Chapter 9 Linear Momentum and Collisions
Chapter 9
Momentum and The Conservation of
Momentum
1,Linear Impulse and Momentum
2,Impulse-Momentum Theorem and
Conservation of Momentum
Chapter 9 Linear Momentum and Collisions
New Words
Impulse 冲量
Momentum 动量
Impulse and linear momentum theorem
Collisions 碰撞
Chapter 9 Linear Momentum and Collisions
9-1 Momentum and Its Relation to Force P200-201
1,momentum（动量）
Momentum of an object is defined as the product
of its mass and its velocity.
Momentum is presented by the symbol,P?
动量
v mp?
Chapter 9 Linear Momentum and Collisions
2 Newton second law of motion in terms of
momentum
m is a constant,so：
dt
PdF
n e t
amF
n e t?
For a particle
t
pF
d
d
am
t
m
t
pF
d
(d
d
d )v
Chapter 9 Linear Momentum and Collisions
9-2 Conservation of linear momentum（ 动量守恒
p202-205）
dt
PdF
n e t
In equation:
Let
0?n etF? P = constant vector
若质点系所受的 合外力为零则系统的总动量 守恒,即 保持 不变,
0exex
i
iFF
i
ipp
动量守恒定律
Chapter 9 Linear Momentum and Collisions
If no net external force acts on a system of
particles,the total linear momentum of the
system cannot change.
Discussion
1）守恒条件 合外力为零当 时，可 略去外力的作用,近似地认为系统动量守恒,例如在碰撞,打击,爆炸等问题中,
0exex
i
iFF
inex FF
Chapter 9 Linear Momentum and Collisions
If the component of the net external force on a closed
system is zero along an axis,then the component of the
linear momentum of the system along that axis cannot
change.
2） 若 某一 方向 合外力为零,则 此 方向动量 守恒,
zizizz
yiyiyy
xixixx
CmpF
CmpF
CmpF
v
v
v
,0
,0
,0
ex
ex
ex
Chapter 9 Linear Momentum and Collisions
9-3 collisions (碰撞 ) and Impulse (冲量 )
Collisions,in a collision,two bodies exert strong
forces on each other for a relatively short time,
1,Collision
This forces are internal to the two-body
system and are significantly larger than any
external force during the collision.
Chapter 9 Linear Momentum and Collisions
2 The impulse (冲量 )
力 的 累积 效应
EWrF
IpttF
,
,)(
对 积累对 积累
1212
2
1
d vv
mmpptFt
t
t
m
t
pF
d
(d
d
d )v )( ddd v mptF
冲量 力对时间的积分（ 矢量 ）
21 dtt tFJ
—– The impulse(冲量 )of force
fitt dtFJ
Chapter 9 Linear Momentum and Collisions
3 Momentum Theorem
研究力和运动的 时间积累过程 关系,
动量定理 在给定的时间内，外力作用在质点上的冲量，等于质点在此时间内动量的增量,
1212
2
1
d vv
mmpptFt
t
Net linear impulse delivered to a particle equals
the change in linear momentum of the particle
Chapter 9 Linear Momentum and Collisions
Descriptions:
(1) F should be the sum of all external forces,
When the force is constant,the direction of J is as
same as the force; if F is variable,the direction of
J is determined by the integral of,? dtF?
(2) Both and are vectors,They have same units
and dimensions,
J? P?
Chapter 9 Linear Momentum and Collisions
(3) In reality,one often use its component form,
kIjIiII zyx
分量形式
zz
t
t
zz
yy
t
t
yy
xx
t
t
xx
mmtFI
mmtFI
mmtFI
12
12
12
2
1
2
1
2
1
d
d
d
vv
vv
vv
Chapter 9 Linear Momentum and Collisions
t
t
F
J
F(t)
(4) Impulse is the area under the curve F(t) --t.
