1
? 3c ?￥
o??
p
?3.1 ?
? ?
(theorem of momemtum)
·1? ? B
HWV??￥ ?rT
§
Hr?
?r?
1.±s?
T
t
p
F
d
d
v
v
=
ptF
v
v
dd =
d?
p￥±s?
T
B aé??
? ?
2
?
HW =￥ ?
1
21
tt →
2.s?
T
ptF
v
v
dd =
∫∫
=
2
1
2
1
dd
p
p
t
t
ptF
v
v
v
v
12
pp
vv
?=
∫
=
2
1
d
t
t
tFI
vv
?l
(impulse)
12
ppI
vv
v
?=∴
é??
? ?
3
(1) ￥Z_
^±
O
?￥Z_ b
tF d
v
I
v
tF d
v
I
v
(2) ü (
?
12
2
1
d
tt
tF
F
t
t
?
=
∫
v
v
12
12
tt
pp
?
?
=
vv
F
1
t
2
t
t
F
4
∫
=
2
1
d
t
t
tFI
vv
12
ppI
vv
v
?=
(3)s
?
T
m?M
H
∫
=
2
1
d)(
t
t
xx
ttFI
xx
mvmv
12
?=
∫
=
2
1
d)(
t
t
yy
ttFI
yy
mvmv
12
?=
∫
=
2
1
d)(
t
t
zz
ttFI
zz
mvmv
12
?=
5
?
vm
v
v
v
a
v
r
v
V?
∫
=
2
1
d
t
t
tFI
vv
3.
?
DV?
V?
?
?
￥1"
12
2
1
d vmvmtFI
t
t
vv
vv
?==
∫
?
? ?o
a¨?8?"
6
y
?
m
α
α
x
o
1
v
v
2
v
v
[ è 1]X?l o m[
? v
1
· : ·a
?1 v
2
(v
l?? v
1
) b p :
?
s￥
b
3 ¨s
E
m
)cos(cos
12
αα mvmvI
x
??=
)sin(sin
12
αα mvmvI
y
???=
αcos2
1
mvI
x
=
0=
y
I
7
y
?
m
α
α
x
o
1
v
v
2
v
v
,cos2
1
αmvI
x
=
0=
y
I
imvI
vv
αcos2
1
=
o
:
s￥
imvII
vvv
αcos2
1
?=?=
o :
8
I??ú
H
?
1
v
v
2
v
v
1
p
v
2
p
v
p
v
?
H?
F
v
?H
F
v
é
F
v
?
F
v
C
?
F
v
E
F
v
9
= aé?" ?
? ?
t
p
fF
d
d
2
212
v
vv
=+
t
p
t
p
ffFF
d
d
d
d
21
211221
vv
vvvv
+=+++
1221
ff
vv
Q ?=
?é?
1
m
2
m
21
f
v
12
f
v
1
F
v
2
F
v
t
p
fF
d
d
1
121
v
vv
=+
10
t
p
t
p
FF
d
d
d
d
21
21
vv
vv
+=+∴
)(
d
d
21
pp
t
vv
+=
n?é?
0=∑∑∴
ij
ji
f
v
y = ?9
^?C
¤
ii
p
t
F
v
v
Σ=Σ
d
d
t
p
F
d
d
v
v
=
?? ?
9?
12
ppI
vv
v
?=∴
11
[ è 2]B
ìú[
?
q v =3m/sV ì?/Y
V b
?
eò
? ?úN￥ ì1 ?m=500Kg
?T
PúN￥
?
q ??M?¨ v￥
? ?$ (úDE?￥ ??9 )
v
v
F
v
md
m
x
3
mmm
ttt
d
d
+?
+?
12
￡ üZ_
mvmmv =?+ 0d
t
H Y
vmm )d( +
t+dt
H Y
mvvmmp ?+= )d(d vm?=d9
? ?
? ?
ptF dd =
vm?= d
v
t
m
F
d
d
=∴ N105.1
3
×=
13
v
v
F
v
md
m
x
63
￡ üZ_?
¨ d?
p
)(
d
d
mv
t
F =
t
v
mv
t
m
d
d
d
d
+=
0=
3500×=
N105.1
3
×=
14
[ è 3]Bé?
s?? ?T¨? ?1
(SI)3)2(210
2
ktjtitF
vvvv
+?+=
pNé?Vá? 7
S 2s =
?
s?? ?
￥
?é? 2s ￥?
b
3 ?
?lμ
∫
=
?
