Чертов А.Г. Решебник
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R = 1, 2 ; ϕ = At + Bt
3
;
A = 0, 5 / ; B = 0, 2 /
3
;
t = 4
a
τ
, a
n
, a
ω =
dϕ
dt
= (At + Bt
3
)
0
= A + 3Bt
2
;
ε =
dω
dt
= (A + 3Bt
2
)
0
= 6Bt;
a
τ
= εR = 6RBt; a
n
= ω
2
R = (A + 3Bt
2
)
2
R;
a =
q
a
2
τ
+ a
2
n
t = 4
a
τ
= 6 · 1, 2 · 0, 2 · 4 = 1, 44 /
2
;
a
n
= (0, 5 + 3 · 0, 2 · 4
2
)
2
· 1, 2 = 16, 43
/
2
.
a =
q
(1, 44)
2
+ (16, 43)
2
= 16, 49 /
2
.
R = 1 ; ξ = At + Bt
2
;
A = 8 / ; B = −1 /
2
;
t = 2 c.
v, a
&%
'$
a
n
A
A
AK
~a
+
R
6
a
τ
v =
dξ
dt
= (At + Bt
2
)
0
= A + 2Bt.
t = 2
v = 8 − 2 · 1 · 2 = 4 / .
a
τ
=
dv
dt
= (A + 2Bt)
0
= 2B = 2 · (−1) = −2
/
2
.
a
n
=
v
2
R
=
4
2
1
= 4
/
2
.
a =
q
a
2
n
+ a
2
τ
=
q
4
2
+ (−2)
2
= 4, 47
/
2
.
x
1
= A
1
+ B
1
t + C
1
t
2
;
x
2
= A
2
+ B
2
t + C
2
t
2
;
A
1
= 10 ; B
1
= 1 ; C
1
= −2
2
;
A
2
= 3
; B
2
= 2 ; C
2
= 0, 2
2
;
τ, a
1
, a
2
v
1
=
dx
1
dt
= (A
1
+ B
1
t + C
1
t
2
)
0
= B
1
+ 2C
1
t;
v
2
=
dx
2
dt
= (A
2
+ B
2
t + C
2
t
2
)
0
= B
2
+ 2C
2
t;
t = τ
B
1
+ 2C
1
t = B
2
+ 2C
2
t; 1 + 2 · (−2)τ = 2 + 2 · 0, 2τ
1 − 4τ = 2 + 0, 4τ; τ = −0, 23 ;
t = 0
a
1
=
dv
1
dt
= (B
1
+ 2C
1
t)
0
= 2C
1
= 2(−2) = 4
2
;
a
2
=
dv
2
dt
= (B
2
+ 2C
2
t)
0
= 2C
2
= 2(0, 2) = 0, 4
2
R = 0, 5 ; ϕ = At + Bt
3
;
A = 2 / ; B = 0, 2 /
3
;
t = 3
.
a
ω =
dϕ
dt
= (At + Bt
3
)
0
= A + 3Bt
2
;
ε =
dω
dt
= (A + 3Bt
2
)
0
= 6Bt;
a
τ
= εR = 6RBt;
a
n
= ω
2
R = (A + 3Bt
2
)
2
R.
a =
q
a
2
τ
+ a
2
n
=
q
(6RBt)
2
+ (A + 3Bt
2
)
4
R
2
.
t = 3
a =
q
(6 · 0, 5 · 2 · 3)
2
+ (2 + 3 · 0, 2 · 3
2
)
4
· (0, 5)
2
= 32, 8 /
2
.
R = 8 ; a
n
= 4 /
2
;
α = 60
◦
.
v, a
τ
&%
'$
a
n
A
A
AK
~a
+
R
6
a
τ
α
a
n
= ω
2
R =
v
2
R
v =
q
a
n
R =
√
4 · 8 = 5, 66 .
a =
a
n
cos α
a
τ
=
q
a
2
− a
2
n
=
s
a
2
n
cos
2
α
− a
2
n
= a
n
s
1 − cos
2
α
cos
2
α
= a
n
tg(α);
a
τ
= 4 · tg(60
◦
) = 6, 93
2
.
x = At + Bt
3
;
A = 6 / ; B = −0, 125 /
3
;
t
1
= 2
; t
2
= 6 .
