Chapter 9. Impulse and
Momentum
Explosions and collisions
obey some surprisingly
simple laws that make
problem solving easier when
comparing the situation
before and after an
interaction.
Chapter Goal: To introduce
the ideas of impulse and
momentum and to learn a
new problem-solving strategy
based on conservation laws.
Chapter 9. Impulse and
Momentum
Topics:
• Momentum and Impulse
• Solving Impulse and Momentum
Problems
• Conservation of Momentum
• Inelastic Collisions
• Explosions
• Momentum in Two Dimensions
Momentum
10 m/s
A
B
After the collision
A
B
v
u
What is the velocity of ball A after the collision? ball B?
What is conserved during the collision?
MOMENTUM
r
r
p = mv
The total momentum is the sum of momentum of ball A and
momentum of ball B.
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r
r
p = mv
Momentum
The total momentum of the system is conserved during the collision:
m Av A,i = m Av + m B u
A
10 m/s
B
A
v
B
u
• Momentum is a vector. It has the same direction as corresponding
velocity.
• General expression for the momentum conservation: the total
momentum before the collision is equal to the total momentum after
the collision
4
r
r
p = mv
Momentum
• General expression for the momentum conservation: the total
momentum before the collision is equal to the total momentum after
the collision
A
B
r
v A,i
r
v A, f
A
r
v B ,i
r
vB, f
B
r
r
r
r
m Av A,i + m B v B ,i = m Av A, f + m B v B , f
Usually this equation is written in terms of components.
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Example:
A
10 m/s
B
m A = 1kg
m B = 4kg
After the collision the balls are moving together
(have the same velocity). What is their velocity?
A
Momentum before the collision:
B
v
pi = m Av A ,i
kg ⋅ m
= 10
s
Momentum after the collision:
p f = ( m A + m B )v = 5v
Conservation of momentum:
pi = p f
10 = 5v
v = 2m / s
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Why do we have conservation of total momentum?
Newton’s second law:
Acceleration:
Then
r
Fnet
After integration
r
r
Fnet = ma
r
r dv
a=
dt
r
r
r
dv d ( mv ) dp
=m
=
=
dt
dt
dt
momentum
t2
r
r
∆p = ∫ Fnet dt
t1
t2
r
r
J = ∫ Fnet dt
t1
The area under
r
Fnet ( t )
curve.
It is called IMPULSE, J.
The impulse of the force is equal to the change of the momentum of
r r
the object.
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∆p = J
pi = mvix
p f = mv fx < 0
J x = p f − pi < 0
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t2
m1v fx ,1 − m1vix ,1 = ∫ Fx ,2 on1dt
t1
t2
m2 v fx ,2 − m2 vix ,2 = ∫ Fx ,1on 2 dt
t1
Newton’s third law:
Fx ,1on 2 = − Fx ,2 on1
Then
t2
∫F
x ,1 on 2
t2
dt = − ∫ Fx ,2 on1dt
t1
(
m2 v fx ,2 − m2 vix ,2 = − m1v fx ,1 − m1vix ,1
t1
)
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(
m2 v fx ,2 − m2 vix ,2 = − m1v fx ,1 − m1vix ,1
m1vix ,1 + m2 vix ,2 = m1v fx ,1 + m2 v fx ,2
pix ,1 + pix ,2 = p fx ,1 + p fx ,2
pix ,total = p fx ,total
The law of conservation of momentum
10
)
r
r
p = mv
Momentum
The law of conservation of momentum:
The total momentum of an isolated system (no external forces) does not
change.
Interactions within system do not change the system’s total momentum
isolated system
A
B
r
v A,i
r
v A, f
A
r
v B ,i
r
vB, f
B
r
r
r
r
m Av A,i + m B v B ,i = m Av A, f + m B v B , f
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r
r
p = mv
Momentum
The ball is dropped onto a hard floor:
The ball is not an isolated system (interaction with the floor)
no conservation of momentum for the ball
Initial momentum is
r
r
pi = mvi
Final momentum (after collision) is
r
r
p f = mv f
The ball+ the floor is an isolated system
The total momentum (ball+floor) is conserved
r
vf
r
vi
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Example: Find
v2 x
Isolated system
Motion with constant
acceleration:
( v1 x , B )2 = 2a x x1 = 16
v1 x , B = 4m / s
Momentum before the “collision”:
pi ,total = m B v1 x , B + mC v1 x ,C = m B v1 x , B = 75 ⋅ 4 = 300
kg ⋅ m
s
Momentum after the “collision”:
p f ,total = m B v 2 x , B + mC v2 x ,C = ( m B + mC )v 2 x = 100v 2 x
Conservation of momentum:
p f ,total = pi ,total
100v 2 x = 300
v 2 x = 3m / s
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Perfectly inelastic collision:
A collision in which the two objects stick together and move with a
common final velocity.
