073-110 ΚΙΝΗΜΑΤΙΚΗ Δ 2007
38
73 7. Περιπτώσεις εφαρμογών Γενικά Το πρώτο και σημαντικότερο που πρέπει να γίνεται στις ασκήσεις της κινηματικής (ειδι- κά στις πιο σύνθετες) είναι ένα πολύ καλό σχήμα, που θα μας βοηθήσει στην κατανόηση αλλά και τη λύση της άσκησης. Ύστερα πρέπει πάντα να γράφουμε τις εξισώσεις που ισχύουν για την ταχύτητα και τη μετατόπιση σε κάθε είδος κίνησης. Αυτές μπορούν να μας οδηγήσουν στη λύση οποιασδήποτε άσκησης. Ειδικές περιπτώσεις 1) Εφαρμογή τύπων στην ευθύγραμμη ομαλή ή στην ευ- θύγραμμη ομαλά μεταβαλλόμενη κίνηση. α. Γράφουμε για κάθε σώμα που αναφέρεται στην άσκηση την εξίσωση της ταχύτητάς του και την εξίσωση της μετατόπισής του, ανάλογα με το είδος της κίνησης που εκτελεί. β. Από τις εξισώσεις που προκύπτουν βρίσκουμε όλους τους αγνώστους. Παράδειγμα_01 : Μια ατμομηχανή έχει μήκος = 20 m, κινείται με ταχύτητα υ = 10 m/s και περνά μία γέφυρα μήκους S = 1980 m. Για πόσο χρόνο θα τμήματα της ατμομηχανής θα βρίσκονται πάνω στη γέφυρα; B A υ υ S υ
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Transcript of 073-110 ΚΙΝΗΜΑΤΙΚΗ Δ 2007
-
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)
/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org) /PDFXTrapped /Unknown
/Description >>> setdistillerparams> setpagedevice
