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Chapter 6
Linear Momentum and
Collisions
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Units of Chapter 6
Linear Momentum
Impulse
Conservation of Linear Momentum
Elastic and Inelastic Collisions
Center of Mass
Jet Propulsion and Rockets
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6.1 Linear Momentum
Definition of linear momentum:
The linear momentum of an object is the product of
its mass and velocity.
Note that momentum is a vector—it has both
a magnitude and a direction.
SI unit of momentum: kg • m/s. This unit has
no special name.
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6.1 Linear Momentum
For a system of objects, the total momentum
is the vector sum of each.
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6.1 Linear Momentum
The change in momentum is the difference
between the momentum vectors.
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6.1 Linear Momentum
If an object’s momentum changes, a force
must have acted on it.
The net force is equal to the rate of change of
the momentum.
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Question 6.2a Momentum and KE I
A system of particles is
known to have a total
kinetic energy of zero.
What can you say about
the total momentum of
the system?
a) momentum of the system is positive
b) momentum of the system is negative
c) momentum of the system is zero
d) you cannot say anything about the
momentum of the system
Question 6.2a Momentum and KE I
A system of particles is
known to have a total
kinetic energy of zero.
What can you say about
the total momentum of
the system?
a) momentum of the system is positive
b) momentum of the system is negative
c) momentum of the system is zero
d) you cannot say anything about the
momentum of the system
Because the total kinetic energy is zero, this means
that all of the particles are at rest (v = 0). Therefore,
because nothing is moving, the total momentum of
the system must also be zero.
Question 6.2c Momentum and KE III
Two objects are known to have
the same momentum. Do these
a) yes
two objects necessarily have the
b) no
same kinetic energy?
Question 6.2c Momentum and KE III
Two objects are known to have
the same momentum. Do these
a) yes
two objects necessarily have the
b) no
same kinetic energy?
If object #1 has mass m and speed v and object #2 has
mass
m and speed 2v, they will both have the same
momentum. However, because KE =
mv2, we see
that object #2 has twice the kinetic energy of object #1,
due to the fact that the velocity is squared.
6.2 Impulse
Impulse is the change in momentum:
Typically, the force
varies during the
collision.
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6.2 Impulse
Actual contact times may be very short.
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6.2 Impulse
When a moving object stops, its impulse
depends only on its change in momentum. This
can be accomplished by a large force acting for
a short time, or a smaller force acting for a
longer time.
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6.2 Impulse
We understand this instinctively—we bend
our knees when landing a jump; a “soft”
catch (moving hands) is less painful than a
“hard” one (fixed hands).
This is how airbags work—they slow down
collisions considerably—and why cars are
built with crumple zones.
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Question 6.3a Momentum and Force
A net force of 200 N acts on a 100-kg
boulder, and a force of the same
magnitude acts on a 130-g pebble.
How does the rate of change of the
boulder’s momentum compare to
the rate of change of the pebble’s
momentum?
a) greater than
b) less than
c) equal to
Question 6.3a Momentum and Force
A net force of 200 N acts on a 100-kg
boulder, and a force of the same
magnitude acts on a 130-g pebble.
How does the rate of change of the
boulder’s momentum compare to
the rate of change of the pebble’s
momentum?
a) greater than
b) less than
c) equal to
The rate of change of momentum is, in fact, the force.
Remember that F = Dp/Dt. Because the force exerted
on the boulder and the pebble is the same, then the
rate of change of momentum is the same.
Question 6.7 Impulse
A small beanbag and a bouncy
rubber ball are dropped from the
same height above the floor.
They both have the same mass.
Which one will impart the greater
impulse to the floor when it hits?
a) the beanbag
b) the rubber ball
c) both the same
Question 6.7 Impulse
A small beanbag and a bouncy
rubber ball are dropped from the
same height above the floor.
They both have the same mass.
Which one will impart the greater
a) the beanbag
b) the rubber ball
c) both the same
impulse to the floor when it hits?