2
0
20
d tFI
vv
∫
+?+=
2
0
2
d]3)2(210[ tktjtit
vvv
sN8420 ?++= kji
vvv
15
0220
ppI
vv
v
?=
?
2
p
v
=
sN8420
20
?++=
?
kjiI
vvvv
??
? ?
sN8420
202
?++==∴
?
kjiIp
vvvv
v
2
p
v
222
8420 ++=
sN9.21 ?=
°=== 24
9.21
20
arccosarccos
I
I
x
α
°= 5.79β °= 6.68γ
16
[ è 4]é
m=1kg￥l oT?? R=2m￥?
???Z?1 (1 ?U
S ), pl oV ?
?
s? ?
￥
b
2
)21( tS π=
s2
1
=t
s2
2
=t
3
[ 011 ?USe?
st 2
1
=
mS π=
1
st 2
2
=
mS π2
2
=
?é RL π2= π4=
A
B
O
17
t
s
v
d
d
=
tπ=
sm2π=∴
A
v
sm2π=
B
v
2
2
1
tS π=
s2s,2
21
== tt
O
A
B
A
v
v
B
v
v
AA
mvp =
smkg2 ?= π
BB
mvp =
smkg2 ?= π
18
O
A
B
A
v
v
B
v
v
ABtt
vmvmI
vv
v
?=∴
?
21
22
21
BAtt
mvmvI +=∴
?
v
smkg6 ?= π
A
vm
v
B
vm
v
21
tt
I
?
v
θ
A
B
mv
mv
=θtan
'4454
o
=θ
2
2
=
19
? a ?
o??
p
(Law of conservation of momentum)
t
p
F
i
i
d
d
v
v
Σ
=Σ
0
d
d
=Σ
i
p
t
v
0=Σ
i
F
v
=Σ=Σ∴
iii
vmp
vv
è
O
20
iii
vmp
vv
Σ=Σ
=è
O
:0=Σ
i
F
v
s
?
T
0=Σ
ix
F
H
ixix
vmp Σ= =è
0=Σ
iy
F
H
iyiy
vmp Σ=
iziz
vmp Σ=
=è
=è
0=Σ
iz
F
H
21
[ è 5]é
1 m￥ |_ Bé
1 M a
é1 l ￥lúB
?á?<_ú￥
6B
p |?lúòM?
? ?(?9 ? )
m
M
v
v
V
v
X
}
x
48476
3
￡ üZ_"d
?
o?
0=+ vmVM
v
v
v
M
m
V
v
v
?=
22
m
M
v
v
V
v
X
}
x
48476
v
M
m
V
v
v
?=
|M?ú￥
?
Vvv
v
vv
?='
v
M
mM
v
+
=
! |
HW t =<?
6B
∫
=
t
tvl
0
d'
∫
+
=
t
tv
M
mM
0
d x
M
mM+
=
l
mM
M
x
+
=∴
xlX ?=
l
mM
m
+
=
23
?3.2 pf?e ?
“
*ú ”|f/
6 b
·Mé
ù5
?
? (v << c)
T? (é
@F )
?
(é
h
)
ú
? (v ～ c)
é
?
?M?
m = m(v)
24
B ap￥
?
y
u
v
v
v
·"d p T8 +
i ?
·Hq ?
s ? ? aE ? (e? )
·98V? i (? ) ?f ( ?
D )
· AB±V?
ttt d+→
? p T8 +
i ?
é
M
? v (p1 )
"d?
Mv
25
? ¨ ?
y
u
v
v
v
é
dm = -dM
? v - u
)(
p1 ?
p ?
1
vvv
vvv
+=
p T8 +
i ?
′
é
M-dm=M + dM
? v + dv (p1 )
"d?
-dM(v - u) + ( M + dM)( v + dv)
26
-dM(v - u) + ( M + dM)( v + dv)
??
o?
)d)(d()(d vvMMuvMMv +++??=
{ ? vM dd ?
0dd =+ vMMu
M
M
uv
d
d ?=
∫∫
?=
f
i
f
i
M
M
v
v
M
M
uv
d
d
f
i
if
M
M
uvv ln+=∴
27
f
i
if
M
M
uvv ln+=
4ú v
f
￥o?
(1)4ú u;
(2)9v N( = M
i
/M
f
)
= ap
?
s￥w ?
?
·ù?` ¨ 8 dm
t
H Y
? v (??8￥
?M] )
?
vdm
t +dt
H Y
? v - u
?
dm(v - u)
28
??
? ? dt = ¨ 8
?
s
mvuvmtF d)(dd ??=
t
m
uF
d
d
?=
?
