Download Chapter 6

Chapter 6.1 Pre-Quiz
Complete on a half-sheet of paper and turn in once completed.
1. What two previously discussed quantities define
momentum?
2. Which has more momentum:
a) 18-Wheeler Truck
b) Child on a tricycle
c) Need more information
3. (True / False) Momentum is a vector
Chapter 6
Momentum and Collisions
6.1 Linear Momentum
Definition of linear momentum:
The linear momentum of an object is the product of its
mass and velocity.
Momentum is a vector— meaning…
• it has both a magnitude and a direction!
SI unit of momentum: kg • m/s
This unit has no special name
Momentum
Linebacker: 108 kg
Velocity: 2.00 m/s
p=m·v
p = 108 kg x 2.00 m/s
p = 216 kg · m/s
6.1 Linear Momentum
For a system of objects, the total momentum is
the vector sum of each.
6.1 Linear Momentum
The change in momentum is the difference
between the momentum vectors.
6.1 Linear Momentum
If an object’s momentum changes, a force must
have acted on it.
The net force is equal to the rate of change of the
momentum.
Momentum
Linebacker: 108 kg
Velocity: 2.00 m/s
Initial Momentum: 216 kg·m/s
Final Momentum: 0 kg·m/s
Time: 0.315 s
Fnet = Δp / Δt
Fnet = (- 216 kg·m/s) / (0.315 s)
Fnet = - 686 N (opposes motion)
6.2 Impulse
Impulse is the change in momentum:
Typically, the force
varies during the
collision.
6.2 Impulse
Real-world contact times may be very short.
Demonstration Time?
Yes.
MUWAHAHA!
6.2 Impulse
• When a moving object stops, its impulse depends only on
its change in momentum.
• This can be accomplished by a large force acting for a
short time, or a smaller force acting for a longer time.
6.2 Impulse
• We understand this instinctively—we bend
our knees when landing a jump; a “soft” catch
(moving hands) is less painful than a “hard”
one (fixed hands).
• This is also how airbags work—they slow
down collisions considerably—and why cars
are built with “crumple zones”.
Audi
Q7 vs Smart
Fiat
Mercedes
C300
Cat
Cat 500
ForTwo
Kinetic Energy and Momentum
• Make an equation for kinetic energy that
uses momentum.
• Hint: It should start out…
KE = 1/2…
Momentum Exit Quiz
Complete on half-sheet of paper and turn in as you leave class.
1. How do we define momentum?
2. During a collision, what does impulse
represent?
3. Explain the effect of changing contact time
for a collision.
Momentum Practice
Page 207-209
Linear Momentum:
1-6*, 8-11, 16, 25**
Impulse:
26-31*, 32-34, 41, 46
* Try these first without your notes!
** Kudos if you can solve this!
KE = ½ Δp · v
KE = ½ Δp2 / m
Momentum Warm-Up
1. Find the momentum of a 1200 kg car travelling at 15
m/s.
2. If a soccer player kicks a 0.43 kg soccer ball, giving it
a velocity of 26 m/s. What is the average force
between their foot and the ball?
3. How is falling 10 feet onto a trampoline, different than
falling 10 feet to the ground? (explain in terms of F, p
and t)
Impulse, Force, and Contact time
For any object coming to rest, the impulse
is not dependent upon how it stops.
Trade-off: Contact time or Average force
Mercedes vs. Smart car
Before Collision:
Red Mercedes C300
Blue Smart Fortwo
1695 kg
730 kg
17.9 m/s East
17.9 m/s West
Watch direction!
After Collision:
Red Mercedes C300
Blue Smart Fortwo
1695 kg
730 kg
8.95 m/s East
2.88 m/s East
Find Δp (impulse) for each car
6.3 Conservation of Linear
Momentum
•
If there is no net force acting on a system, its total
momentum cannot change.
•
This is the law of conservation of momentum.
•
If there are internal forces, the momenta of individual
parts of the system can change, but the overall
momentum stays the same.
6.3 Conservation of Linear Momentum
In this example, there is no external force, but the individual
components of the system do change their momenta:
6.3 Conservation of Linear Momentum
Before/after any event* the momentum of the objects
involved will be conserved.
Therefore, the sum of momenta before equals the
sum of the momenta after.
*event = collision, objects pushing apart, anything that
changes one/both/all objects’ velocity.
m1 = 1695 kg
v1o = 17.9 m/s
v1 = 0 m/s
m2 = 730 kg
v2o = -17.9 m/s
v2 = ? m/s
Cannon
m1 = 100 kg
v1 = ? m/s
Cannonball
m2 = 5.00 kg
v2 = 40.0 m/s
Zorba the Dog
m1 = 156 kg
v1o = 4.2 m/s
v1 = ? m/s
Boat
m2 = 140 kg
v2o = 0 m/s
v2 = ? m/s
6.3 Conservation of Linear Momentum
Classifying Collisions
• In this class we will examine three types
of collisions:
• Elastic Collisions
• Inelastic Collisions
• Perfectly Inelastic Collisions
6.4 Elastic and Inelastic Collisions
For an elastic collision, both the kinetic
energy and the momentum are conserved:
This is the same as saying…
6.4 Elastic and Inelastic Collisions
Before
After
p1o +
p2o =
p1
+
p2
K1o +
K2o =
K1
+
K2
Total momentum before = Total momentum after
Total kinetic energy before = total kinetic energy after
6.4 Elastic and Inelastic Collisions
Collisions may take
place with the two
objects approaching
each other, or with
one overtaking the
other.
