Momentum Defined

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Momentum Defined
p m v
p momentum vector m mass v velocity vector
2
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 km /s
26º
p 45 kg m /s at 26º N of E
5 g
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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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Impulse Defined
Impulse is defined as the product force acting on
an object and the time during which the force
acts. Impulse F t Example A 50 N force is
applied to a 100 kg boulder for 3 s. The impulse
of this force is Impulse (50 N) (3 s) 150 N
s.
Note that we didnt need to know the mass of the
object in the above example.
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Impulse - Momentum Theorem
The impulse due to all forces acting on an object
(the net force) is equal to the change in
momentum of the object
Fnet t ? p
We know the units on both sides of the equation
are the same (last slide), but lets prove the
theorem formally
Fnet t m a t m (? v / t) t m ? v
? p
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Stopping Time
F t F t
Imagine a car hitting a wall and coming to rest.
The force on the car due to the wall is large
(big F ), but that force only acts for a small
amount of time (little t ). Now imagine the
same car moving at the same speed but this time
hitting a giant haystack and coming to rest. The
force on the car is much smaller now (little F
), but it acts for a much longer time (big t ).
In each case the impulse involved is the same
since the change in momentum of the car is the
same. Any net force, no matter how small, can
bring an object to rest if it has enough time. A
pole vaulter can fall from a great height without
getting hurt because the mat applies a smaller
force over a longer period of time than the
ground alone would.
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Impulse - Momentum Example
A 1.3 kg ball is coming straight at a 75 kg
soccer player at 13 m/s who kicks it in the exact
opposite direction at 22 m/s with an average
force of 1200 N. How long are his foot and the
ball in contact?
answer Well use Fnet t ? p. Since the
ball changes direction, ? p m ? v m (vf
- v0) 1.3 22 - (-13) (1.3 kg) (35 m/s)
45.5 kg m /s. Thus, t 45.5 / 1200
0.0379 s, which is just under 40 ms.
During this contact time the ball compresses
substantially and then decompresses. This
happens too quickly for us to see, though. This
compression occurs in many cases, such as hitting
a baseball or golf ball.
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Fnet vs. t graph
Fnet (N)
Net area ? p
t (s)
6
A variable strength net force acts on an object
in the positive direction for 6 s, thereafter in
the opposite direction. Since impulse is Fnet
t, the area under the curve is equal to the
impulse, which is the change in momentum. The net
change in momentum is the area above the curve
minus the area below the curve. This is just
like a v vs. t graph, in which net
displacement is given area under the curve.
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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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Proof of Conservation of Momentum
The proof is based on Newtons 3rd Law. Whenever
two objects collide (or exert forces on each
other from a distance), the forces involved are
an action-reaction pair, equal in strength,
opposite in direction. This means the net force
on the system (the two objects together) is zero,
since these forces cancel out.
F
M
F
m
force on M due to m
force on m due to M
For each object, F (mass) (a) (mass) (?v /
t ) (mass ?v) / t ?p / t. Since the force
applied and the contact time is the same for each
mass, they each undergo the same change in
momentum, but in opposite directions. The result
is that even though the momenta of the individual
objects changes, ?p for the system is zero.
The momentum that one mass gains, the other
loses. Hence, the momentum of the system before
equals the momentum of the system after.
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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!
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Sample Problem 3
An apple is originally at rest and then dropped.
After falling a short time, its moving pretty
fast, say at a speed V. Obviously, momentum is
not conserved for the apple, since it didnt have
any at first. How can this be?
answer Gravity is an external force on the
apple, so momentum for it alone is not conserved.
To make gravity internal, we must define a
system that includes the other object responsible
for the gravitational force--Earth. The net
force on the apple-Earth system is zero, and
momentum is conserved for it. During the fall
the Earth attains a very small speed v. So, by
conservation of momentum
apple
m
V
F
v
EarthM
F
m V M v
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Sample Problem 4
A crate of raspberry donut filling collides with
a tub of lime Kool Aid on a frictionless surface.
Which way on how fast does the Kool Aid rebound?
answer Lets draw v to the right in the
after picture.
3 (10) - 6 (15) -3 (4.5) 15 v
v -3.1 m/sSince v came out negative, we
guessed wrong in drawing v to the right, but
thats OK as long as we interpret our answer
correctly. After the collision the lime Kool Aid
is moving 3.1 m/s to the left.
before
6 m/s
10 m/s
3 kg
15 kg
after
4.5 m/s
v
3 kg
15 kg