73

7.

(-

) ,

.

.

.

1) -

. .

, .

. .

_01 : A = 20 m, = 10 m/s

S = 1980 m.

;

B A

G G

S

G

A

74

-

, .

, S = S + A

x = S ( ,

). :

x = t x S St t

+ = = = = A t = 200 s

_02 : -

2 m/s2. :

. 15 s.

. .

. : = 0 + t = t = 30 m/s . : x = S = 0t + t2 S = t2 S = 225 m

_03 : 0 = 10 m/s -

= 2 m/s2.

. -

;

. -

;

75

. :

= 0 - t (1) x = 0t - t2 (2)

(1) 0 0 0 00 0 10

- t t - t t t 2 2 2 2 4

= = = = = t = 2,5 s

(2) => x = S = 0t - t2 => 1x 10 2,5 - 2 6,252

= =>

=> x = 25 - 6,25 => x = 18,75 m

. ( = 0). :

(1) => 0 = 0 - t => t = 0 => 0 10t t t 5 s 2= = =

(2) => x = 0t - t2 => 1x 10 5 - 2 252= =>

=> x = 50 - 25 => x = 25 m

76

77

2) . . -

.

. t t-1 t -

. -

xt, xt -1. t :

xt = xt - xt - 1

-

.

_04 : -

= 4 m/s2.

;

, -

:

x = 0t + t2 => x x0 = t2 => x = t2 (1)

4 t = 3 s, t = 4 s. (1) :

t = 3 s: x3 = 432 = 29 = 18 m t = 4 s: x4 = 442 = 216 = 32 m :

x34 = x4x3 =(32 18)m x3 = 14 m

_05 : -

, -

69 m. :

78

.

.

.

. , -

:

x = 0t + t2 => x x0 = t2 => x = t2 (1) 12 t = 11 s, t = 12 s. (1) :

t = 11 s: x11 = 112 = 121 = 60,5 t = 12 s: x12 = 122 = 144 = 72 :

x1112 = x12 - x11 = 72 60,5 x1112 = 11,5 = =2x 69

11,5 11,5

=> = 6 m/s2

. 4 :

x0-4 = t2 => x0-4 = 616 => x0-4 = 48 m

_06 : -

. :

. 10 s.

. 2 -

.

. 0 , - .

x = S = => S = 1020 => S = 100 m . , -

(t = 0 => 0 = 0). :

79

= t => = t

=

=> = 2 m/s2 (

t = 10 s => = 20 m/s = f(t))

t, t

= 1 s t = 2 s.

t = 1 s: 1 = 21 = 2 m/s t = 2 s: 2 = 22 = 4 m/s

S = x = E = (4 2) 12+

=> S = 3 m

_07 : =10m/s

, to xo=0 -

2m/s2.

.

o (t t ) (1)= - , (=0).

(1) =10m/s, =2m/s2 to=0 :

0 10 2(t 0) t 5s= =

(m/s)

20

4

20 1 2 10 t (s)

ttto=0

ouG

ttt 4 ttt5 54x

G

5xG

4xG

80

5, -

5 4t t t 5 4 1s= = = . 1 :

: 21

x t t2

= x4 x5 ,

t4=4s t5=5s.

:

24 4 4

1x 10 4 2 4 x 40 16 x 24m

2= = =

25 5 5

1x 10 5 2 5 x 50 25 x 25m

2= = =

5 :

4 5 5 4x x x 1m = =

2 : -

-t.

t4=4s. :

o (t t )= 4 10 2(4 0)= 4 4 10 8 2m/s= = .

- :

-

- x -

, :

1x E 2 (5 4) x 1m2

= = =

4 5

2

10

(m/s)

t(s)

81

3)

.

t2.

.

x1 .

. -

t2 x2 -

.

. , -

:

x = x1 + x2

d . x > d,

, x < d,

.

.

.

_08 : 72km/h. -

23,5m.

. t=0,3sec,

-5m/s2.

.

. ; ( )

82

. , ;

. , ,

;

.