?
?
q
? u ]_
·p
?
s￥w ?
t
m
uFF
d
d
=?=
p
? u Q_
kg/s1038.1
d
d
4
×=
t
m
m/s1094.2
3
×=u
(N)1006.4
7
×=? F
29
? a )p
·?)p
km/s11
0
15
km/s1.4
=?
?
?
?
?
?
=
=
=
f
i
v
v
N
u
I
n1 o? ?? b E ?
L=
?oμ
km/s7
km/s9.7<
(?Bó?
? )
· )p (B?1 ?) )
332211
lnlnln NuNuNuv
f
++=
30
?3.3 é? é??? ?
B aé?
(center of mass)
m
i
?
i
r
v
m
2
?
2
r
v
?
m
1
1
r
v
? C
y
c
x
c
c
r
v
·é?"
M
rm
m
rm
r
ii
i
ii
c
vv
v
Σ
=
Σ
Σ
=
é?ê
O F ? ü (
é?
^+?? b
y
x
o
z
31
M
rm
m
rm
r
ii
i
ii
c
vv
v
Σ
=
Σ
Σ
=
·é
??s?
mr
M
r
M
c
∫
= d
1
vv
y
x
o
z
md
r
v
M
xm
x
ii
c
Σ
=
M
mx
x
M
c
∫
=
d
s
T
,L=
c
y L=
c
z
( ?
? V¤ )
32
[ è 6] p￡B é?￥é?ê? C ?
￥?? b
3
!?é1 l é
1 m ?êé
é
1
λ
xm dd λ=
x
l
m
d=
∫
=∴
m
C
mx
m
x d
1
xx
l
m
m
l
l
d
1
2
2
∫
?
=
0=
x0
2l? 2l
md
xd
x
33
[ è 7]B??? ( M
′
??1 R p é?ê? b
R
x
y
0
θd
θ
ld
lm dd λ=
3 é? y à
m
my
y
m
C
∫
=
d
m
lyλ
l
∫
=
d
θsinRy=Q
θdd Rl =
Rm λπ=
π
θθ
π
∫
=∴
0
dsinR
y
C
π
R2
=
34
[ è 8]B??1 R￥ é? ? 7
B??1 r ￥? d
???
0 a0’M 1 d p é? b
R
x
0
r
'0
3 é
m
1
￥í dv ? +
é
-m
2
￥l ?
!é
?
á1
σ
2
1
Rm σπ=
é?US
0
1
=x
é
v ?
{
35
22
22
)(0)(
rR
drR
σπσπ
σπσπ
?
×?+×
=
22
2
rR
dr
?
?=
2
2
rm σπ?=
é?US
dx =
2
é
l ?
{
M
xm
x
ii
c
Σ
=
R
x
0
r
'0
36
M
rm
r
ii
c
v
v
Σ
=
= aé??? ?
1.é?￥
??F
?
M
t
r
m
i
i
d
d
v
Σ
=
M
vm
ii
v
Σ
=
(1)
?
t
r
v
c
d
d
v
v
=
(2)F
?
t
v
a
c
c
d
d
v
v
=
M
am
ii
v
Σ
=
37
pvmvM
iic
vvv
=Σ=
3.é??? ?
2.é?￥?
M
vm
v
ii
c
v
v
Σ
=
M
am
a
ii
c
v
v
Σ
=
iic
amaM
vv
Σ=
)(
ii
fF
vv
+Σ=
??=Σ
i
f
v
t
p
aMF
ci
d
d
v
v
v
==∑ é??? ?
é?F
? %??é?"
?
s￥?? ?
38
4.?? ?1
,
Hé?￥?
t
p
aMF
ci
d
d
v
v
v
==∑
0=∑
i
F
v
è
O
=
c
v
v
·il ?é?"￥?? ?1
,
H
é?á?T
?°L?
39
[ è 9]
￡ ü
?
μBf??
bB
( o o￥é
1 M=0.3kg b|?_
· ?
Hμ f =0.1N￥ ? ?T¨ o
b p? o o?F
??Vá? 7
S￥ 2
e = o?M
?M?￥ ? b
f
v
C
C
a
v
C
Maf =
3
?é??? ?
M
f
a
C
=∴
2
sm2.0=
40
7
S 2
e = o?
?￥ ?1
2
2
1
tas
CC
=
2
22.0
2
1
××=
m4.0=
f
v
C
C
a
v
41
?3.4 é?￥??
(?
)? ?
B aé?￥??
é? m
H Y￥?
