x )

2

+ ( y y )

2

2.5 2.7, .

2.6x (1.162s ) = 1.736 1.162 = 2.017232 m y (1.162s ) = 9.848 1.162 4.905 (1.1622 ) = 4.82043m.: R (1.162s ) = ( 2.017232 m ) i + ( 4.82043m ) j t = 1.796s.

(

)

2.7 . , x() .

213

2.8 X , . , Y () , , . , , Y , .

2.9Vy(1.162s) = 1.55122m/s, Vy(1.796s) = 7.77076m/s. x . t = 1.162s :

|V(1.162s) | = [(1.736)2 + (1.55122)2]1/2m/s 2.326m/s. X (2.5) : cos = (1.736 ) 2.32 = 0.748(2.113)

, . : 270 < < 360 (2.113) 318.42. t = 1.796s.

2.10 (2.23) :V ( t ) = (1.736 m s ) i + 9.848m s ( 9.81m s 2 ) t j

(

)

(2.114)

(2.114) . t=0.3s V(0.3s)=(1.736 m/s) i+(6.905 m/s) j t=0.5s V(0.5s)=(1.736 m/s) i+(4.943 m/s) j. :

Vy ( t ) = 9.848m s ( 9.81m s 2 ) t j

(

)

(2.115)

j . , , Vy j. (2.115) :

Vy ( 0.9s ) = (1.019 m s ) j , Vy (1.7s ) = ( 6.829 m s ) j

214

:

2.11 B . . F .

2.12 .

2.13 . Ts = s N = B sin (2.116)

, . : Tk = k N < B sin (2.116) (2.117): s > k . (2.117)

2.14m a = B sin k N N = B cos m a = B sin k B cos

a = g ( sin k cos )

(2.118)

( [0,90]), . , (2.118), .

2.15 (.. ) . . , , . , . , , . , , .

215

. ( , ). : , , , . , .

2.16 (2.39) .

2.17 .

2.18 (2.39) .

2.19 .

2.20 !

2.21 .

2.22 (2.53) .

2.23 (2.55) 2 = 90.

2.24 (2.57) .

216

:

2.25 : V0 sin ,

(V0 sin )

2

+ 2 g cos h g cos ( 0 90

)

. . , 1.21 1 .

2.26 (2.63), (2.64) (2.65). ) = 0, sin = 0, cos = 1 ) = 0, = = 0 ) = 0, h = 0. (2.52) (2.55) . : Vx = V0cos Vy = V0sin gt

2.27 20 .

2.28 (2.71) V r = V/r. , . , , , . , .

2.29 . .

2.30 (2.81) (2.82) :

217

Fr = G T=

M m

( R + h)

2

= m3

4 2 ( R + h)T2

4 2 ( R + h) GM

.

2.31 m , M R , (2.83) :

G

M m

( R)

2

= m 2 R M = ( 2 R 3 ) G

(2.119)

, :

Fr = G

M m

( R)

2

= m ar ar = G

M

( R)

2

(2.120)

, . : 1N = 1kg 1m/s2

2.32 (2.99) V(t) .

2.33 . L : t = L (V u ) (2.121) . (2.121) ( ) . , ( ) . , , , .

2.34 .

218

:

2.35 , F, . , , , .

2.36 (), () . , :

B = T + m a T = m g m a = m(g a ), 3 : F = T (2.122) (2.123): F = m ( g a )

(2.