, ,

:

1

mx 20 0,3s 6m

s = =

. -

, : 2 2

02

u u 0 400x m 40m

2 a 10 = = =

:

1 2x x 6m 40m 46m d 23,5m + = + = > = .

.

, ;

: ( )

: '2 1x d x 17,5m = =

2

2 2' 0u ux

2 a =

2 20u u 2 a x= +

u 400 2 ( 5) 17,5 m / s= + u 225 m / s= u 15m / s= .

(),

, 46m.

83

_09 :

0=72Km/h. 50 m .

t = 0,7 s (

).

. -

10 m/s2.

t = 0,7 s, 0 = 72 m/s.

:

x1 = 0t => x1 = 14 m

:

= 0 - t (1) x = 0t - t2 (2)

= 0, :

(1) => 0 - t = 0 => = 0t

=> t = 2 s

(2) => x2 = 202 - 1022 => x2 = 20 m

:

x = x1 + x2 = 14 + 20 =>

x = 34 m < 50 m, .

x2 x1

x1 x2

0G

G

(t = 0) A (t = t = 0,7 s) B

0G

= 0

84

_10 : 0,2s.

=40m/s

, , ,

10m/s2.

-

.

,

.

-

-

.

2 2 1t t t= , :

2 t= , =10m/s2 , =40m/s =0, :

20 40 10t= 2t 4s= 2 2 1t t t= - , :

22

1x 40 4 10 42

= 22 2 21x t (t )2= 2 2x 160 80 x 80m= =

1 1 o 1t t t t 0 0,2s= = = (= ) , :

1 1 1x t 40 0,2 x 8m= = = , :

1 2x x x 88m= + =

to=0

ouG

ouG

t1 t2

=0

G

1xG 2x

G

85

to=0

(t=t2), -

-

-, :

1 2 1 o 2 11x E E (t t ) (t t ) (1)2

= + = +

to=0, t1=0,2s t2=t1+t2=4,2s

(1) :

1x 40 0,2 40 4 x 8 80 x 88m2

= + = + =

4) 1 :

1 :

.

, 1) , 2) .

2 :

,

.

3 :

x1 x2

. .

1) 1 2d x x= 2) 1 2d x x= +

_11 :

d = 500 m

= 30 m/s,

= 20 m/s. :

(m/s)

to t1 t2

1 2 t(s)

86

. t -

.

. .

. t .

-

:

: x = t x = t (1)

: x = t xM = t (2)

:

x - x = d ( - ) t = d =

dt -

t = 50 s

. : x = S S = 1 t S = 1500 m

2 :

1.

. 2. ,

, : - (t=t) , -

G

t =

t = 0

x

d

t = 0

t = tx

G

87

- . o(A)t 0= 0)( =Aox , 0)( >Ao ( 0)( >Aa ), tt Bo =)( .

- , . . 0)()( == BoAo tt 0)( =Aox , 0)( >Ao ( 0)( >Aa ).

3. , .

: o o ox x (t t )= + () : o oa(t t ) = +

: 2o o o o1

x x (t t ) a(t t )2

= + +

-. , - .

d, o(B)x d= , - - , .

4. (t t )= , (x ) : A Bx x x= = -

t .

-

t . -

.

5. - -

, .

88

_12 : . .

60km/h. -

0,008h,

100km/h, 100km/h.

.

1. ;

. 0,0048h . 0,008h . 0,005h . 0,006h

2. ( 1), ;

. 0,2km . 0,15km . 0,3km . 0,144km

3. 0,012 ;

4. ;

. 0,012h . 0,005h . 0,008h . 0,01h

5. , ;

. 0,3km . 0,6km . 0,72km . 0,45km

1.

: () 0,0048h, :

.

: 2Au 100km /h

a 12500km /ht 0,008h

= = =

60km/h, :

A A

60u a t 60km /h t h t 0,0048h

12500= = = =

89

2.

: () 0,144km, :

, :

A

60 0,0048x km 0,144km

2 = =

Bx 60 0,0048km 0,288km = = :

B Ad x x 0,144km= = 3.