1
vmp
vv
=
?
H Y ?? o￥
O?1 r
v
prL
vv
v
×=
vmr
vv
×=
o
r
v
m
?
p
v
·
L
v
?
5N
H Y é? m%?? o￥??
1
42
o
m
r
v
?
p
v
·
L
v
?
vmrL
vv
v
×=
?sinrmvLL ==
v
vl
Z_
￥·
m
?è -éZ_
pr
vv
→
?ê /smkg
2
? sJ?
·é?T
?
q??
???
￥v
l aZ_ (?M b
?
L
v
p
v
r
v
m
mrvL=
43
r
v
'r
v
v
v
m
0
'0
θ
L
v
'L
v
vmrL
vv
v
×=
Z_?M
0
vmrL
vv
v
×
′
=
0’
Z_??M
44
= aé?￥??
? ?
tpF /dd
v
v
=
)(
d
d
d
d
pr
tt
L
vv
v
×=
t
p
rp
t
r
d
d
d
d
v
vv
v
×+×=
Frpv
v
vvv
×+×=
Fr
v
v
×=
0=
FrM
v
v
v
×=
? ? ? à Q (moment of force)
---- ? I?￥ ?
F
v
45
o
r
v
m
?
F
v
·
M
v
α
⊥
r
MFr
t
L
vv
v
v
=×=
d
d
FrrFM
⊥
== αsin
?
^??
?
?￥ ±s?
T
t
L
M
d
d
v
v
=
é? ??￥??
￥
HWM?
q
??é?
?
s?? ???￥ ?
46
· ? ? T¨B
HWV? 5μ
t
L
M
d
d
v
v
=
12
2
1
2
1
dd LLLtM
L
L
t
t
vvvv
v
v
?==
∫∫
( ? ￥ ?T¨rT )
??
? ?￥s?
T B
HWV
??
? ?￥s?
T B
HWV
??é?
?
s￥
??é??
??é?
?
s￥
??é??
?
￥9
b
47
? aé?￥??
o??
p
t
L
M
d
d
v
v
=
0=M
v
0
d
d
=
t
L
v
è
O
=L
v
(1)??
o??
p
^t ?D￥'?
p-
B b??
a¨??48"9
a¨?±
4"d b
(2) μ? ?T¨/é?￥??
o? b
μ? ? é?
s ?
S?·_ ( ? 7 )
B??? ( ?? ) b
48
f
v
r
w
·μ? ? ??￥ ?
1
, ? é? ??￥
??
B?
o? (?é
?
s ??1
,?
i
?
o? ) b
· ??? ?
t ?
t ??
o?
?? ?1 o …… … 1? ……………
è0 ?? …… … ? ………………
49
1 aé?"￥??
? ?
i
LL
vv
Q Σ=
ii
pr
vv
×Σ=
=? ii
i
FF
t
p
vv
v
+=
d
d
t
L
t
L
i
d
d
d
d
vv
Σ=∴
t
pr
ii
d
)d(
vv
×
Σ=
t
p
rp
t
r
i
ii
i
d
d
d
d
v
vv
v
×Σ+×Σ=
)(
=? iii
FFr
vv
v
+×Σ=
? =
MM
vv
+=
0=
=
M
v
Q
?
M
t
L
v
v
=∴
d
d
50
1
v
2
v
1
m
2
m
R
o
[ è 10]
?]"×￥l
Yò" tVá
}
70
￥
B?Y0¨ ?
_
?
6B?5t
70?? b?9á
}é
?à
? '?l
Y5?rá
}$ ?
l
Y×
??
H f ??
??$
51
1
v
2
v
1
m
2
m
R
o
3 : ,"d O
???
o?
mmm ==
21
0
21
=? RmvRmv
21
vv =∴
]
H?rá
}
21
mm ≠
gRmgRmM
12
?=
?
! 7
S
H 0=L
52
1
v
2
v
1
m
2
m
R
o
gRmgRmM
12
?=
?
RvmRvmL
2211
?=
???
? ?
gRmm
t
L
M )(
d
d
12
?==
?
21
mm > 0
d
d
<
t
L
0<L
2211
vmvm <
21
vv <
53
21
mm < 0
d
d
>
t
L
0>L
2211
vmvm >
21
vv >
9
^8 ^￥lY5?rá
}
1
v
2
v
1
m
2
m
R
o
21
mm >
21
vv <
gRmm
t
L
M )(
d
d
12
?==
?
54
?3-5 ???
q
???
q (Work and Energy)
? bW￥ ?r? ---?