< v >
v =
dx
dt
= (At + Bt
3
)
0
= A + 3Bt
2
= 6 − 0, 375t
2
;
t
0
t
0
6 − 0, 375t
2
0
= 0; t
0
=
s
6
0, 375
= 4
.
t
1
t
0
s
1
= (At
0
+ Bt
3
1
) − (At
0
+ Bt
3
1
) = (6 · 4 − 0, 125 · 4
3
) − (6 · 2 − 0, 125 · 2
3
) = 4, 5 .
t
0
t
2
s
2
= (At
0
+ Bt
3
0
) − (At
2
+ Bt
3
2
) = (6 · 4 − 0, 125 · 4
3
) − (6 · 6 − 0, 125 · 6
3
) = 7 .
< v >=
s
1
+ s
2
t
2
− t
1
=
4, 5 + 7
6 − 2
= 2, 88
/ .
x = At + Bt
3
;
A = 3 / ; B = 0, 06 /
3
;
t
1
= 0; t
2
= 3
.
v
x
, a
x
, < v
x
>, < a
x
>
v
x
=
dx
dt
= (At + Bt
3
)
0
= A + 3Bt
2
= 3 + 0, 18t
2
.
a
x
=
dv
x
dt
= (A + 3Bt
2
)
0
= 6Bt = 0, 36t.
t
1
= 0
v
x1
= 3 + 0, 18 · (0)
2
= 3 / ; a
x1
= 0, 36 · 0 = 0.
t
2
= 3
v
x2
= 3 + 0, 18 · (3)
2
= 4, 62 / ; a
x2
= 0, 36 · 3 = 1, 08 /
2
.
t
1
t
2
s = x
2
− x
1
= (At
2
+ Bt
3
2
) − (At
1
+ Bt
3
1
) = (3 · 3 + 0, 06 · 3
3
) − 0 = 10, 62
.
< v
x
>=
s
t
2
− t
1
=
10, 62
3 − 0
= 3, 54
/ .
< a
x
>=
v
x2
− v
x1
t
2
− t
1
=
4, 62 − 3
3 − 0
= 0, 54
/
2
.
R = 0, 2; ϕ = A + Bt + Ct
3
;
A = 3
; B = −1 ;
t = 10 ; C = 0, 1
3
.
a
τ
, a
n
, a
ω =
dϕ
dt
= (A + Bt + Ct
3
)
0
= B + 3Ct
2
;
ε =
ω
dt
= (B + 3Ct
2
)
0
= 6Ct;
a
n
= ω
2
R = (B + 3Ct
2
)
2
R
a
τ
= εR = 6CtR
t = 10
a
n
= (−1 + 3 · 0, 1 · 10
2
)
2
· 0, 2 = 168, 2 /c
2
; a
τ
= 6 · 0, 1 · 10 · 0, 2 = 1, 2 /c
2
.
a =
q
a
2
n
+ a
2
τ
=
q
(162, 2)
2
+ (1, 2)
2
≈ 168, 2 /c
2
.
h = 2 ; h
1
= 0, 5 ;
m = 200 .
p
g
?
6
?
h
g
6
v
1
h
v =
√
2gh,
v
1
=
√
2gh
1
.
mv
mv
1
p = mv + mv
1
= m(v + v
1
)
p = m(
√
2gh +
√
2gh
1
) = m
√
2g · (
√
h +
√
h
1
)
p = 0, 2 ·
√
2 · 9, 8 · (
√
2 +
√
0, 5) = 1, 88
·
.
m = 8 ; v = 250 /
m
1
= 6 ; u
1
= 400 /
u
2
-
v
m
-
u
2
m
2
-
u
1
m
1
mv = m
1
u
1
+ m
2
u
2
m
2
= m − m
1
= 8 − 6 = 2 .
u
2
=
mv − m
1
u
1
m
2
=
8 · 250 − 6 · 400
2
= −200
/c.
u
2
v
1
= 3 / ; m
1
= 210 ;
m
2
= 70 ; u
1
= 4 / .