:

, :

A

100 0,008x km (0,012 0,008) 100km 0,4km 0,4km 0,8km

2 = + = + =

Bx 60 0,012km 0,72km = = :

A Bd x x 0,8km 0,72km 0,08km= = = 4.

: () 0,01h, :

,

. 0,008h,

:

, :

A

100 0,008x km 0,4km

2 = =

Bx 60 0,008km 0,48km = = t , :

B A

100 0,008x x 60 t (t 0,008) 100

2 = = +

60 t 0, 4 100 t 0,8 t 0,01h = + = 5.

: () 0,6km, :

t 0,01h = : A B kmx x 60 0,01h 0,6kmh = = =

90

5)

. , -

.

.

.

x=x1+x2+x3+... t=t1+t2+t3+...

_13 : to=0

0 (xo=0), 1 = 2 m/s -

t1 = 2 s.

2s t = 2 s -

. -

S = 20 m. :

. .

. .

. =f (t), =f (t), x=f (t)

.

u0=0 1uG

1 2u u=G G u3=0 a2=0

x=x t=t ()

3 3

x=xt=4sec (B)

2x=x t=2sec (A)

1 x=x=0 t = t= 0 (O)

x

1 3

(+)

91

. A0 ( )sec20 t :

:

0 1 0

20 0 o 1 0

0 0 0

a (t t )

1x x (t t ) a (t t )

20, t 0, x x

= + = + + = = =

1

21

a t(1)

1x a t (2)

2

==

t=2 s

==

m4x)2(sec/m4)1(

1

1

BA ( )sec42 t :

=======++=

+=

0a,m4xxsec,2ttsec,/m4

)tt(a21)tt(xx

)tt(

101010

201o00

00

1

1 1 1

2m / sec .

x x (t t ) x 4 4(t 2)(3)

= = = = + = +

t=2s m12x 2 = B ( )34 tt :

m20xxS 33 == 0 0

20 0 o 1 0

0 2 0 2 0 1 3

(t t )

1x x (t t ) a (t t )

2x x 12m, t t 4 sec, 2m / sec,a a

= + = + + = = = = = = =

1 3 2

22 1 2 3 2

a (t t )(4)

1x x (t t ) a (t t ) (5)

2

= + = + +

t=t3 : m20xx0

3 ===

92

( )(4)

3 3 3 3

53 3

3 3 3

3 33 3 3 3

0 0 4 (t 4) (t 4) 41

8 4(t 4) (t 4)( 4)1 2x 20m 20 12 4(t 4) (t 4)2

t t12 (t 4) (t 4) 2 t 4 2 4 t 8 sec

2 2 2

= = + = = + = = + + = = + = =

== 4.44)4t( 333 23 sec1ma =

I. sec83 == tt

II. : 2

2

2

2m / sec , 0 t 2 sec

a 0m / sec , 2 t 4 sec

1m / sec , 4 t 8 sec

= = =

1 1

1 3 2

u t 2t, 0 t 2 sec4m / s , 2 t 4 secu (t t ) 4 (t 4), 4 t 8 sec

= = = = + =

2 21

1 1

2 22 1 1 3 2

1a t t , 0 t 2 sec

2x x (t t ) 4 4(t 2), 2 t 4 sec

1 1x (t t ) a (t t ) 12 4(t 4) )t 4) , 4 t 8 sec

2 2

= = + = + + + = +

t(sec)

(m/sec )2

o 2 4 6

2

(m/sec)

t(sec)o 2 4

4

6 8

t(sec)o 2 4

4

8

12

20

x(m)

93

6)

...

,

. -

.

:

. , -

-

.

. . -

.

. .

,

.

.

-

.

[ ][ ] [ ]usec

mtx ==

x (m)

t (sec)

[ ][ ] [ ]asec

s/mtu ==

u (m/s)

t (sec)

94

.1. , , .

:

utx

ttxx

12

12 ==

=

.2. , ,

:

atu =

=

.

_14 :

.

-

:

) t=3sec;

)

t=0sec t=6s;

0 1 2 3 4 5 6 7 805

1015202530

t(sec)

x m)(

x (m)

t (sec) t1 t2

x1 x2

u (m/s)

t (sec) t

u

95

) t=2s

t=6s;

) .