?
? ? a
] ? a? ?? ? a? ?
o??
p?
B
B
a?
?
(work)
rFA
r
v
dd ?=
rF
v
dcosθ=
|d| rF
r
v
=
í?
?1S
àμZ_?μ?μ
F
v
r
F
o
r
v
d
θ
55
F
v
r
v
d
2
r
v
1
r
v
0
a
b
rFA
v
v
dd ?=
∫
=
b
a
AA d
?
?
a M??
M??
b
∫
?=
b
a
rF
v
v
d
)ddd( zFyFxF
zy
b
a
x
++=
∫
kFjFiFF
zyx
vvvv
++=
kzjyixr
vvv
v
dddd ++=
56
=
=
aB ?￥?
1
d r
v
2
d r
v
'd r
v
1
m
2
m
1
r
v
2
r
v
12
f
v
21
f
v
1
d r
v
2
d r
v
x
y
z
O
2112
ff
vv
?=
12
dd'd rrr
vvv
?=
m
2
M?
M?
m
1
￥êM
￥êM
1121
dd rfA
v
v
?=
2212
dd rfA
v
v
?=
57
12
dd'd rrr
vvv
?=
,dd
1121
rfA
v
v
?=
2212
dd rfA
v
v
?=
,
2112
ff
vv
?=
T¨ ??QT¨ ?￥í?-?
T¨ ??QT¨ ?￥í?-?
)d(d
1221
rrf
vv
v
??=
21
ddd AAA +=
'd
21
rf
v
v
?=
? ?￥9?
? ?￥9?
(dA)oDMoT¨ ?
oDMoT¨ ?
#MêMμ1
#MêMμ1
----D? I"￥ê4í1
D? I"￥ê4í1
58
·il
é?W￥B ?T?-???B
?é?
s￥ ??é?M?
6B
é?M?￥
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a1B?é?á?ê 1USe
?o9
?N"? ?
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￥?' V b
pB ?T?ù5 → pB? ?T?ù5
59
? a?
q (power)
t
rF
d
d
v
v
?
=
vF
v
v
?=
t
A
P
d
d
=
60
[ è 1]B |V
B |V
10m
'￥
￡Yü
'￥
￡Yü
10Kg￥
￥
￡
?4
???a
P
￡
?
6ú
1m
P 0.2Kg ùü
￡VY￥
￡
?4?
ùü
￡VY￥
￡
?4?
Y g3S
? $ (?9a￥é
)
3
3
xm 2.010?=
mgF =
∫
=
10
0
d xFA
∫
?=
10
0
d)2.010( xgx
J88=
x
x
F
o
61
[ è 2]é? ?
é? ?
￥T¨
￥T¨
/VUSe???
/VUSe???
P(2,3)? p
? p
?￥? b
jixF
vvv
23 +=
)dd( yFxF
y
b
a
x
+=
∫
∫∫
+=
3
0
2
0
d2d3 yxx
J12=
∫
?=
b
a
rFA
v
v
d
3
3
62
?3-6 ? ?? ?
] ?
(kinetic energy theorem)
B a? ?? ?
sF dcos?=
sF
t
d=
sma
t
d=
s
t
v
m d
d
d
=
vmv d=
rFA
v
v
dd ?=
a
v
t
a
v
n
a
v
a
b
F
v
r
v
d
?
1.é?
é?
63
2
2
1
mvE
k
=?l ?l
----
? ?
22
2
1
2
1
ab
mvmv ?=
∫
=
b
a
v
v
vmvA d
vmvA dd =
kakb
EEA ?=∴
----é?? ?? ?
64
[ è 3]é
1
é
1
1Kg￥
￥
t8
t8
xà
à
Z
Z
_￥M ?T¨/V x = 0)?á? 7
)?á? 7
S? b
!M ?
S? b
!M ?
FD
D
x ￥1" ?m
U
￥1" ?m
U
pt8
pt8
x=4m )￥
?
q b
∫
= xFA d
(N)F
0
2
2
4
(m)x
R
3
3
(J)4 π+=
kakb
EEA ?=
kb
E=
m
v
)4(2 π+
=∴
π+= 4
2
1
2
mv
65
2.é?"
é?"
1
a
1
b
1
m
2
a
2
b
2
m
12
f
v
21
f
v
1
F
v
2
F
v
∫
?+=
1
1
11211
d)(
b
a
rfFA
v
vv
11 kakb
EE ?=
m
1
m
2
∫
?+=
2
2
21222
d)(
b
a
rfFA
v
vv
22 kakb
EE ?=
66
TMF
TMF
)()(
2121 kakakbkb
EEEE +?+=
? ?? A
?
