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Name___________________
Chapter 9: Linear Momentum and Collisions
The ___________________ of a particle of mass m and
velocity v is defined as
The linear momentum is a vector quantity.
It’s direction is along v.
The components of the momentum of a particle:
Momentum Facts
• _________________
• Momentum is a _________ quantity!
• _________ and _________vectors point in the _____
direction.
• SI unit for momentum: ____________
• Momentum is a _________ quantity
• __________________________________________________
• Momentum is ________ proportional to both ____ and _____.
• Something big and slow could have the same momentum as
something small and fast.
Momentum Examples
10 kg
3 m/s
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.
26º
5g
p=
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
The train, bus, and car all have different masses and speeds,
but their ___________ are the _______ in _________.
The difficulty in bringing each vehicle to rest--in terms of a
combination of the force and time required--would be the
_______, since they each have the same momentum.
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:
Example: A 50 N force is applied to a 100 kg
boulder for 3 s.
Note that we didn’t need to know the mass of the
object in the above example.
Impulse Units
J = F t shows why the SI unit for impulse is the _________________
Impulse and momentum have the ___________________
______________________________
The impulse due to all forces acting on an object (the net force) is
equal to the change in momentum of the object:
We know the units on both sides of the equation are the same
(last slide), but let’s prove the theorem formally:
Stopping
Time
Ft = Ft
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.
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?
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.
Fnet (N)
Fnet vs. t graph
____________
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.
As long as there are
__________________ acting
on a system of particles,
collisions between the particles
will exhibit conservation of
linear momentum.
This means that the vector sum
of the momenta ________
collision is _______ to the
vector sum of the momenta of
the particles _________.
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.
before: ______________
v2
v1
m1
m2
after: _______________
va
m1
m2
vb
Directions after a collision
On the last slide the boxes were drawn going in the opposite direction
after colliding. This isn’t always the case. 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
m1
m2
vb
Simple Examples of Head-On Collisions
(___________________________________________)
Collision between two objects _______________________________
Collision between two objects________________________________
Collision between two objects__________________________________
Simple Examples of Head-On Collisions
(___________________________________________)
Collision between two objects _______________________________
Collision between two objects________________________________
Collision between two objects__________________________________
___________ is conserved in any collision,
________________________________________
_____________ is _____ conserved in elastic collisions.
_____________________________: After colliding, particles
stick together. There is a loss of energy (deformation).
_____________________________: Particles bounce off
each other without loss of energy.
_____________________________: Particles collide with
some loss of energy, but don’t stick together.
Notice that p and v are
vectors and, thus have
a direction (+/-)
Ki  Eloss  K f
1
1
1
2
2
2
m1v1i  m2v2i  (m1  m2 )v f  Eloss
2
2
2
There is a
loss in energy
Eloss
For
collisions in one dimension:
Suppose we know the initial masses and velocities.
Then:
Note, that these are pretty specialized equations, (elastic collision in one
dimension, known initial velocities, and masses)
Black board example
9.2
Two carts collide elastically on a frictionless track. The first
cart (m1 = 1kg) has a velocity in the positive x-direction
of 2 m/s; the other cart (m = 0.5 kg) has velocity in the
negative x-direction of 5 m/s.
(a) Find the speed of both carts after the collision.
(b) What is the speed if the collision is perfectly inelastic?
(c) How much energy is lost in the inelastic collision?
Black board example 9.3 and
demo
Determining the speed of a bullet
A bullet (m = 0.01kg) is fired into a block (0.1 kg) sitting at the edge of a table.
The block (with the embedded bullet) flies off the table (h = 1.2 m) and
lands on the floor 2 m away from the edge of the table.
a.) What was the speed of the bullet?
b.) What was the energy loss in the bullet-block collision? (skip)
vb = ?
h = 1.2 m
x=2m
Motion of a System of Particles.
Newton’s second law for a System of Particles
The center of mass of a system of particles (combined mass M)
moves like one equivalent particle of mass M would move under
the influence of an external force.