u
21
j j
m
1
-
v
1
m
2
j j
m
1
-
u
1
B
B
C
C
u
2
(m
1
+ m
2
)v
1
u
1
u
2
m
1
u
1
− m
2
u
2
(m
1
+ m
2
)v
1
= m
1
u
1
− m
2
u
2
u
2
=
m
1
m
2
· u
1
−
m
1
m
2
+ 1
v
1
u
21
= u
1
+ u
2
= u
1
+
m
1
m
2
· u
1
−
m
1
m
2
+ 1
v
1
=
m
1
m
2
+ 1
(u
1
− v
1
)
u
21
=
210
70
+ 1
(4 − 3) = 4
/
m
1
= 60 ; u
1
= 480 / ;
α = 30
◦
; m
2
= 18
;
u
2
w
3
m
1
, u
1
m
2
u
2
α
m
1
u
1
cos α − m
2
u
2
= 0
u
2
=
m
1
u
1
cos α
m
2
=
60 · 480 · cos(30
◦
)
18 · 10
3
= 1, 4
/ .
m = 300 ; v = 8 / .
α = 60
◦
.
∆p
6
-
x
>
@
@
@I
l
~v
~v
0
α
v
x
= v sin α
v
0
x
= −v sin α.
∆p
x
= m(v
0
x
− v
x
) = −2mv sin α
∆p = −∆p
x
= 2mv sin α;
∆p = 2 · 0, 3 · 8 · sin 60
◦
= 4, 16 · / .
m
1
= 60 ; m
2
= 20 /
v = 2 / .
u
2
j
-
~u
1
~u
2
u
1
u
2
~u
1
~u
2
v = u
1
+ u
2
;
m
1
u
1
− m
2
u
2
= 0
u
1
= v − u
2
; m
1
(v − u
2
) − m
2
u
2
= 0;
u
2
=
m
1
m
1
+ m
2
· v =
60
60 + 20
· 1 = 0, 75
/
v = 400 / ;
m
1
= 0, 4m; u
1
= 150 / .
u
2
-
~v
m
u
1
m
1
-
u
2
m
2
m
2
u
2
− m
1
u
1
= mv
u
1
, u
2
m
1
m
2
m
m
1
= 0, 4m; m
2
= m − 0, 4m = 0, 6m
0, 6mu
2
− 0, 4mu
1
= mv; 0, 6u
2
− 0, 4u
1
= v;
u
2
=
v + 0, 4u
1
0, 6
=
400 + 0, 4 · 150
0, 6
= 767
/ .
l = 1, 8 ; m
1
= 8 ;
m
2
= 4
; α = 3
◦
v
u-
m
2
~v
C
C
C
C
C
C
C
?
6
h
α
m
1
l l
u
m
2
v = (m
1
+ m
2
)u; u =
m
2
m
1
+ m
2
v;
m
1
+ m
2
u =
m
2
m
1
v;
h
m
1
u
2
2
= m
1
gh; u
2
= 2gh;
h α
h = l(1 − cos α);
u
2
= 2gl(1 − cos α);
m
2
m
1
v
2
= 2gl(1 − cos α);
v =
m
1
m
2
q
2gl(1 − cos α);
v =
8
0, 004
q
2 · 9, 8 · 1, 8(1 − cos 3
◦
) = 440
/
m
1
= 300 ; m
2
= 80 ;
η
m
2
?
m
1
v u
m
2
u = (m
1
+ m
2
)v
T
1
=
m
2
u
2
2
T
2
=
(m
1
+ m
2
)v
2
2
;
v =
m
2
u
m
1
+ m
2
;
T
2
=
(m
1
+ m
2
)
2
·
(m
2
u)
2
(m
1
+ m
2
)
2
=
m
2
2
u
2
2(m
1
+ m
2
)
.
T
1
− T
2
η =
T
1
− T
2
T
1
= 1 −
T
2
T
1
=
= 1 −
m
2
2
u
2
2(m
1
+ m
2
)
·
2
m
2
u
2
= 1 −
m
2
m
1
+ m
2
=
m
1
m
1
+ m
2
η =
300
300 + 8
= 0, 974.
m
1
= 1 ; v
1
= 4 / ;
m
2
= 2 ; v
2
= 3 / .
u
1
, u
2
-
m
1
v
1
v
2
m
2
-
u
1
u
2
m
1
v
1
− m
2
v
2
= −m
1
u
1
+ m
2
u
2
(1)
m
1
v
2
1
2
+
m
2
v
2
2
2
=
m
1
u
2
1
2
+
m
2
u
2
2
2
(2)
u
2
u
2
=
m
1
m
2
(v
1
+ u
1
) − v
2