∫∫
?+?
2
2
1
1
2211
dd
b
a
b
a
rFrF
v
v
v
v
∫∫
?+?+
2
2
1
1
221112
dd
b
a
b
a
rfrf
v
v
v
v
∫
?+=
1
1
11211
d)(
b
a
rfFA
v
vv
11 kakb
EE ?=
∫
?+=
2
2
21222
d)(
b
a
rfFA
v
vv
22 kakb
EE ?=
= ?? A
=
)()(
2121 kakakbkb
EEEEAA +?+=+∴
=?
kakb
EE ?=
67
w<?
w<?
n?é?￥é?"
kakb
EEAA ?=+
=?
----é?"
é?"
? ?? ?
?é?￥é?"
= ? ??M
??M
"d￥
"d￥
9?
?
?
?M
?M
"d￥
"d￥
9? ?
68
[ è 4]%
7B
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1
%
7B
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1
m ￥l o
￥l o
,
6B
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Pl o
5
Pl o
?
[
?
?
[
?
v
1
??1
??1
r
1
￥??
?
￥??
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? ?adè?
¤1|
7_/ ?
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¤1|
7_/ ?
P??
hl?
hl?
r
2
b
!l oD
?￥ ??9 p
!l oD
?￥ ??9 p
N
Hl o￥
?? ? ?
N
Hl o￥
?? ? ?
T l o
?T￥? b
O
1
r
2
r
69
3
3
0 ?
?
??
o?
??
o?
2211
vmrvmr
vvvv
×=×
-
{l o
7Z_￥
?
-
{l o
7Z_￥
?
2211
mvrmvr =∴
2
1
12
r
r
vv =
2
1
2
2
2
1
2
1
mvmvA ?=
]1)[(
2
1
2
2
1
2
1
?=
r
r
mv
O
1
r
2
r
70
= a
] ?
o ?
o ?
(conservative force)
? ?t8
?S￥?oD"d￥
? ?t8
?S￥?oD"d￥
S ê?μ17D?￥
^üí
S ê?μ17D?￥
^üí
1?"￥ ??1
o ? b
1?"￥ ??1
o ? b
o ?
o ?
?B>?
^?
?B>?
^?
l
0d =?
∫
l
rF
v
v
71
2.
] ? (potential energy)
?
o ?￥?? ?
] ?
?
o ?￥?? ?
] ?
(1)× ?
] ?
gmP
v
v
=
× ?
x
y
O
a
b
r
v
d
P
v
θ
rPA
v
v
Q dd ?=
θcosd rmg
v
=
ymg d?=
jmgP
vv
?=
72
)(
ba
yymg ?=
?
?
0=
b
y
∫
?=∴
b
a
y
y
ab
ymgA d
5
5
∫
?=
0
0
d
a
y
a
ymgA
a
mgy=
?l
?l `
mgyE
p
=
apa
mgyE =∴
0a
A=
ymgA dd ?=
x
y
O
a
b
r
v
d
P
v
θ
73
(2)??
] ?
0
x
m
F
v
kxF ?=
?? ?
?? ?
rFA
v
v
Q dd ?= xkxd?=
a
x
a
b
x
b
∫
?=∴
b
a
x
x
ab
xkxA d
22
2
1
2
1
ba
kxkx ?=
?
?
ab
xx >
5
5
0<
ab
A
ab
xx <
?
?
5
5
0>
ab
A
74
0=
b
x
?
?
H
H
∫
?=
0
0
d
a
x
a
xkxA
2
2
1
a
kx=
2
2
1
apa
kxE =∴
0a
A=
?l
?l
2
2
1
kxE
p
= ----??
] ?
22
2
1
2
1
baab
kxkxA ?=
?T)?1 ?é
H1??
] ?
,?
?T)?1 ?é
H1??
] ?
,?
75
(3)? ?
] ?
0
2
r
r
mM
GF
v
v
?=
M
a
b
a
r
b
r
r
v
rr
vv
d+
m
α
rd
F
v
r
v
d
rFA
v
v
Q dd ?=
)cos(d
2
απ?= r
r
GmM
v
r
r
GmM
d
2
?=
rd?
76
`
∫
?=∴
b
a
r
r
ab
r
r
GmMA
2
d
)
11
(
ba
rr
GmM ??=
? ?
] ?
? ?
] ?
r
mM
GE
p
?=
?
?