)

;

:

) ,

3sec,

x=20m

.

) m10m10m0xxx 06 === ) , -

2sec 4sec m0m20m20xxx 2442 === , 4sec 6sec m20m20m0xxx 4664 === . ,

m20xxs 6442 =+= . ) ( )

0sec 2sec . 2sec 4sec -

. 4sec 6sec -

(

).

) :

s/m10sm

220

sm

46200

tx

u,s6ts4

s/m0sm

20

sm

242020

tx

u,s4ts2

s/m5sm

210

sm

021020

txu,s2t0

3

33

2

22

1

11

===

=

===

=

===

=

-

10m/s.

0 10 20 30-10

x x(m)

96

_15 : ,

.

.

:

,

, -

:

tua =

, -

,

:2

m20 msa 54 s s

= =

. .

-

, .

[ ] [ ] [ ]xmssmtu ===

. - ( t):

) ,

.

) : = t

-

.

) -

.

0 1 2 3 4 5 6 7 8 9 10 0 5

10 15 20 25 30 35 40 45 50

t( sec)

um/sec) (

u (m/s)

t (sec)t1 t2

u1 u2

x

97

v). , ,

: - , -

, - , - .

. - ( - t):

) ,

.

) , -

( = .t = )

.

) -

() - -

.

. , :

S

t=

. : - , - , - ,

.

. - -

, .

98

_16 :

-

aG

. 10s

.

-

. ,

-

, -

.

1. :

2. ;

. 10s . 50s . 70s . 134,807s

3. ;

. 21000m . 10000m . 7000m . 3000m

4. -

;

99

1.

: 600m/s, :

() 600m/s.

600m/s, , -

.

2.

: () 70s, :

10s ,

(-

). 70s, -

, ( ).

3.

: () 21000m, :

, 0s 70s, :

( )600m / s 70sx 21000m

2

= =

4.

:

-

. -

, -

, -

.

_17 : -

.

. 25s.

. ,

5 s.

100

. -

25 s .

. 25 s.

.

0 t 5 s -

5 t 15 s 15 t 20 s -

20 t 25 s , .

.

= = = = 20 - 10 10 t 5 5

2 2 m/s

. -

= f(t) .

S = 1 + 2 + 3 + 4 => S = (10 20) 5 1 1

10 20 20 5 20 52 2 2

+ + + + =>

=> S = 75 + 200 + 50 + 50 => S = 375 m

1, 2, 3, 4.

x = 1 + 2 + 3 - 4 => x = (10 20) 5 1 110 20 20 5 - 20 52 2 2

+ + + =>

=> x = 75 + 200 + 50 - 50 => x = 275 m

.

= =

S 375

t 25 = 15 m/s

E3 E1

(m/s)

20

10

-20

0 5 15 20 25 t (s)

E2

E4

101

( ): 1) , t1 = 6 s -

x1 = 24 m. :

. 16 s

. 100 m.

2)

126 km/h 875 m. 90

km/h, ;

3) L1 = 200 m

=20m/s. L2 = 2000 m.

. ;

. ;

4) .

1 = 10 Km/h -

2 = 30 Km/h.

1 . :

. .

. .

( ): 5) . -

, t = 8 s -

= 72 Km/h.

. .

102

. ;

6) 35 m/sec 15 m/sec, -

500 m. :

. .

. .

7) t0 = 0 -

= 4m/s2.

t1 1 = 20m/s. t2

t1;

8) 100

m 10 s. :

.

.

. 64/100

.

9) , t0 = 0, x0 = 0

0 = 40 m/s. -

4 m/s2. :

. .

. .

10) 0 = 20 m/s.

t=0 = 4 m/s2. :

. ;

. ;

11) 0 = 5 m/s,

= 2 m/s2 10 s. :

.

. .

. - -

10 s.

103

12) 0, -

= 10 m/s2 . -

80 m, :

. 0

. 0/2, -

0/4.

: 13) 108 Km/h, -

90 m.

