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Chapter 9
Linear Momentum and
Collisions
9-1 Linear Momentum
Momentum is a vector; its direction is the
same as the direction of the velocity.
9-1 Linear Momentum
Change in momentum:
(a) mv
(b) 2mv
ConcepTest 9.2b Momentum and KE II
A system of particles is known to
have a total momentum of zero.
Does it necessarily follow that the
total kinetic energy of the system
is also zero?
1) yes
2) no
ConcepTest 9.2b Momentum and KE II
A system of particles is known to
have a total momentum of zero.
Does it necessarily follow that the
1) yes
2) no
total kinetic energy of the system
is also zero?
Momentum is a vector, so the fact that ptot = 0 does
not mean that the particles are at rest! They could be
moving such that their momenta cancel out when you
add up all of the vectors. In that case, since they are
moving, the particles would have non-zero KE.
ConcepTest 9.2c Momentum and KE III
Two objects are known to have
the same momentum. Do these
1) yes
two objects necessarily have the
2) no
same kinetic energy?
ConcepTest 9.2c Momentum and KE III
Two objects are known to have
the same momentum. Do these
1) yes
two objects necessarily have the
2) no
same kinetic energy?
If object #1 has mass m and speed v and object #2 has
mass 1/2 m and speed 2v, they will both have the same
momentum. However, since KE = 1/2 mv2, we see that
object #2 has twice the kinetic energy of object #1, due
to the fact that the velocity is squared.
9-2 Momentum and Newton’s Second Law
Newton’s second law, as we wrote it before:
is only valid for objects that have constant
mass. Here is a more general form, also
useful when the mass is changing:
9-3 Impulse
Impulse is a vector, in the same direction
as the average force.
9-3 Impulse
We can rewrite
as
So we see that
The impulse is equal to the change in
momentum.
9-3 Impulse
ConcepTest 9.3a Momentum and Impulse
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.
Both forces act for the same mount
of time. How does the change of the
boulder’s momentum compare to
the change of the pebble’s
momentum?
1) greater than
2) less than
3) equal to
ConcepTest 9.3a Momentum and Impulse
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.
Both forces act for the same mount
of time. How does the change of the
boulder’s momentum compare to
the change of the pebble’s
momentum?
1) greater than
2) less than
3) equal to
ConcepTest 9.3b Velocity and Impulse
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.
Both forces act for the same mount
of time. How does the change of the
boulder’s velocity compare to the
change of the pebble’s velocity?
1) greater than
2) less than
3) equal to
ConcepTest 9.3b Velocity 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 velocity compare to the
rate of change of the pebble’s
velocity?
1) greater than
2) less than
3) equal to
ConcepTest 9.7 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?
1) the beanbag
2) the rubber ball
3) both the same
ConcepTest 9.7 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
1) the beanbag
2) the rubber ball
3) 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
Example
• To make a bounce pass, a player throws a
0.60 kg basketball toward the floor. The
ball hits the floor with a speed of 5.4 m/s
and rebounds with the same speed and
angle.
– What impulse does the floor deliver to the
basketball?
– If the ball is in contact with the floor for 0.1 s,
what average force did the floor exert?
9-4 Conservation of Linear Momentum
The net force acting on an object is the rate
of change of its momentum:
If the net force is zero, the momentum does not
change
9-4 Conservation of Linear Momentum
Internal Versus External Forces:
Internal forces act between objects within the
system.
As with all forces, they occur in action-reaction
pairs. As all pairs act between objects in the
system, the internal forces always sum to zero:
Therefore, the net force acting on a system is
the sum of the external forces acting on it.
9-4 Conservation of Linear Momentum
Furthermore, internal forces cannot change the
momentum of a system.
However, the momenta of components of the
system may change.
9-4 Conservation of Linear Momentum
An example of internal forces moving
components of a system:
ConcepTest 9.14a Recoil Speed I
Amy (150 lbs) and Gwen (50 lbs) are
standing on slippery ice and push off
each other. If Amy slides at 2 m/s,
what speed does Gwen have?
(1) 1 m/s
(2) 2 m/s
(3) 3 m/s
(4) 4 m/s
(5) 6 m/s
150 lbs
50 lbs
ConcepTest 9.14a Recoil Speed I
Amy (150 lbs) and Gwen (50 lbs) are
standing on slippery ice and push off
each other. If Amy slides at 2 m/s,
what speed does Gwen have?
(1) 1 m/s
(2) 2 m/s
(3) 3 m/s
(4) 4 m/s
(5) 6 m/s
The initial momentum is zero,
so the momenta of Amy and
Gwen must be equal and
opposite. Since p = mv,
then if Amy has 3 times
more mass, we see that
Gwen must have 3 times
more speed.