∞→
b
r
∫
∞
∞
?=
a
r
a
r
r
GmMA
2
d
a
r
GmM
?=
r
r
GmM
A dd
2
?=
77
r
mM
GE
p
?=
a
pa
r
GmM
E ?=∴
∞
=
a
A
? ?
] ?
,? | í kù)
? ?
] ?
,? | í kù)
1
] ?oμMil
] ?-μ
] ?oμMil
] ?-μ
μ 'il b
(2)oμ
o ?? ?? ?
] ?￥à Q
oμ
o ?? ?? ?
] ?￥à Q
78
(3)
o ?￥???"d
] ?9
￥
o ?￥???"d
] ?9
￥
μ′
μ′
"d
] ?￥h
"d
] ?￥h
'
'
)
12 pp
EEA ??=
p
E??=
(4)"d ?Bê?
H￥
] ???
?
"d ?Bê?
H￥
] ???
?
V?ê?M?à
] ?
,?
H
o
V?ê?M?à
] ?
,?
H
o
?
?S￥?
?
?S￥?
∫
=
? I
,?
a
pa
rFE dcosθ
79
?3-7 ? ?e ? ? ?
o??
p
B a? ?e ? (work-energy theorem)
?é?"? ?? ?
?é?"? ?? ?
kakb
EEAA ?=+
=?
= ?
o = ? ad
o = ?
= ?
o = ? ad
o = ?
d = = =
AAA +=∴
p
EA ??=
=
Q
)(
papb
EE ??=
80
d = =? =?
AAAAA ++=+
)(
papb
EEAA ??+=
d =?
kakb
EE ?=
)()(
pakapbkb
EEEEAA +?+=+∴
d =?
? ?
? ?
? ?
? ?
:
pk
EEE +=
81
)()(
pakapbkb
EEEEAA +?+=+
d =?
pk
EEE +=
ab
EEAA ?=+∴
d =?
"d? ?Dd
o = ?￥?-?
"d? ?Dd
o = ?￥?-?
??"d? ?￥9
b
??"d? ?￥9
b
----"d￥? ?e ?
"d￥? ?e ?
82
= a? ?
o??
p
( law of conservation of mechanical energy)
oμ
o = ?S?
H
oμ
o = ?S?
H
0=+
d =?
AA
ab
EE =5
5
----"d￥? ?
o?
83
[ è 5]é
1
é
1
m￥é?" B
%?￥
70
￥é?" B
%?￥
70
Y?
￡ ü
?
T??1
Y?
￡ ü
?
T??1
R￥??? b
￥??? b
?
??B?
H?
?
?
??B?
H?
?
v
o
hl1
hl1
v
o
/2 b
p
p
:(1) ? ?T￥? ;(2)á? ?"
? ;(3)á
á
? -é??
?
? -é??
?
?
3
3
(1)? ? ?? ? ? ?￥?
? ? ?? ? ? ?￥?
2
0
2
2
1
2
1
mvmvA ?=
2
0
2
0
2
1
)
2
(
2
1
mv
v
m ?=
2
0
)83( mv?=
84
(2)y ? ?
mgf μ=
Z_D?Z_MQ
Z_D?Z_MQ
∫
=∴
R
sfA
π
θ
2
0
dcos
∫
?=
R
smg
π
μ
2
0
d
Rmg πμ 2??=
2
0
8
3
mv?=
V¤
Rg
v
π
μ
16
3
2
0
=
2
0
)83( mvA ?=
85
(3)
!é??
!é??
n ?
?
∫
?
∴
Rn
sf
π
θ
2
0
dcos
∫
?
?=
Rn
smg
π
μ
2
0
d
Rmgn πμ 2??=
?? ?? ?
?? ?? ?
2
0
2
1
02 mvRmgn ?=?? πμ
V¤
3
4
=n
( ?
?
)
86
[ è 6] ?mé
1
?mé
1
m￥ ! vD??"
?
1
1
k￥ ^?T · ! v|?Ta
ê
x
ü
b
! ! vD|
?-W￥ ?"
?1 0 ù
ù
7
S ·
H ! v
?
q v1 v
1 v
?