0,5 s 10 m/s2,

.

14) 90 Km/h,

65 m , .

0,3 s 5 m/s2 -

;

15)

,

t=0,4s. 108 km/h

5 m/s2.

,

.

16) 72 Km/h -

50 m.

, 0,2 s -

-5 m/s2 .

.

104

: 17) = 2 m/s2.

-

;

18) ,

46 m.

:

.

. .

19) t0 = 0 0 = 12 m/s

x0 = 0. 2 m/s2 -

:

. .

. .

20) ,

= 3 m/s2. 9 m -

. .

21) .

5 9 -

44 m 60 m .

12 .

: 22) , 90 m -

.

5 m/s ,

4 m/s.

. ;

. ;

. ;

105

23) - ( )

6 Km. t = 0 , -

,

. 1 = 20 m/s (

) 2 = 40 m/s.

;

24) S=10Km. -

1=40Km/h

2=60Km/h . :

. ;

. ;

:

,

.

(. 0,5h 20km 4km)

25) 1

= 1,5 m/s 2 = 1 m/s.

, 30 m .

30 m ; -

;

26) = 20 m/s

= 50 m/s, . -

100 m . -

200 m ;

27) , 120 Km/h,

180 Km/h.

72 Km/h, 4

m/s2, .

;

106

28) -

24 m. = 1 m/s2

=2m/s2.

. ;

. ;

. - - -

.

29) 1 = 20 m/s

2 = 30 m/s 1 = 1 m/s2

2 = -1 m/s2 . :

. .

. .

. -

.

30) , 1 = 20 m/s , -

t0 = 0 x0 = 0.

64 m ,

= 2 m/s2

. :

. .

. .

. - -

.

31) ,

1=40m/s 2 = 60 m/s, . -

.

2610 m

1m/s2, .

107

: 32) = 5 m/s2.

10 m/s, . -

8 s.

. .

. - - .

33) t1

1 =2 m/s2

2=8 m/s2, . t = 50 s. :

. .

. -

.

34) t1

1 = 1 m/s2, = 10 m/s.

t2

t3 2 = 5 m/s2, -

. 100 m. :

. .

. 5 s.

. - .

35) 4 s

1 m/s2. 2 s

, 20 m, ,

.

. .

. .

. .

36) ,

S = 400 m. , -

1 = 2,5 m/sec2. 2 = 10 m/sec

2=1m/sec2.

108

.

.

. - , - - -

.

37) 1 = 2

m/s2 7 s. 2

4 s. 3 = 8

m/s2 . t = 14 s. -

:

. 2.

. .

. t = 9 s.

38) , -

1 6 s, 1 = 3 m/s.

-

3 = 30 cm/s2 .

x = 42 m. :

. () 1.

. , -

. .

. t = 14 s.

39) -

10 m/sec 0,4 m/sec2. -

5 sec .

= (t) x = x(t).

40) x . -

t0 = 0 x0 = 0, 1 m/s2

3 s, -1 m/s2 .

. .

109

. .

. .

41)

.

. .

. - .

. .

42) - ,

.

. .

. - .

. - , x0 = 0.

.

.

. -

.

.

: 2 s, 5 s, 7 s.

43) H

, . t0 = 0 -

x0 = -10 m.

. -

.

.

10 s 40 s.

. .

. -t x-t.

(m/s)

10

0 2 5 6 t (s)

(m/s)

40

20

0 10 20 40 t (s)

(m/s)

20 10

-10 0 2 4 6 8 10 t (s)

110

44) t0 = 0 0 = 0.

- :

. .

. - .

. , x0 = 0.

. -

t = 15 s.

45) -

. t0=0, x0=20m,

0 = 4 m/s

20s .

. -.

. -t,

20s , t0=0

t=20s.

. ;

46) -

- -

,

.

t0=0,

x0=0.

.

.

. - -.

. ,

.

(m/s2)

10

5

-5 0 10 20 30 t (s)

(m/s2)

2

-3

0 4 8 12 16 20 t (s)

(m/s)

20

10

-10 0 1 2 3 4 5 6 7 t (s)

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