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 describes the tendency of a mass to
  keep going in the same direction with the same
  speed.
• Something big and slow could have the same
  momentum as something small and fast.

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Vocabulary
angular momentum collision law of conservation of
momentum elastic collision gyroscope impulse inela
stic collision linear momentum momentum
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Momentum

• The momentum of a ball depends on its mass and
  velocity.
• Ball B has more momentum than ball A.

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Momentum and Inertia

• Inertia is another property of mass that resists
  changes in velocity however, inertia depends
  only on mass.
• Inertia is a scalar quantity.
• Momentum is a property of moving mass that
  resists changes in a moving objects velocity.
• Momentum is a vector quantity.

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Momentum

• Ball A is 1 kg moving 1m/sec,
• Ball B is 1kg at 3 m/sec.
• A 1 N force is applied to deflect each ball's
  motion.
• What happens?
• Does the force deflect both balls equally?

• Ball B deflects much less than ball A when the
  same force is applied because ball B had a
  greater initial momentum.

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Calculating Momentum

• The momentum of a moving object is its mass
  multiplied by its velocity.
• That means momentum increases with both mass and
  velocity.

Momentum (kg m/sec)
Velocity (m/sec)
Mass (kg)
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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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Comparing momentum
A car is traveling at a velocity of 13.5 m/sec
(30 mph) north on a straight road. The mass of
the car is 1,300 kg. A motorcycle passes the car
at a speed of 30 m/sec (67 mph). The motorcycle
(with rider) has a mass of 350 kg. Calculate and
compare the momentum of the car and motorcycle.

1. You are asked for momentum.
2. You are given masses and velocities.
3. Use p m v
4. Solve for car p (1,300 kg) (13.5 m/s) 17,550
   kg m/s
5. Solve for cycle p (350 kg) (30 m/s) 10,500
   kg m/s
6. The car has more momentum even though it is going
   much slower.

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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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Review Questions

• What is momentum equal to?
• A. Mass x Velocity
• B. Current x Voltage
• C. Force x Time
• D. Frequency x Wavelength

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Review Questions

• Why is momentum a vector quantity?
• Mass is a vector
• Velocity is a vector
• Time is a vector

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Do Momentum Problems
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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. The symbol for impulse is J. So, by
definition J F t Example A 50 N force is
applied to a 100 kg boulder for 3 s. The impulse
of this force is J (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 Units
J F t shows why the SI unit for impulse is the
Newton second. There is no special name for
this unit, but it is equivalent to a kg m /s.
proof 1 N s 1 (kg m /s2) (s) 1
kg m /s

Fnet m a shows this is equivalent to a newton.
Therefore, impulse and momentum have the same
units, which leads to a useful theorem.
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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
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Impulse
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Impulse Ft ?mv
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Impulse Ft ?mv
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Impulse Ft ?mv
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Impulse Ft ?mv
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Impulse
Ft ?mv
F t ?mv
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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)
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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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Review Questions

• What is Impulse?
• A. Mass x Velocity
• B. Current x Voltage
• C. Force x Time
• D. Frequency x Wavelength

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Review Questions

• What can you do to increase momentum?
• Increase Time
• Increase Force
• Increase Impulse
• All of these

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Do Impulse Problems
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Conservation of Momentum

• The law of conservation of momentum states when a
  system of interacting objects is not influenced
  by outside forces (like friction), the total
  momentum of the system cannot change.

If you throw a rock forward from a skateboard,
you will move backward in response.
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Collisions in One Dimension

• A collision occurs when two or more objects hit
  each other.
• During a collision, momentum is transferred from
  one object to another.
• Collisions can be elastic

or inelastic.
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Collisions
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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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Elastic collisions
Two 0.165 kg billiard balls roll toward each
other and collide head-on. Initially, the 5-ball
has a velocity of 0.5 m/s. The 10-ball has an
initial velocity of -0.7 m/s. The collision is
elastic and the 10-ball rebounds with a velocity
of 0.4 m/s, reversing its direction. What is
the velocity of the 5-ball after the collision?
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Elastic Collisions

• You are asked for 10-balls velocity after
  collision.
• You are given mass, initial velocities, 5-balls
  final velocity.
• Diagram the motion, use m1v1 m2v2 m1v3 m2v4
• Solve for V3
• (0.165 kg)(0.5 m/s) (0.165 kg) (-0.7 kg)(0.165
  kg) v3 (0.165 kg) (0.4 m/s)
• V3 -0.6 m/s

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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
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
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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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Inelastic Collisions
A train car moving to the right at 10 m/s
collides with a parked train car. They stick
together and roll along the track. If the moving
car has a mass of 8,000 kg and the parked car has
a mass of 2,000 kg, what is their combined
velocity after the collision?