Both objects reach the same speed at the floor. However, while
the beanbag comes to rest on the floor, the ball bounces back
up with nearly the same speed as it hit. Thus, the change in
momentum for the ball is greater, because of the rebound.
The impulse delivered by the ball is twice that of the beanbag.
For the beanbag:
For the rubber ball:
Dp = pf – pi = 0 – (–mv ) = mv
Dp = pf – pi = mv – (–mv ) = 2mv
Follow-up: Which one imparts the larger force to the floor?
A 0.17 kg baseball is thrown with a speed of
38.m/s and it is hit straight back to the pitcher
with a speed of 62.m/s. What is the magnitude of
the IMPULSE exerted upon the ball by the bat?
Jennifer hits a stationary 200. gram ball and it
leaves the racket at 40. m/s. If time lapse
photography shows that the ball was in
contact with the racket for 40. ms:
(a) What average force was exerted on the
racket?
(b) What is the ratio of this force to the
weight of the ball?
A 0.32 kg ball is moving horizontally 30. m/s
just before bouncing off a wall, thereafter
moving 25. m/s in the opposite direction.
(a) What is the magnitude of its change in
momentum?
(b) What percentage of the kinetic energy
was lost in the collision?
6.3 Conservation of Linear
Momentum
If there is no net force acting on a system, its
total momentum cannot change.
This is the law of conservation of momentum.
If there are internal forces, the momenta of
individual parts of the system can change, but
the overall momentum stays the same.
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6.3 Conservation of Linear
Momentum
In this example, there is no external force, but
the individual components of the system do
change their momenta:
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6.3 Conservation of Linear
Momentum
Collisions happen quickly enough that any
external forces can be ignored during the
collision. Therefore, momentum is conserved
during a collision.
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An empty coal-car (mass 20,000.kg) of a train
coasts along at 10.m/s. An unfortunate 3000. kg
elephant falls from a bridge and drops vertically
into the car. Determine the speed of the car
immediately after the elephant is added to its
contents.
6.4 Elastic and Inelastic Collisions
In an elastic
collision, the total
kinetic energy is
conserved.
Total kinetic energy
is not conserved in
an inelastic
collision.
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6.4 Elastic and Inelastic Collisions
A completely inelastic
collision is one where the
objects stick together
afterwards.
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6.4 Elastic and Inelastic Collisions
The fraction of the total kinetic energy that is
left after a completely inelastic collision can
be shown to be:
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6.4 Elastic and Inelastic Collisions
For an elastic collision, both the kinetic
energy and the momentum are conserved:
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6.4 Elastic and Inelastic Collisions
Collisions may take
place with the two
objects approaching
each other, or with
one overtaking the
other.
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Question 6.12a Inelastic Collisions I
A box slides with initial velocity 10 m/s on a
a) 10 m/s
frictionless surface and collides inelastically with
b) 20 m/s
an identical box. The boxes stick together after the
c) 0 m/s
collision. What is the final velocity?
d) 15 m/s
e) 5 m/s
vi
M
M
M
M
vf
Question 6.12a Inelastic Collisions I
A box slides with initial velocity 10 m/s on a
a) 10 m/s
frictionless surface and collides inelastically with
b) 20 m/s
an identical box. The boxes stick together after the
c) 0 m/s
collision. What is the final velocity?
d) 15 m/s
e) 5 m/s
The initial momentum is:
M vi = (10) M
vi
M
M
The final momentum must be the same!!
The final momentum is:
Mtot vf = (2M) vf = (2M) (5)
M
M
vf
As shown in Fig. 6-2, a car (mass = 1500 kg)
and a small truck (mass = 2000 kg) collide at
right angles at an icy intersection. The car was
traveling East at 20 m/s and the truck was
traveling North at 20 m/s when the collision
took place. What is the speed of the combined
wreck, assuming a completely inelastic
collision?
In space, a 4.0 kg metal ball moving 30. m/s
has a head-on collision with a stationary 1.0
kg second ball. After the elastic collision,
what are the velocities of the balls?