Because?t is very short,to simplify the problem,
the average force often be used.
t
P
t
dtF
t
J
F
f
i
t
t
a v g
tFJ a v g
t
t
F
F avg
J
Chapter 9 Linear Momentum and Collisions
A pitched 140g baseball,in horizontal flight with
a speed of 39m/s,is struck by a bat and the ball
leaves the bat with a speed of 45m/s,at an
upward angle of 30o (a) what impulse act on the
ball? (b) If the impact time is 1.2ms,what
average force act on the baseball.
)( ifif vvmppJSolution,(a)
)/(15.392.10 smkgjiJ
o
xy
yx
JJ
JJJ
16)/(t a n
)m / skg(4.11
1
22
o x
y
iv?
iv
fv?
v30o
EXAMPLE:
Chapter 9 Linear Momentum and Collisions
(b)
)(9 5 0 00 0 1 2.0 4.11 NtJFJtF a v ga v g
Discussion,Compare the Favg with the Fext.
o x
y
iv?
iv
fv?
v30o
Chapter 9 Linear Momentum and Collisions
4 Momentum Theorem of Particle System:
There are N particles in a system,The equation
of motion for particle i is,
质点系
1m
2m
12F
21
F?
1F
2F
)()(d)( 2021012211212
1
vvvv mmmmtFFt
t
20222212 d)(
2
1
vv mmtFFtt
10111121 d)(
2
1
vv mmtFFt
t
因为内力,故 0
2112 FF
Chapter 9 Linear Momentum and Collisions
质点系动量定理 作用于系统的合外力的冲量等于系统动量的增量,
n
i
iii
n
i
i
t
t
mmtF
1
0
1
ex2
1
d vv
0ppI
Chapter 9 Linear Momentum and Collisions
注意 内力不改变质点系的动量
gb m2?m000 bg vv
初始速度 则
00?p?
bg vv 2? 0?p
推开后速度 且方向相反 则推开前后系统动量不变
0pp
Chapter 9 Linear Momentum and Collisions
1v
m
2v
m
v?m?
12
12
12
2
1
d
tt
mm
tt
tF
F
t
t
vv
动量定理常应用于碰撞问题
F?
1t
F
mF
2t
F
t
o
越小，则 越大,
例如人从高处跳下、飞机与鸟相撞、打桩等碰撞事件中，作用时间很短，冲力很大,
注意
t? F
在 一定时p
Chapter 9 Linear Momentum and Collisions
例 1 一质量为 0.05kg、速率为 10m·s -1的刚球,以与钢板法线呈 45o角的方向撞击在钢板上,并以相同的速率和角度弹回来,设碰撞时间为 0.05s.求在此时间内钢板所受到的平均冲力,
1v
m
2v
m
x
y
解 建立如图坐标系,由动量定理得
c o s2 vm?
0s ins invv mαm
F
N1.14c o s2?
t
mFF
x
v方向沿 轴反向x
xxx mmtF 12 vv )c o s(c o s vv mm
yyy mmtF 12 vv
Chapter 9 Linear Momentum and Collisions
Conservation
of MomentumCpP N
i
i
1
,0?n etFIf? then
If no net external force acts on a particles’
system,total linear momentum of the system
cannot change.
No exceptions to this law have been discovered,
So it is considered to be more fundamental than
N-II law in both macroscopic and microscopic
physical worlds.
Chapter 9 Linear Momentum and Collisions
9-4&5&6&7 Momentum and kinetic energy in
collisions(P208-214)
CpFF
i
i
inex
碰撞 两物体互相接触时间极短而互作用力较大的相互作用,
CEEE 2k1kk
完全弹性碰撞 两物体碰撞之后,它们的动能之和不变,
非弹性碰撞 由于非保守力的作用,两物体碰撞后，使机械能转换为热能、声能，化学能等其他形式的能量,
完全非弹性碰撞 两物体碰撞后,以同一速度运动,
Chapter 9 Linear Momentum and Collisions
Elastic collision（ 弹性碰撞 ）,kinetic energy of
the system is conserved.
Inelastic collision （ 非弹性碰撞 ）,some kinetic
energy transferred to other forms of energy,
such as thermal energy or energy of sound,So
the kinetic energy of the system is not conserved.
Completely inelastic collision （ 完全非弹性碰撞 ）,the bodies always stick together and lose
kinetic energy,So the kinetic energy of the
system is not conserved.
Chapter 9 Linear Momentum and Collisions
完全弹性碰撞
（五个小球质量全同）
Chapter 9 Linear Momentum and Collisions
20v
例 设有两个质量分别为 和,速度分别为和 的弹性小球作对心碰撞,两球的速度方向相同,若碰撞是完全弹性的,求碰撞后的速度 和,P210,9-8
2m1m 10v
1v
2v
2211202101 vvvv mmmm
解 取速度方向为正向，由动量守恒定律得由机械能守恒定律得
2
22
2
11
2
202
2
101 2
1
2
1
2
1
2
1 vvvv mmmm
A
1m 2m
10v
20v
B
1v
2v
A B
碰前碰后
Chapter 9 Linear Momentum and Collisions
2
22
2
11
2
202
2
101 2
1
2
1
2
1
2
1 vvvv mmmm
)()( 220222212101 vvv-v mm
)()( 20221101 vvvv mm
2211202101 vvvv mmmm
解得
,
2)(
21
2021021
1 mm
mmm
vv
v
21
1012012
2
2)(
mm
mmm
vvv
A
1m 2m
10v
20v
B
1v
2v
A B
碰前碰后
Chapter 9 Linear Momentum and Collisions
（ 1）若
21 mm?
则
102201,vvvv
（ 2）若 且
0 20?v12 mm
则
0,2101 vvv
0 20?v12 mm
（ 3）若 且
102101 2,vvvv
则讨 论
21
2021021
1
2)(
mm
mmm
vv
v
21
1012012
2
2)(
mm
mmm
vv
v
A
1m 2m
10v
20v
B
1v
2v
A B
碰前碰后
Chapter 9 Linear Momentum and Collisions
Summary for Chapter Nine
See P223 and 224
Chapter 9
Momentum and The Conservation of
Momentum
1,Linear Impulse and Momentum
2,Impulse-Momentum Theorem and
Conservation of Momentum
Chapter 9 Linear Momentum and Collisions
New Words
Impulse 冲量
Momentum 动量
Impulse and linear momentum theorem
Collisions 碰撞
Chapter 9 Linear Momentum and Collisions
9-1 Momentum and Its Relation to Force P200-201
1,momentum（动量）
Momentum of an object is defined as the product
of its mass and its velocity.
Momentum is presented by the symbol,P?
动量
v mp?
Chapter 9 Linear Momentum and Collisions
2 Newton second law of motion in terms of
momentum
m is a constant,so：
dt
n e t
amF
n e t?
For a particle
t
pF
d
d
am
t
m
t
pF
d
(d
d
d )v
Chapter 9 Linear Momentum and Collisions
9-2 Conservation of linear momentum（ 动量守恒
p202-205）
dt
n e t
In equation:
Let
0?n etF? P = constant vector
若质点系所受的 合外力为零则系统的总动量 守恒,即 保持 不变,
0exex
i
iFF
i
ipp
动量守恒定律
Chapter 9 Linear Momentum and Collisions
If no net external force acts on a system of
particles,the total linear momentum of the
system cannot change.
Discussion
1）守恒条件 合外力为零当 时，可 略去外力的作用,近似地认为系统动量守恒,例如在碰撞,打击,爆炸等问题中,
0exex
i
iFF
inex FF
Chapter 9 Linear Momentum and Collisions
If the component of the net external force on a closed
system is zero along an axis,then the component of the
linear momentum of the system along that axis cannot
change.
2） 若 某一 方向 合外力为零,则 此 方向动量 守恒,
zizizz
yiyiyy
xixixx
CmpF
CmpF
CmpF
v
v
v
,0
,0
,0
ex
ex
ex
Chapter 9 Linear Momentum and Collisions
9-3 collisions (碰撞 ) and Impulse (冲量 )
Collisions,in a collision,two bodies exert strong
forces on each other for a relatively short time,
1,Collision
This forces are internal to the two-body
system and are significantly larger than any
external force during the collision.