Fnet  MaCM
Fnet , x  MaCM , x
Fnet , y  MaCM , y
Fnet , z  MaCM , z
Center of mass
Center of mass for many particles:
Black board example 9.6
Where is the center of mass
of this arrangement of
particles.
(m3 = 2 kg; m1 = m2 = 1 kg)?
Velocity of the center of mass:
Acceleration of the center of mass:
A rocket is shot up in the air and explodes.
Describe the motion of the center of mass before and after
the explosion.
A method for finding the center of mass of any
object.
- Hang object from
two or more points.
- Draw extension of
suspension line.
- Center of mass is at
intercept of these lines.
A change in momentum is called “impulse”:
During a collision, a force F acts on
an object, thus causing a change in
momentum of the object:
tf


p  J   F (t )dt
ti
For a constant (average) force:
Think of hitting a soccer ball: A force F acting over a time t
causes a change p in the momentum (velocity) of the ball.
Black board example 9.6
A soccer player hits a ball (mass m =
440 g) coming at him with a
velocity of 20 m/s. After it was
hit, the ball travels in the
opposite direction with a velocity
of 30 m/s.
(a) What impulse acts on the ball
while it is in contact with the
foot?
(b) The impact time is 0.1s. What
average force is the acting on the
ball?
(c) How much work was done by the
foot? (Assume and elastic
collision.) (skip)
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
v=?
4 cm/s
7 kg
Sample Problem 2
7 kg
35 g
700 m/s
v=0
Same as the last problem except this time it’s 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
Note: Once again we’re 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.
Black board example
9.1
You (100kg) and your skinny friend (50.0 kg)
stand face-to-face on a frictionless, frozen
pond. You push off each other. You move
backwards with a speed of 5.00 m/s.
(a) What is the total momentum of the youand-your-friend system?
(b) What is your momentum after you pushed
off?
(c) What is your friends speed after you pushed
off?
(d) How much energy (work) did you and your
friend expend?(skip)
Sample Problem 3
An apple is originally at rest and then dropped. After falling a short
time, it’s moving pretty fast, say at a speed V. Obviously, momentum
is not conserved for the apple, since it didn’t have any at first. How
can this be?
apple
m
V
F
v
Earth
M
F
mV = M v
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?
After the collision the lime Kool Aid is moving _______________
before
3 kg
10 m/s
6 m/s
15 kg
after
4.5 m/s
3 kg
15 kg
v
Conservation of Momentum in 2-D
To handle a collision in 2-D, we conserve momentum in each
dimension separately.
Choosing down & right as positive:
m1
v1
m2
before:
2 v
2
1
after:
a
m1
va
m2
vb
b
Conservation of momentum equations:
Conserving Momentum w/ Vectors
B
E m1
1
F
O
R
E
A
F
T
E
R
a
m1
2
m2
p before
m2
b
p after
This diagram shows momentum vectors, which are parallel to
their respective velocity vectors. Note p1 + p 2 = p a + p b
and p before = p after as conservation of momentum demands.
Exploding Bomb
Acme
after
before
A bomb, which was originally at rest, explodes and shrapnel flies
every which way, each piece with a different mass and speed. The
momentum vectors are shown in the after picture.
continued on next slide
Exploding Bomb (cont.)
Since the momentum of the bomb was zero before the
explosion, it must be zero after it as well. Each piece does
have momentum, but the total momentum of the exploded
bomb must be zero afterwards. This means that it must be
possible to place the momentum vectors tip to tail and form a
closed polygon, which means the vector sum is zero.
If the original momentum of
the bomb were not zero,
these vectors would add up
to the original momentum
vector.
Two-dimensional collisions (Two particles)
Conservation of momentum:
Split into components:
If the collision is elastic, we can also use conservation of
energy.
Velocity Components in Projectile Motion
(In the absence of air resistance.)
Note that the ____________ component
of the velocity remains the __________ if air
resistance can be ignored.
Example of Non-Head-On Collisions
(Energy and Momentum are Both Conserved)
Collision between two objects of the same mass. One mass is at rest.
If you vector add the total momentum after collision,
you get the total momentum before collision.
2-D Sample Problem
152 g
before
40
34 m/s
0.3 kg
5 m/s
A mean, old dart strikes an innocent
mango that was just passing by
minding its own business. Which
way and how fast do they move off
together?
Working in grams and taking left & down as + :
after
452 g

v
Dividing equations :
Substituting into either of the first two
equations :
Black board example
9.5
Accident investigation. Two automobiles
of equal mass approach an intersection. One
vehicle is traveling towards the east with 29
mi/h (13.0 m/s) and the other is traveling
north with unknown speed. The vehicles
collide in the intersection and stick together,
leaving skid marks at an angle of 55º north
of east. The second driver claims he was
driving below the speed limit of 35 mi/h
(15.6 m/s).
13.0 m/s
??? m/s
a) Is he telling the truth?
b) What is the speed of the “combined vehicles” right after the
collision?
c) How long are the skid marks (mk = 0.5)?
ROTATIONAL INERTIA &
ANGULAR MOMENTUM
• For every type of linear quantity we
have a rotational quantity that does
much the same thing
Linear Quantities
Rotational Quantities
Rotational Inertia(I)
• AKA (not really but could be) Rotational
Mass
• _________________________________
– Objects that are rotating about an axis tend to
stay rotating, objects not rotating tend to remain
at rest, unless an outside torque is applied
• _________________________________
• It’s the rotational equivalent to mass,
– Harder to give an ang. acc. to an object w/ a
larger I
Moment of Inertia
Any moving body has inertia.
_______________________________________________________
_______________________________________________________
Single point-like mass
m
r
Q
System of masses
m2
m1
r1
r2
Q
Moment of Inertia Example
Two merry-go-rounds have the same mass and are spinning with the
same angular velocity. One is solid wood (a disc), and the other is a
metal ring. Which has a bigger moment of inertia relative to its
center of mass?
r
r