A 50-gram ball moving +10 m/s collides headon with a stationary ball of mass 100.g. The
collision is elastic. What is the speed of each
ball immediately after the collision?
6.5 Center of Mass
Definition of the center of mass:
The center of mass is the point at which all of the
mass of an object or system may be considered to be
concentrated, for the purposes of linear or
translational motion only.
We can then use Newton’s second law for the
motion of the center of mass:
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6.5 Center of Mass
The momentum of the center of mass does
not change if there are no external forces on
the system.
The location of the center of mass can be
found:
This calculation is straightforward for a
system of point particles, but for an
extended object calculus is necessary.
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6.5 Center of Mass
The center of mass of a flat object can be
found by suspension.
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6.5 Center of Mass
The center of mass may be located outside a
solid object.
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Question 6.19 Motion of CM
Two equal-mass particles
(A and B) are located at
some distance from each
other. Particle A is held
stationary while B is
moved away at speed v.
What happens to the
center of mass of the
two-particle system?
a) it does not move
b) it moves away from A with speed v
c) it moves toward A with speed v
d) it moves away from A with speed ½v
e) it moves toward A with speed ½v
Question 6.19 Motion of CM
Two equal-mass particles
(A and B) are located at
some distance from each
other. Particle A is held
stationary while B is
moved away at speed v.
What happens to the
center of mass of the
two-particle system?
a) it does not move
b) it moves away from A with speed v
c) it moves toward A with speed v
d) it moves away from A with speed ½v
e) it moves toward A with speed ½v
Let’s say that A is at the origin (x = 0) and B is at
some position x. Then the center of mass is at x/2
because A and B have the same mass. If v = Dx/Dt
tells us how fast the position of B is changing,
then the position of the center of mass must be
changing like D(x/2)/Dt, which is simply v.
Question 6.20 Center of Mass
The disk shown below in (1) clearly has its
center of mass at the center.
a) higher
b) lower
Suppose the disk is cut in half and the pieces
arranged as shown in (2).
Where is the center of mass of (2) as
compared to (1) ?
c) at the same place
d) there is no definable
CM in this case
(1)
X
CM
(2)
Question 6.20 Center of Mass
The disk shown below in (1) clearly has its
center of mass at the center.
a) higher
b) lower
Suppose the disk is cut in half and the pieces
arranged as shown in (2).
c) at the same place
Where is the center of mass of (2) as
compared to (1) ?
The CM of each half is closer
to the top of the semicircle
than the bottom. The CM of
the whole system is located
at the midpoint of the two
semicircle CMs, which is
higher than the yellow line.
d) there is no definable
CM in this case
(1)
X
CM
(2)
CM
6.6 Jet Propulsion and Rockets
If you blow up a balloon and then let it go,
it zigzags away from you as the air shoots
out. This is an example of jet propulsion.
The escaping air exerts a force on the
balloon that pushes the balloon in the
opposite direction.
Jet propulsion is another example of
conservation of momentum.
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6.6 Jet Propulsion and Rockets
This same phenomenon explains the
recoil of a gun:
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6.6 Jet Propulsion and Rockets
The thrust of a rocket works
the same way.
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6.6 Jet Propulsion and Rockets
Jet propulsion can be used to slow a rocket
down as well as to speed it up; this involves
the use of thrust reversers. This is done by
commercial jetliners.
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Summary of Chapter 6
Momentum of a point particle is defined as
its mass multiplied by its velocity.
The momentum of a system of particles is
the vector sum of the momenta of its
components.
Newton’s second law:
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Summary of Chapter 6
Impulse–momentum theorem:
In the absence of external forces,
momentum is conserved.
Momentum is conserved during a collision.
Kinetic energy is also conserved in an
elastic collision.
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Summary of Chapter 6
The center of mass of an object is the point
where all the mass may be considered to be
concentrated.
Coordinates of the center of mass:
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