Lecture 11: Conservation of
Momentum
Today’s Concept:
Inelastic Collisions
How did the chicken cross the frictionless road?
It laid an egg and threw it backwards.
Mechanics Lecture 11, Slide 1
Announcements
§  Exam 2 on Wednesday, October 21: SFEBB 1110
Ø  Covers Units 6-10
Ø  80 minutes: If you have a conflict, you need to email me by
Friday night so I can make arrangements
Ø  Post Exam Review: W (SFEBB 180); TH (WEB L114)
Ø  Same rules as last time.
§  You have until Monday, October 19 to turn in a regrade form for
Exam 1
Mechanics Lecture 11, Slide 2
Recap
!
!
Ptot = ∑ mi vi
i
Mechanics Lecture 11, Slide 3
Stuff you asked about:
What is the different between conservation of Momentum and conservation
of energy? How do I know if the Momentum is conserve or energy is
conserve?
Momentum (Unit 11)
Energy (Unit 8)
Mechanics Lecture 11, Slide 4
ACT: Bullet and Block
A
B
C
D
A wood block rests on a table. A bullet shot into the block stops inside, and
the bullet plus block start sliding on the frictionless surface. The momentum
of the bullet plus block remains constant:
Before
A) Before the collision.
B) During the collision
C) After the collision
D) All of the above
E) Only A and C above
After
As long as there are no external
forces acting on the system, then
dP~tot
= 0 =) P~tot = const.
dt
Mechanics Lecture 11, Slide 5
Homework Problem
p1
p2
pinit
p2
p1
pfinal
θ
Mechanics Lecture 11, Slide 6
CheckPoint: Cart Propulsion 1
Suppose you are on a cart initially at rest that rides on a
frictionless track. If you throw a ball off the cart towards the left,
will the cart be put into motion?
Le# A) Yes, and it moves to the right.
B) Yes, and it moves to the left.
C) No, it remains in place.
Right Conservation of momentum
means the cart must move to
the right since the ball moves
to the left.
Mechanics Lecture 11, Slide 7
ACT: Cart Propulsion
A
B
C
D
Suppose you are on a cart that is moving at a constant
speed v toward the left on a frictionless track. If you throw a
massive ball straight up (relative to the cart), how will the
speed of the cart change?
Left
Right
v
A) Increase
B) Decrease
C) Will not change
The net external force in the horizontal
direction is zero:
èPtotal is conserved in x direction
Mechanics Lecture 11, Slide 8
CheckPoint: Cart Propulsion 2
Suppose you are on a cart which is iniAally at rest that rides on a fricAonless track. You throw a ball at a verAcal wall that is firmly aFached to the cart. If the ball bounces straight back as shown in the picture, will the cart be put into moAon a#er the ball bounces back from the wall? Le# Right A) Yes, and it moves to the right. B) Yes, and it moves to the le#. C) No, it remains in place. Mechanics Lecture 11, Slide 9
ACT: Elastic vs. Inelastic
A
B
C
D
Two equal-­‐mass balls swing down and hit idenAcal bricks while traveling at idenAcal speeds. Ball A bounces back, but ball B just stops when it hits the brick. Which ball is more likely to knock the brick over? A B A) A B) B C) They both have the same chance. Mechanics Lecture 11, Slide 10
+x
A B Momentum of ball before collision: Case A: Case B: pi = +mv
pi = +mv
Momentum of ball a#er collision: Case A: Case B: pf = mv
pf = 0
Case A: Change in ball’s momentum: Case B: pf
pi
pi p f
p = 2mv
p = mv
Mechanics Lecture 11, Slide 11
CheckPoint: Box and Ball Collisions
A
B
C
D
Two balls of equal mass are thrown horizontally with the same iniAal velocity. They hit idenAcal staAonary boxes resAng on a fricAonless horizontal surface. The ball hiSng box 1 bounces back, while the ball hiSng box 2 gets stuck. Which box ends up moving faster? A) Box 1 B) Box 2 C) same 1
2
Mechanics Lecture 11, Slide 12
CheckPoint Revote
A
B
C
D
Which box ends up moving faster? A) Box 1 B) Box 2 C) same 1 2 A) The system in box 2 has the same momentum, but more mass, which means a smaller velocity. B) with box one, half the momentum from the ball will be transferred to the box and the other half will carry the ball in the opposite direcAon. with box two, all the momentum will be transferred to the box C) I have convinced myself that all three answers are right. I await enlightenment in the lecture. :-­‐). Think of a 2-­‐step “bounce” Mechanics Lecture 11, Slide 13
Example problem: Ballistic Pendulum
m
v
M
H
A projecAle of mass m moving horizontally with speed v strikes a staAonary mass M suspended by strings of length L. Subsequently, m + M rise to a height of H. If you measure a max height H, what is the iniAal speed v of the projecAle? Mechanics Lecture 11, Slide 14
Breaking it down into steps
before m during a#er V v M H splat Which quanAAes are conserved before the collision? A) Momentum B) mechanical energy C) both momentum and mechanical energy Mechanics Lecture 11, Slide 15
Breaking it down into steps
before m during a#er V v M H splat Which quanAAes are conserved during the collision? A) momentum E = WN C
B) mechanical energy C) both momentum and mechanical energy mv = (m + M )V
Mechanics Lecture 11, Slide 16
Breaking it down into steps
before m during a#er V v M H splat Which quanAAes are conserved a#er the collision A) momentum B) mechanical energy E = WN C = 0
1
(m + M )V 2 = (m + M )gH
2
C) both momentum and mechanical energy V = 2 gH
2
Mechanics Lecture 11, Slide 17
Ballistic Pendulum
m v M mv = (m + M )V
V = 2 gH
2
⎛m+M ⎞
v = 2 gH ⎜
⎟
⎝ m ⎠
H m
V=
v
(m + M )
⎛ M⎞
v = 2 gH ⎜1 + ⎟
m⎠
⎝
Mechanics Lecture 11, Slide 18
Ballistic Pendulum Demo
L L m v L H M L M + m d In the demo we measure forward displacement d, not H: L-­‐H H L d L2 = d 2 + ( L − H ) 2
H = L − L2 − d 2
g
⎛ M⎞
v = ⎜ 1+ ⎟ d
⎝
m⎠
L
for d << L Mechanics Lecture 11, Slide 19