150 lbs
50 lbs
9-5 Inelastic Collisions
A completely inelastic collision:
9-5 Inelastic Collisions
For collisions in two dimensions, conservation
of momentum is applied separately along each
axis:
ConcepTest 9.12a Inelastic Collisions I
A box slides with initial velocity 10 m/s
1) 10 m/s
on a frictionless surface and collides
2) 20 m/s
inelastically with an identical box. The
3) 0 m/s
boxes stick together after the collision.
4) 15 m/s
What is the final velocity?
5) 5 m/s
vi
M
M
M
M
vf
ConcepTest 9.12a Inelastic Collisions I
A box slides with initial velocity 10 m/s
1) 10 m/s
on a frictionless surface and collides
2) 20 m/s
inelastically with an identical box. The
3) 0 m/s
boxes stick together after the collision.
4) 15 m/s
What is the final velocity?
5) 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
Example
• A car (m1 = 950 kg) heading due E at 20.0 m/s collides
with a minivan (m2 = 1300 kg) headed due N (see
figure.) After the collision, the wreckage moves at 40.0˚
above the x-axis. Find the initial speed of the minivan
and the final speed of the wreckage.
9-6 Elastic Collisions
In elastic collisions, both kinetic energy and
momentum are conserved.
One-dimensional elastic collision:
ConcepTest 9.10a Elastic Collisions I
Consider two elastic collisions:
1) a golf ball with speed v hits
a stationary bowling ball head-on.
2) a bowling ball with speed v
hits a stationary golf ball head-on.
In which case does the golf ball
have the greater speed after the
collision?
v
1) situation 1
2) situation 2
3) both the same
at rest
at rest
1
v
2
ConcepTest 9.10a Elastic Collisions I
Consider two elastic collisions:
1) a golf ball with speed v hits a
stationary bowling ball head-on.
2) a bowling ball with speed v
hits a stationary golf ball head-on. In
which case does the golf ball have the
greater speed after the collision?
Remember that the magnitude of the
relative velocity has to be equal before
and after the collision!
1) situation 1
2) situation 2
3) both the same
v
1
In case 1 the bowling ball will almost
remain at rest, and the golf ball will
bounce back with speed close to v.
In case 2 the bowling ball will keep going
with speed close to v, hence the golf ball
will rebound with speed close to 2v.
v
2v
2
9-6 Elastic Collisions
Two-dimensional collisions can only be solved if
some of the final information is known, such as
the final velocity of one object:
Example
• A 722-kg car stopped at an intersection is
rear-ended by a 1620-kg truck moving with
a speed of 14.5 m/s. If the car was in
neutral and the brakes were off, so that
the collision is approximately elastic, find
the final speed of both vehicles after the
collision.
9-7 Center of Mass
The center of mass of a system is the point where
the system can be balanced in a uniform
gravitational field.
9-7 Center of Mass
For two objects:
The center of mass is closer to the more
massive object.
9-7 Center of Mass
The center of mass need not be within the object:
9-7 Center of Mass
Motion of the center of mass:
9-7 Center of Mass
The total mass multiplied by the acceleration of
the center of mass is equal to the net external
force:
The center of mass
accelerates just as
though it were a point
particle of mass M
acted on by
ConcepTest 9.18 Baseball Bat
Where is center of mass
of a baseball bat located?
1) at the midpoint
2) closer to the thick end
3) closer to the thin end (near handle)
4) it depends on how heavy the bat is
ConcepTest 9.18 Baseball Bat
Where is center of mass
of a baseball bat located?
1) at the midpoint
2) closer to the thick end
3) closer to the thin end (near handle)
4) it depends on how heavy the bat is
Since most of the mass of the bat is at the thick end,
this is where the center of mass is located. Only if
the bat were like a uniform rod would its center of
mass be in the middle.
Summary of Chapter 9
• Linear momentum:
• Momentum is a vector
• Newton’s second law:
• Impulse:
• Impulse is a vector
• The impulse is equal to the change in
momentum
• If the time is short, the force can be quite
large
Summary of Chapter 9
• Momentum is conserved if the net external
force is zero
• Internal forces within a system always sum to
zero
• In collision, assume external forces can be
ignored
• Inelastic collision: kinetic energy is not
conserved
• Completely inelastic collision: the objects
stick together afterward
Summary of Chapter 9
• A one-dimensional collision takes place along
a line
• In two dimensions, conservation of
momentum is applied separately to each
• Elastic collision: kinetic energy is conserved
• Center of mass:
Summary of Chapter 9
• Center of mass:
Summary of Chapter 9
• Motion of center of mass: