Chapter 8:
Momentum Conservation
Impulse
Work
Distance, l
K = (1/2) m v2
Work-Energy Theorem
Energy Conservation
p=mv
Impulse-Momentum Theorem
Momentum Conservation
Momentum Conservation
Definitions
Momentum Conservation
Examples of 1D Collisions
M
m
M
m
Momentum Conservation
Elastic Collision
Momentum :
Kinetic Energy :
m1v1  m2 v2  m1v1'  m2 v2'
2
2
1
1
1
1
m1v12  m2 v 22  m1v1'  m2 v 2'
2
2
2
2
Momentum Conservation
Energy Conservation
K 1,i  K 2,i  K 1, f  K 2, f
K 1,i  K 2,i  K 1, f  K 2, f  Q
Loss of energy as thermal and
other forms of energy
Momentum Conservation
Example 2
After collision
Before collision
(totally inelastic collision)
m v1 + m v2 = m v1’ + m v2’
v1’ = v2’
Momentum Conservation
Railroad cars, locking up after the collision
How to fire a rifle to reduce recoil
Momentum Conservation
Elastic collision
Momentum Conservation
Elastic Collision between different mass balls
m(A)=m(B)
Momentum Conservation
v(ax)=0 v(bx)=v(x)=v(i)
billiard balls
Remark on relative velocity
Momentum Conservation
Inelastic Collision
Elastic Collision
Momentum Conservation
Elastic Collision on a air track
Momentum Conservation
Momentum Conservation
Inelastic Collision on an air track
Momentum Conservation
Impulsive Force
[Example] an impulsive force on
a baseball that is struck with a bat
has:
Impulsive Force
[Note] The “impulse’’ concept
is most useful for impulsive
forces.
Momentum Conservation
Very short time
Very large magnitude
<F> ~ 5000 N & Dt ~ 0.01 s
Impulse-Momentum Theorem
F (t )
F
|J |
  

Dp p f - pi
F 

Dt
t f - ti

 
J  F  ( t f - ti )  p f - pi
Momentum Conservation
Momentum Conservation
(A) Momentum Conservation
Ballistic Pendulum
Express v and v’ in terms of
m, M, g, and h.
(A) mv = (m+M) v’
(B) K1+Ug1 = K2+Ug2
2
1
(B) Energy Conservation
Momentum Conservation
Ballistic Pendulum (cont.)
• 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?
Two parts: 1-collision (momentum is conserved)
2-from low point (after collision) to high
point: conservation of energy
1st part:
x : m v 0  (M  m) v'
mv
 v' 
(M  m)
y : 00 00
2nd part:
E bottom  E top


1
(M  m)(v' ) 2  0  0  (M  m)gh
2
1
m 2 v2
2
 h  (v' ) 
2g
2g(m  M) 2
Ballistic Pendulum numerical example
=0.767 m/s
K(bullet)=236J
Momentum Conservation
K(block+bullet)=0.6J
Momentum Conservation
Example 8.8 Accident analysis
Momentum Conservation
Throwing a package overboard
Momentum Conservation
N
Momentum Conservation
Center of Mass (CM)
What is the “Center of Mass?”
• More importantly “Why do we
care?”
• This is a special point in space
where “it’s as if the object could
be replaced by all the mass at
that one little point”
Center of mass
Center of Mass (c.m. or CM)
The overall motion of a mechanical system
can be described in terms of a special point
called “center of mass” of the system:


Fsystem  M system acm

where Fsystem is the vector sum of all the
forces exerted on the system.
Momentum Conservation
How do you calculate CM?
1. Pick an origin
2. Look at each “piece of mass” and figure
out how much mass it has and how far it
is (vector displacement) from the origin.
Take mass times position
3. Add them all up and divide out by the
sum of the masses
The center of mass is a displacement vector
“relative to some origin”
Spelling out the math:



m1x1 m2x2
Xcmfor2part icles

m1 m2




m1x1 m2x2 m3x3
Xcmfor3part icles


m1 m2 m3



m1x1 m2x2 m3x3
M
etc...

Notethatx is the3- D vector
displaceme
nt
Momentum Conservation
CM Position (2D)
m3
ycm = 0.50 m X
m1 + m2
m1
X
xcm = 1.33 m m2 + m3
Total momentum in terms of mass





Mvcm  ma va  mb vb  mc vc  ...  p
Motion of center of mass





Macm  ma aa  mb ab  mc ac  ...   Fext
Momentum Conservation
Momentum Conservation
8.52
Walking in a boat
M(lady)=45kg
M(boat)=60 kg
The center of mass does not move, since there is no net horizontal force
Momentum Conservation