MOMENTUM!

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MOMENTUM!
Momentum Impulse Conservation of Momentum in
1 Dimension Conservation of Momentum in 2
Dimensions Angular Momentum Torque Moment of
Inertia
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Momentum Defined
p m v
p momentum vector m mass v velocity vector
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Momentum Facts

• p m v
• Momentum is a vector quantity!
• Velocity and momentum vectors point in the same
  direction.
• SI unit for momentum kg m /s (no special
  name).
• Momentum is a conserved quantity (this will be
  proven later).
• A net force is required to change a bodys
  momentum.
• Momentum is directly proportional to both mass
  and speed.
• Something big and slow could have the same
  momentum as something small and fast.

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Momentum Examples
3 m /s
30 kg m /s
10 kg
10 kg
Note The momentum vector does not have to be
drawn 10 times longer than the velocity vector,
since only vectors of the same quantity can be
compared in this way.
9 m /s
26º
p 45 kg m /s at 26º N of E
5 kg
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Equivalent Momenta
Car m 1800 kg v 80 m /s p
1.44 105 kg m /s
Bus m 9000 kg v 16 m /s p
1.44 105 kg m /s
Train m 3.6 104 kg v 4 m /s
p 1.44 105 kg m /s
continued on next slide
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Equivalent Momenta (cont.)
The train, bus, and car all have different masses
and speeds, but their momenta are the same in
magnitude. The massive train has a slow speed
the low-mass car has a great speed and the bus
has moderate mass and speed. Note We can only
say that the magnitudes of their momenta are
equal since theyre arent moving in the same
direction. The difficulty in bringing each
vehicle to rest--in terms of a combination of the
force and time required--would be the same, since
they each have the same momentum.
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Conservation of Momentum in 1-D
Whenever two objects collide (or when they exert
forces on each other without colliding, such as
gravity) momentum of the system (both objects
together) is conserved. This mean the total
momentum of the objects is the same before and
after the collision.
(Choosing right as the direction, m2 has -
momentum.)
before p m1 v1 - m2 v2
v2
v1
m1
m2
m1 v1 - m2 v2 - m1 va m2 vb
after p - m1 va m2 vb
va
vb
m1
m2
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Directions after a collision
On the last slide the boxes were drawn going in
the opposite direction after colliding. This
isnt always the case. For example, when a bat
hits a ball, the ball changes direction, but the
bat doesnt. It doesnt really matter, though,
which way we draw the velocity vectors in after
picture. If we solved the conservation of
momentum equation (red box) for vb and got a
negative answer, it would mean that m2 was
still moving to the left after the collision. As
long as we interpret our answers correctly, it
matters not how the velocity vectors are drawn.
v2
v1
m1
m2
m1 v1 - m2 v2 - m1 va m2 vb
va
vb
m1
m2
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Sample Problem 1
35 g
7 kg
700 m/s
v 0
A rifle fires a bullet into a giant slab of
butter on a frictionless surface. The bullet
penetrates the butter, but while passing through
it, the bullet pushes the butter to the left, and
the butter pushes the bullet just as hard to the
right, slowing the bullet down. If the butter
skids off at 4 cm/s after the bullet passes
through it, what is the final speed of the
bullet?(The mass of the rifle matters not.)
35 g
7 kg
4 cm/s
v ?
continued on next slide
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Sample Problem 1 (cont.)
Lets choose left to be the direction use
conservation of momentum, converting all units to
meters and kilograms.
35 g
7 kg
p before 7 (0) (0.035) (700) 24.5
kg m /s
700 m/s
v 0
35 g
p after 7 (0.04) 0.035 v 0.28
0.035 v
7 kg
4 cm/s
v ?
p before p after 24.5 0.28 0.035
v v 692 m/s
v came out positive. This means we chose the
correct direction of the bullet in the after
picture.
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Sample Problem 2
35 g
7 kg
700 m/s
v 0
Same as the last problem except this time its a
block of wood rather than butter, and the bullet
does not pass all the way through it. How fast
do they move together after impact?
v
7. 035 kg
(0.035) (700) 7.035 v v 3.48
m/s
Note Once again were assuming a frictionless
surface, otherwise there would be a frictional
force on the wood in addition to that of the
bullet, and the system would have to include
the table as well.
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Conservation of Momentum applies only in the
absence of external forces!
In the first two sample problems, we dealt with a
frictionless surface. We couldnt simply
conserve momentum if friction had been present
because, as the proof on the last slide shows,
there would be another force (friction) in
addition to the contact forces. Friction
wouldnt cancel out, and it would be a net force
on the system.
The only way to conserve momentum with an
external force like friction is to make it
internal by including the tabletop, floor, or the
entire Earth as part of the system. For example,
if a rubber ball hits a brick wall, p for the
ball is not conserved, neither is p for the
ball-wall system, since the wall is connected to
the ground and subject to force by it. However,
p for the ball-Earth system is conserved!