Chapter 9 Linear Momentum and Collisions
2 The impulse (冲量 )
力 的 累积 效应
EWrF
IpttF
,
,)(
对 积累对 积累
1212
2
1
d vv
mmpptFt
t
t
m
t
pF
d
(d
d
d )v )( ddd v mptF
冲量 力对时间的积分（ 矢量 ）
21 dtt tFJ
—– The impulse(冲量 )of force
fitt dtFJ
Chapter 9 Linear Momentum and Collisions
3 Momentum Theorem
研究力和运动的 时间积累过程 关系,
动量定理 在给定的时间内，外力作用在质点上的冲量，等于质点在此时间内动量的增量,
1212
2
1
d vv
mmpptFt
t
Net linear impulse delivered to a particle equals
the change in linear momentum of the particle
Chapter 9 Linear Momentum and Collisions
Descriptions:
(1) F should be the sum of all external forces,
When the force is constant,the direction of J is as
same as the force; if F is variable,the direction of
J is determined by the integral of,? dtF?
(2) Both and are vectors,They have same units
and dimensions,
J? P?
Chapter 9 Linear Momentum and Collisions
(3) In reality,one often use its component form,
kIjIiII zyx
分量形式
zz
t
t
zz
yy
t
t
yy
xx
t
t
xx
mmtFI
mmtFI
mmtFI
12
12
12
2
1
2
1
2
1
d
d
d
vv
vv
vv
Chapter 9 Linear Momentum and Collisions
t
t
F
J
F(t)
(4) Impulse is the area under the curve F(t) --t.
Because?t is very short,to simplify the problem,
the average force often be used.
t
P
t
dtF
t
J
F
f
i
t
t
a v g
tFJ a v g
t
t
F
F avg
J
Chapter 9 Linear Momentum and Collisions
A pitched 140g baseball,in horizontal flight with
a speed of 39m/s,is struck by a bat and the ball
leaves the bat with a speed of 45m/s,at an
upward angle of 30o (a) what impulse act on the
ball? (b) If the impact time is 1.2ms,what
average force act on the baseball.
)( ifif vvmppJSolution,(a)
)/(15.392.10 smkgjiJ
o
xy
yx
JJ
JJJ
16)/(t a n
)m / skg(4.11
1
22
o x
y
iv?
iv
fv?
v30o
EXAMPLE:
Chapter 9 Linear Momentum and Collisions
(b)
)(9 5 0 00 0 1 2.0 4.11 NtJFJtF a v ga v g
Discussion,Compare the Favg with the Fext.
o x
y
iv?
iv
fv?
v30o
Chapter 9 Linear Momentum and Collisions
4 Momentum Theorem of Particle System:
There are N particles in a system,The equation
of motion for particle i is,
质点系
1m
2m
12F
21
F?
1F
2F
)()(d)( 2021012211212
1
vvvv mmmmtFFt
t
20222212 d)(
2
1
vv mmtFFtt
10111121 d)(
2
1
vv mmtFFt
t
因为内力,故 0
2112 FF
Chapter 9 Linear Momentum and Collisions
质点系动量定理 作用于系统的合外力的冲量等于系统动量的增量,
n
i
iii
n
i
i
t
t
mmtF
1
0
1
ex2
1
d vv
0ppI
Chapter 9 Linear Momentum and Collisions
注意 内力不改变质点系的动量
gb m2?m000 bg vv
初始速度 则
00?p?
bg vv 2? 0?p
推开后速度 且方向相反 则推开前后系统动量不变
0pp
Chapter 9 Linear Momentum and Collisions
1v
m
2v
m
v?m?
12
12
12
2
1
d
tt
mm
tt
tF
F
t
t
vv
动量定理常应用于碰撞问题
F?