pi ,total = m1vix ,1 + m2 v ix ,2
p f ,total = ( m1 + m2 )v fx
m1vix ,1 + m2 vix ,2 = ( m1 + m2 )v fx
vix ,2
 m1

m1
3
3
=
vix ,1 = − ⋅ 0.5 − = −2.25m / s
+ 1  v fx −
m2
2
2
 m2

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Chapter 9. Summary Slides
General Principles
General Principles
Important Concepts
Important Concepts
Applications
Applications
Chapter 9. Questions
The cart’s change of momentum is
A. 30 kg m/s.
B. 10 kg m/s.
C.–10 kg m/s.
D.–20 kg m/s.
E.–30 kg m/s.
The cart’s change of momentum is
A. 30 kg m/s.
B. 10 kg m/s.
C.–10 kg m/s.
D.–20 kg m/s.
E.–30 kg m/s.
A 10 g rubber ball and a 10 g clay ball are
thrown at a wall with equal speeds. The
rubber ball bounces, the clay ball sticks.
Which ball exerts a larger impulse on the
wall?
A. They exert equal impulses because they
have equal momenta.
B. The clay ball exerts a larger
impulse because it sticks.
C. Neither exerts an impulse on the
wall because the wall doesn’t move.
D. The rubber ball exerts a larger impulse
because it bounces.
A 10 g rubber ball and a 10 g clay ball are
thrown at a wall with equal speeds. The
rubber ball bounces, the clay ball sticks.
Which ball exerts a larger impulse on the
wall?
A. They exert equal impulses because they
have equal momenta.
B. The clay ball exerts a larger
impulse because it sticks.
C. Neither exerts an impulse on the
wall because the wall doesn’t move.
D. The rubber ball exerts a larger impulse
because it bounces.
Objects A and C are made of different materials, with different
“springiness,” but they have the same mass and are initially at rest.
When ball B collides with object A, the ball ends up at rest. When ball B
is thrown with the same speed and collides with object C, the ball
rebounds to the left. Compare the velocities of A and C after the
collisions. Is vA greater than, equal to, or less than vC?
A. vA > vC
B. vA < vC
C. vA = vC
Objects A and C are made of different materials, with different
“springiness,” but they have the same mass and are initially at rest.
When ball B collides with object A, the ball ends up at rest. When ball B
is thrown with the same speed and collides with object C, the ball
rebounds to the left. Compare the velocities of A and C after the
collisions. Is vA greater than, equal to, or less than vC?
A. vA > vC
B. vA < vC
C. vA = vC
The two particles are both moving to the right. Particle 1 catches
up with particle 2 and collides with it. The particles stick together
and continue on with velocity vf. Which of these statements is
true?
A. vf = v2.
B. vf is less than v2.
C. vf is greater than v2, but less than v1.
D. vf = v1.
E. vf is greater than v1.
The two particles are both moving to the right. Particle 1 catches
up with particle 2 and collides with it. The particles stick together
and continue on with velocity vf. Which of these statements is
true?
A. vf = v2.
B. vf is less than v2.
C. vf is greater than v2, but less than v1.
D. vf = v1.
E. vf is greater than v1.
An explosion in a rigid pipe shoots out
three pieces. A 6 g piece comes out the
right end. A 4 g piece comes out the left
end with twice the speed of the 6 g
piece. From which end does the third
piece emerge?
A. Right end
B. Left end
An explosion in a rigid pipe shoots out
three pieces. A 6 g piece comes out the
right end. A 4 g piece comes out the left
end with twice the speed of the 6 g
piece. From which end does the third
piece emerge?
A. Right end
B. Left end