• You are asked for the final velocity.
• You are given masses, and initial velocity of
  moving train car.

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Inelastic Collisions

• Diagram the problem
• Use m1v1 m2v2 (m1v1 m2v2) v3
• Solve for v3 m1v1 m2v2
• (m1v1 m2v2)
• v3 (8,000 kg)(10 m/s) (2,000 kg)(0 m/s)
• (8,000 2,000 kg)
• v3 8 m/s
• The train cars moving together to right at 8 m/s.

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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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Bouncing

• Alfred went on a date with Mabel. When Alfred
  dropped off Mabel after the date, he was anxious
  to play Angry Birds, so he forgot to kiss her on
  the cheek good night. She went up to her room,
  opened the window and threw a flower pot at
  Alfred. On of three things could happen.
• 1. The flower pot head collision is elastic
• 2. The flower pot head collision is inelastic
• 3. The flower pot bounces off his head
• Which will hurt more?????

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Elastic Collision
Before
After
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Elastic Collision

• Alfred Flower pot Alfred Flower pot
• m1v1 m2v2 m1v3 m2v4
• 100kg(0m/s) 10kg (15 m/s) 100kg (v3) 10kg
  (0m/s)
• 150kgm/s 100kg (v3)
• 150kgm/s 100kg (v3)
• 100kg 100kg
• 1.5 m/s v3 (elastic)

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Inelastic Collision
Before
After
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Inelastic Collision

• Alfred Flower pot (Alfred Flower pot)
• m1v1 m2v2 (m1 m2)v3
• 100kg(0m/s) 10kg(15 m/s) (100kg 10kg) (v3)
• 150kgm/s 110kg(v3)
• 150kgm/s 110kg(v3)
• 110kg 110kg
• 1.36 m/s v3 (inelastic)
• 1.5 m/s v3 (elastic)

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Bouncing
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Elastic Collision

• Alfred Flower pot Alfred Flower pot
• m1v1 m2v2 m1v3 m2v4
• 100kg(0m/s) 10kg(15 m/s) 100kg(v3)
  10kg(-5m/s)
• 150kgm/s 100kg(v3) 50kgm/s
• 200kgm/s 100kg(v3)
• 100kg 100kg
• 2.0 m/s v3 (bouncing)
• 1.5 m/s v3 (elastic)
• 1.36 m/s v3 (inelastic)

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Do Collision Problems
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Angular Momentum

• Momentum resulting from an object moving in
  linear motion is called linear momentum.
• Momentum resulting from the rotation (or spin) of
  an object is called angular momentum.

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Conservation of Angular Momentum

• Angular momentum is important because it obeys a
  conservation law, as does linear momentum.
• The total angular momentum of a closed system
  stays the same.

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Calculating Angular Momentum
Angular momentum is calculated in a similar way
to linear momentum, except the mass and velocity
are replaced by the moment of inertia and angular
velocity.
Moment of inertia (kg m2)
Angular momentum (kg m/sec2)
Angular velocity (rad/sec)
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Calculating Angular Momentum

• The moment of inertia of an object is the average
  of mass times radius squared for the whole
  object.
• Since the radius is measured from the axis of
  rotation, the moment of inertia depends on the
  axis of rotation.

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Gyroscopes Angular Momentum

• A gyroscope is a device that contains a spinning
  object with a lot of angular momentum.
• Gyroscopes can do amazing tricks because they
  conserve angular momentum.
• For example, a spinning gyroscope can easily
  balance on a pencil point.

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Gyroscopes

• A gyroscope on the space shuttle is mounted at
  the center of mass, allowing a computer to
  measure rotation of the spacecraft in three
  dimensions.
• An on-board computer is able to accurately
  measure the rotation of the shuttle and maintain
  its orientation in space.

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Comparison Linear Angular Momentum

• Linear Momentum, p
• Tendency for a mass to continue moving in a
  straight line.
• Parallel to v.
• A conserved, vector quantity.
• Magnitude is inertia (mass) times speed.
• Net force required to change it.
• The greater the mass, the greater the force
  needed to change momentum.

• Angular Momentum, L
• Tendency for a mass to continue rotating.
• Perpendicular to both v and r.
• A conserved, vector quantity.
• Magnitude is rotational inertia times
  angular speed.
• Net torque required to change it.
• The greater the moment of inertia, the
  greater the torque needed to change angular
  momentum.