α
v
v
x
1
h
2
h
N
v
gm
v
f
v
k
f
v
3
3
! ·
H#
! ·
H#
a
êKv
H ! v
a
êKv
H ! v
úsY1
úsY1
h
1
ah
2
)()
1122 pkpk
EEEEA +?+=∴
d
87
α
v
v
x
1
h
2
h
N
v
gm
v
f
v
k
f
v
Nf
k
μ=
αμ cosmg=
xfA
k
?=
xmg ??= αμ cos
xmg ?? αμ cos
)
2
1
()
2
1
1
22
2
mghmvkxmgh +?+=
'
'
88
xmg ?? αμ cos
)
2
1
()
2
1
1
22
2
mghmvkxmgh +?+=
m
hhmgmgxkx
v
)(2cos2
12
2
??+
=
αμ
m
mgxmgxkx ααμ sin2cos2
2
?+
=
89
[ è 7]é
1
é
1
m
1
m
2
(m
2
>m
1
)￥
!e
A?
?
B ¨ ^?T ? B ? m
?
m
?
U bù :(1)à
3¨ v￥a ?
à
3¨ v￥a ?
FF?
F?
e? ? ? ?ü ?a z
P
e? ? ? ?ü ?a z
P
m
2
? 71
? ?(2) ?
?
m
1
m
2
?Dê?2
?Dê?2
T ??
T ??
? 1
m
2
m
90
1
x?
2
x?
3
3
!
@F
!
@F
Fa?T1eé
a?T1eé
ê
ê
, Fü ?a?T
ü ?a?T
%é
%é
P
P
m
2
4
4
1
x?
2
x?
1
m
2
m
F
gm
1
? |
|
m
1
)?
)?
K??ê?
K??ê?
1× ?
] ?￥
,?
1× ?
] ?￥
,?
"d? ?
o?
"d? ?
o?
2
1
)(
2
1
xk ?
)()(
2
1
211
2
2
xxgmxk ?+?+?=
91
k
gmF
x
1
1
+
=?∴
k
gm
x
2
2
=?
3¤
3¤
)(
21
mmF +=
11
xkgmF ?=+Q
22
xkgm ?=
2
1
)(
2
1
xk ? )()(
2
1
211
2
2
xxgmxk ?+?+?=
1
x?
2
x?
1
m
2
m
F
gm
1
? m
1
m
2
?Dê?2T?M b
?Dê?2T?M b
92
?3-8 · (collision)
? · · -a￥
??" o?
￥ ?L ( ·8 VT o8 )
m
1
m
2
· -
10
v
v
20
v
v
m
1
m
2
·a
1
v
v
2
v
v
x
93
B a?? · (elastic collision)
· -a9? ?
o?
! · x _
?
o?
?
o?
2211202101
vmvmvmvm +=+
? ?
o?
? ?
o?
2
22
2
11
2
202
2
101
2
1
2
1
2
1
2
1
vmvmvmvm +=+
94
21
2021021
1
2)(
mm
vmvmm
v
+
+?
=
21
1012012
2
2)(
mm
vmvmm
v
+
+?
=
(1) ?
?
m
1
= m
2
201
vv =
102
vv =
(2)?
?
m
2
>>m
1
O
O
v
20
=0
H
H
101
vv ?=
0
2
=v
(3)?
?
m
2
<<m
1
O
O
v
20
=0
H
H
101
vv =
102
2vv =
95
= .? ?d?? ·
(perfcct inelastic collision)
·a
o[]B
??
·a
o[]B
??
?
o?
?
o?
2211202101
vmvmvmvm +=+
21
202101
mm
vmvm
v
+
+
=
21
vv = v=
96
[ è 8]? l $￥e ?s￡
ü
? l $￥e ?s￡
ü
bθθ >
′
o
1
m
2
3
4
5
v
v
v
v
′
θ
θ
′
l
l
′
3 m -- |￥é?
|￥é?
1→2 |á
?/
P
μr?é
μr?é
om?
?
l ′M1
M1
l
2→3 /?V?
? ?
o?
)cos1(
2
1
2
θ?= mglmv
97
o
1
m
2
3
4
5
v
v
v
v
′
θ
θ
′
l
l
′
)cos1(
2
1
2
θ?= mglmv3→4 |á
?_
|á
?_
0=
?O
M
O???
o?
???
o?
mvllvm =
′′
4→5
?V?
?V?
? ?
o?
)cos1(
2
1
2
θ
′
?
′
=
′
lmgvm
98
o
1
m
2
3
4
5
v
v
v
v
′
θ
θ
′
l
l
′
)cos1(
2
1
2
θ?= mglmv
mvllvm =
′′
)cos1(
2
1
2
θ
′
?
′
=
′
lmgvm
1
cos1
cos1
3
3
>
′
=
?
′
?
l
l
θ
θ
θθ
′
< coscos
θθ
′
>
| ? ú ?
' ú ?$