1t
F
mF
2t
F
t
o
越小，则 越大,
例如人从高处跳下、飞机与鸟相撞、打桩等碰撞事件中，作用时间很短，冲力很大,
注意
t? F
在 一定时p
Chapter 9 Linear Momentum and Collisions
例 1 一质量为 0.05kg、速率为 10m·s -1的刚球,以与钢板法线呈 45o角的方向撞击在钢板上,并以相同的速率和角度弹回来,设碰撞时间为 0.05s.求在此时间内钢板所受到的平均冲力,
1v
m
2v
m
x
y
解 建立如图坐标系,由动量定理得
c o s2 vm?
0s ins invv mαm
F
N1.14c o s2?
t
mFF
x
v方向沿 轴反向x
xxx mmtF 12 vv )c o s(c o s vv mm
yyy mmtF 12 vv
Chapter 9 Linear Momentum and Collisions
Conservation
of MomentumCpP N
i
i
1
,0?n etFIf? then
If no net external force acts on a particles’
system,total linear momentum of the system
cannot change.
No exceptions to this law have been discovered,
So it is considered to be more fundamental than
N-II law in both macroscopic and microscopic
physical worlds.
Chapter 9 Linear Momentum and Collisions
9-4&5&6&7 Momentum and kinetic energy in
collisions(P208-214)
CpFF
i
i
inex
碰撞 两物体互相接触时间极短而互作用力较大的相互作用,
CEEE 2k1kk
完全弹性碰撞 两物体碰撞之后,它们的动能之和不变,
非弹性碰撞 由于非保守力的作用,两物体碰撞后，使机械能转换为热能、声能，化学能等其他形式的能量,
完全非弹性碰撞 两物体碰撞后,以同一速度运动,
Chapter 9 Linear Momentum and Collisions
Elastic collision（ 弹性碰撞 ）,kinetic energy of
the system is conserved.
Inelastic collision （ 非弹性碰撞 ）,some kinetic
energy transferred to other forms of energy,
such as thermal energy or energy of sound,So
the kinetic energy of the system is not conserved.
Completely inelastic collision （ 完全非弹性碰撞 ）,the bodies always stick together and lose
kinetic energy,So the kinetic energy of the
system is not conserved.
Chapter 9 Linear Momentum and Collisions
完全弹性碰撞
（五个小球质量全同）
Chapter 9 Linear Momentum and Collisions
20v
例 设有两个质量分别为 和,速度分别为和 的弹性小球作对心碰撞,两球的速度方向相同,若碰撞是完全弹性的,求碰撞后的速度 和,P210,9-8
2m1m 10v
1v
2v
2211202101 vvvv mmmm
解 取速度方向为正向，由动量守恒定律得由机械能守恒定律得
2
22
2
11
2
202
2
101 2
1
2
1
2
1
2
1 vvvv mmmm
A
1m 2m
10v
20v
B
1v
2v
A B
碰前碰后
Chapter 9 Linear Momentum and Collisions
2
22
2
11
2
202
2
101 2
1
2
1
2
1
2
1 vvvv mmmm
)()( 220222212101 vvv-v mm
)()( 20221101 vvvv mm
2211202101 vvvv mmmm
解得
,
2)(
21
2021021
1 mm
mmm
vv
v
21
1012012
2
2)(
mm
mmm
vvv
A
1m 2m
10v
20v
B
1v
2v
A B
碰前碰后
Chapter 9 Linear Momentum and Collisions
（ 1）若
21 mm?
则
102201,vvvv
（ 2）若 且
0 20?v12 mm
则
0,2101 vvv
0 20?v12 mm
（ 3）若 且
102101 2,vvvv
则讨 论
21
2021021
1
2)(
mm
mmm
vv
v
21
1012012
2
2)(
mm
mmm
vv
v
A
1m 2m
10v
20v
B
1v
2v
A B
碰前碰后
Chapter 9 Linear Momentum and Collisions
Summary for Chapter Nine
See P223 and 224