m
answer:

m
The big idea
• Rotational Inertia depends _______________
• If either one of these is large, then rotational
inertia is ______, and object will be harder to
______
• Different types of objects have different
equations for rotational inertia
• But all equations have ___________ in them.
Rotational Inertia
• Some objects have
more rotational
inertia than others
__________________
________________
________________
________________
________________
________________
________________
________________
________________
Angular Momentum, L
Depends on ___________________________________ from a
particular point.
If _____________________ then the magnitude of angular momentum w/
resp. to point Q is given by _____________________
In this case L points out of the page. If the mass were moving in the
opposite direction, L would point into the page.
Unit: ___________
v
r
Q
A _______ is needed to change L,
just a _____ is needed to change p.
m Anything spinning has angular has
angular momentum.
____________________________
____________________________
____________________________
Angular Momentum: General Definition
If r and v are _________________ then the angle between these
two vectors must be taken into account. The general definition of
angular momentum is given by a vector cross product:
This formula works regardless of the angle. As you know from
our study of cross products, the magnitude of the angular
momentum of m relative to point Q is:
In this case, ______________________, L points out of the
page. If the mass were moving in the opposite direction,
v
L would point into the page.

r
Q
m
Comparison: Linear & Angular Momentum
Linear Momentum, p
Angular Momentum, L
• _________________________ • _________________________
___________________________ ___________________________
•__________________________ •__________________________
• _________________________
• _________________________
•__________________________ •__________________________
___________________________ ___________________________
•__________________________ •__________________________
___________________________ ___________________________
•__________________________ •__________________________
___________________________ ___________________________
___________________________ ___________________________
Angular Acceleration
Angular acceleration occurs when a spinning object spins faster or
slower.
Note how this is very similar to a =  v /t for linear acceleration.
Ex: If a wind turbine spinning at 21 rpm speeds up to 30 rpm over
10 s due to a gust of wind, its average angular acceleration is
9 rpm/10 s. This means every second it’s spinning 9 revolutions
per minute faster than the second before. Let’s convert the units:
Torque & Angular Acceleration
Newton’s 2nd Law, as you know, is _________
The 2nd Law has a rotational analog: ___________
A force is required for a body to undergo acceleration. A
“turning force” (a torque) is required for a body to undergo
angular acceleration.
Both m and I are measures of a body’s inertia
(resistance to change in motion).
Linear Momentum & Angular Momentum
Recall, angular momentum’s magnitude is given by:
r
v
m
So, if a net torque is applied, angular velocity
must change, which changes angular momentum.
proof:
So net torque is the rate of change of angular
momentum, just as net force is the rate of change of
linear momentum.
From the formula v = r , we get
Why does a tightrope walker
carry a long pole?
• _________________________________________
_________________________________________
• _________________________________________
• _________________________________________
• http://www.youtube.com/watch?v=w8Tfa5fHr3s
Sports Connection
• Running
–
• Gymnastics/Diving
–
Angular
Momentum
“inertia of rotation”
•
• Ang. Momentum= Rotational Inertia X
Rotational Speed
– L=Iω
Conservation of Angular Momentum
• _________________________________
_________________________________
• _________________________________
_________________________________
• _________________________________
_________________________________
I – represents rotational inertia
ω -represents angular speed
Sports Connection…
• Ice skating
–
– http://www.youtube.com/watch?v=AQLtcEAG9v0
– http://www.youtube.com/watch?v=NtEnEeEyw_s
–
–
• Gymnastics (pummel horse or
floor routine)
–
Do cats violate physical law?
• Video
• _____________
_____________
_____________
_____________
_____________
_____________
_____________
_____________
_____________
_____________
_____________
_____________
_____________
59
• Helicopter tail rotor failure
• Tail rotor failure #2
Universe Connection
• Rotating star shrinks
radius…. What
happens to
rotational speed??
–
• Rotating star
explodes outward….
What happens to
rotational speed??
–
Make a list of equations