Page 196
Ex 6.11
6.4 Elastic and Inelastic Collisions
1. Elastic Collision
•
Kinetic energy is conserved (Kf = Ki)
•
Objects retain their initial shapes.
6.4 Elastic and Inelastic Collisions
1. Elastic Collision
•
Kinetic energy is conserved (Kf = Ki)
•
Objects retain their initial shapes.
6.4 Elastic and Inelastic Collisions
1. Elastic Collision
•
Kinetic energy is conserved (Kf = Ki)
•
Objects retain their initial shapes.
6.4 Elastic and Inelastic Collisions
1. Elastic Collision
•
Kinetic energy is conserved (Kf = Ki)
•
Objects retain their initial shapes.
6.4 Elastic and Inelastic Collisions
1. Elastic Collision
•
Kinetic energy is conserved (Kf = Ki)
•
Objects retain their initial shapes.
6.4 Elastic and Inelastic Collisions
1. Elastic Collision
•
Kinetic energy is conserved (Kf = Ki)
•
Objects retain their initial shapes.
6.4 Elastic and Inelastic Collisions
An inelastic collision is exemplary of real-world
collisions where kinetic energy is lost.
How is energy “lost”?
We can still get by using the elastic collision
equations/assumptions though because for many
small-scale collisions the KE lost is negligible.
6.4 Elastic and Inelastic Collisions
A perfectly inelastic collision
is one where the objects stick
together afterwards.
= mv
6.4 Elastic and InelasticpCollisions
Everyday collisions are imperfect inelastic
collisions. Often times one or both of the objects
colliding will be dented/damaged.
Such physical changes require work (energy) to be
done and thus kinetic energy is not conserved.
Kinetic energy may also be lost to things such as
friction and sound.
Types of Collisions
|-------------------------------------- Momentum is always conserved --------------------------------------|
Perfectly
Elastic
partially elastic
partially inelastic
Perfectly
Inelastic
|--KE conserved --||---------------------------------------Some KE lost---------------------------------------|
Equations for Each Type
• Inelastic:
m1v1o + m2v2o = m1v1 + m2v2
Two objects collide and do not remain stuck together, therefore all
we know is that the overall momentum is conserved.
• Perfectly Inelastic:
m1v1o + m2v2o = (m1 + m2)v
Two objects collide and remain stuck together, so there is only one
final velocity (helpful in reducing the # of variables).
Equations for Each Type
Allows solving for
final velocities
when both v1 and
v2 are unknown.
(When you only
know the starting
info!)
Elastic Collision Practice
A 120 kg rock is flying through space at 20.0
m/s when it undergoes an elastic collision
with another 90.0 kg rock that is travelling at
15 m/s in the same direction. What is the final
velocity of each rock?
Elastic Collision Practice
A 140 kg bumper car is travelling at 4.5 m/s
to the East when it experiences a head-on
elastic collision with a 160 kg bumper car that
was travelling at 6.0 m/s. What is the final
velocity of each car?
6.6 Jet Propulsion and Rockets
If you blow up a balloon and then let it go, it
zigzags away from you as the air shoots out.
This is an example of jet propulsion. The
escaping air exerts a force on the balloon that
pushes the balloon in the opposite direction.
Jet propulsion is another example of
conservation of momentum.
6.6 Jet Propulsion and Rockets
This same phenomenon explains the
recoil of a gun:
6.6 Jet Propulsion and Rockets
The thrust of a rocket works
the same way.
6.6 Jet Propulsion and Rockets
Jet propulsion can be used to slow a rocket
down as well as to speed it up; this involves
the use of thrust reversers. This is done by
commercial jetliners.
Center of Mass
Definition of the center of mass:
The center of mass is the point at which all of the
mass of an object or system may be considered to be
concentrated, for the purposes of linear or
translational motion only.
6.5 Center of Mass
The momentum of the center of mass does
not change if there are no external forces on
the system.
The location of the center of mass can be
found:
This calculation is straightforward for a
system of point particles, but for an
extended object calculus is necessary.
6.5 Center of Mass
The center of mass of a flat object can be
found by suspension.
6.5 Center of Mass
The center of mass may be located outside a
solid object.
6.5 Center of Mass
Top
Mass = 130 g
Loc = (0.0 m, 0.3 m)
Middle
Mass = 70.0 g
Loc = (0.0 m, 0 m)
XCM
Bottom
Mass = 80.0 g
Loc = (0.0 m, -0.30 m)