Download Lecture 14

Physics 218
Lecture 14
Dr. David Toback
Physics 218, Lecture XIV
1
Chapter 8: Momentum
Want to deal with more
complicated systems
• Collisions
• Explosions
Newton’s laws still work, but
we need some new ideas
Physics 218, Lecture XIV
2
Physics 218, Lecture XIV
3
Today’s Lecture
• Begin with a definition of
Linear Momentum
• Then show that conservation
of momentum helps us solve
certain types of problems
–Things colliding
–Things exploding
Physics 218, Lecture XIV
4
Definition of Linear Momentum
Vector equation!


P  mV

Psystem 

m
V
 ii
Physics 218, Lecture XIV
5
Restating Newton’s Second Law
“The rate of change of momentum of an object is equal
to the net force applied to it”


dP

F
 dt
Do a check for constant mass :

dP
dt


d ( mV )
dt

d (V )
dt
m

 ma √
If we exert a net force on a body, the momentum of the
body changes
Physics 218, Lecture XIV
6
What if SF=0?
If SF=0, then dp/dt = 0,  p = constant
Momentum doesn’t change


mv  m' v'
momentum before = momentum after
Physics 218, Lecture XIV
7
Conservation of Momentum
For a system, by Newton’s laws, SF=0

Conservation of Momentum

' '
 mi v i   mi v i
i
i
Sum of all
Sum of all
momentum before = momentum after
True in X and Y directions separately!
Physics 218, Lecture XIV
8
Problem Solving
For Conservation of Momentum problems:
1. BEFORE and AFTER
2. Do X and Y Separately
Physics 218, Lecture XIV
9
Before
Y
X
Physics 218, Lecture XIV
10
After
X
Y
Physics 218, Lecture XIV
11
So what?
Momentum is useful when we don’t
know anything about the forces
Examples from everyday life:
–When ice skating, if you push
someone, why do you go
backwards?
–Why does a gun recoil when you
shoot it?
Physics 218, Lecture XIV
12
Everyday Experience?
Question: Why do you go
backwards when you push
someone on the ice?
Newton’s Law’s answer: When you
exert a force on another person,
then, by Newton’s law, the person
exerts an equal and opposite force
on you.
Physics 218, Lecture XIV
13
Everyday Experience? Cont…
Question: Why do you go backwards when
you push someone on the ice?
Momentum Conservation Answer:
• Before:
– The system starts with zero momentum
(nobody is moving)
• After:
– The system ends with zero momentum.
You and your friend move in opposite
directions
Physics 218, Lecture XIV
14
Simple Gun Example
A gun of mass MG is sitting at rest
with a bullet of mass MB inside it.
You shoot the gun and the bullet
comes out with a speed V at
angle Q.
What is the recoil velocity of the
gun?
Physics 218, Lecture XIV
15
Weird example
Ball of mass m is dropped from a
height h:
• What is the momentum before
release?
• What is the momentum before it
hits the ground?
• Is momentum conserved?
Physics 218, Lecture XIV
16
What if we add the Earth?
• What is the force on the ball?
• What is the force on the earth?
• Is there any net force in this
system?
• Is momentum conserved?
SF=0, then dp/dt = 0, → p = constant
Physics 218, Lecture XIV
17
Momentum for a system is Conserved
• Momentum is ALWAYS conserved for a
SYSTEM, you just have to look at a big
enough system to see it correctly.
– Not conserved for a single ball
• A ball falling is not a big enough system. You
need to consider what is making it fall.
– Newton’s Law: For every action there is an
equal and opposite reaction
• Add up all the momentums in the problem
– The forcer and the forcee
Physics 218, Lecture XIV
18
Energy and Momentum in Collisions
Definitions:
• Elastic collision = kinetic
energy is conserved
• Inelastic collision =
kinetic energy is not
conserved.
• Momentum conserved?
• Total Energy conserved?
Physics 218, Lecture XIV
19
Inelastic Collisions
• By definition:
Inelastic =
mechanical energy not conserved =
kinetic energy not conserved
• Inelastic Example: Two trains which
collide and stick together
Physics 218, Lecture XIV
20
Colliding Trains: 1 Dimension
The train car on the left, mass m1, is moving with
speed Vo when it collides with a stationary car of
mass m2. The two stick together.
1. What is their speed after the collision?
2. Show that this is inelastic
Physics 218, Lecture XIV
21
Ballistic Pendulum
A bullet of mass m and
velocity Vo plows into
a block of wood with
mass M which is part
of a pendulum.
How high, h, does the
block of wood go?
Is the collision elastic or
inelastic?
Physics 218, Lecture XIV
22
Bottom line: When to use Momentum
• When you don’t know the
forces in the system
• When you are studying all
of the pieces of the system
which are doing the forcing
Before and After Problems
Physics 218, Lecture XIV
23
Coming up…
• Next time: Finish Chapter 8
• Next week:
–Start Chapter 9 on Rotation
th
–Exam 2, Thursday October 26
–Homework 7 due
Physics 218, Lecture XIV
24
Physics 218, Lecture XIV
25
Notes
• Exam coming up next time….:
– Here. Usual class time: Covering:
• Exam 1 material, Chapter 3(9)
• Chapter 4(1-8), Chapter 5(1-3)
• Chapter 6(1-8), Chapter 7(1-4), Calculus 2
• Today’s material NOT ON EXAM
5 Bonus points on the mini-practice exam
II. Requires a 100 on all 10 math quizzes,
all HW and HW quizzes up through and
including Chapter 7
Physics 218, Lecture XIV
26
Next time:
• Exam coming up next time….:
– Here. Usual class time: Covering:
• Exam 1 material, Chapter 3(9)
• Chapter 4(1-8), Chapter 5(1-3)
• Chapter 6(1-8), Chapter 7(1-4), Calculus 2
• 5 bonus points for getting a 100 on mini-practice exam II
– Must complete everything to Chap 7.
• Reading for next lecture:
– Rest of Chapter 9 on Momentum
– Reading was due today, but I’ll grant an extension.
Questions: 1 & 14
Physics 218, Lecture XIV
27
Head On Collision
A ball of mass m1 collides head on
(elastically) with a second ball at
rest and rebounds (goes in the
opposite direction) with speed
equal to ¼ of its original speed.
What is the mass of the second
ball m2?
Physics 218, Lecture XIV
28
Next time:
• Exam Thursday
– Extra credit if you have 100’s on all
HW’s, HW quizzes and math quizzes
before the exam
• Reading for Tuesday:
– Rest of Chapter 9 on Momentum
Physics 218, Lecture XIV
29
Two Balls Collide
Two billiard balls of equal and known mass
m are traveling with known velocities V1
and V2. They collide elastically
What are the velocities after the collision?
Physics 218, Lecture XIV
30
A Ball collides with a Stationary Ball
•
We have two billiard balls
of different and
known masses m1 and m2. Ball one is
traveling with known velocity V1. They
collide elastically
What are the velocities after the collision?
Physics 218, Lecture XIV
31
Collisions and Impulse
Physics 218, Lecture XIV
32
Playing Pool: 2 Dimensions
Before the collision,
ball 1 moves with
speed V1 in the x
direction, while ball
2 is at rest. Both
have equal mass.
After the collision,
the balls go off at
angles Q and –Q.
What are v’1 and v’2
after the collision?
Q
-Q
Physics 218, Lecture XIV
33